B. G. TEUBNER'S SAMMLUNG VON LEHRBUCHERN
AUF DEM GEBIETE DER
MATHEMATISCHEN WISSENSCHAFTEN
MIT EINSCHLUSS IHRER ANWENDUNGEN.
BAND VI.
LINEAR GROUPS
WITH AN EXPOSITION
OF THE GALOIS FIELD THEORY
BY
LEONARD EUGENE DICKSON, PH. D.,
ASSISTANT PKOFESSOK OF MATHEMATICS IN THE UNIVEESITY OF CHICAGO.
1
LEIPZIG
PUBLISHED BY B. G. TEUBNER
1901.
Qfl
PBINTED BY B. G. TBUBNEB, DBESDEN.
ELECTRONIC VERSION
AVAILABLE
PREFACE.
Since the appearance in 1870 of the great work of Camille Jordan
on substitutions and their applications, there have been many important
additions to the theory of finite groups. The books of Netto, Weber
and Burnside have brought up to date the theory of abstract and
substitution groups. On the analytic side, the theory of linear groups
has received much attention in view of their frequent occurrence in
mathematical problems both of theory and of application. The theory
of collineation groups will be treated in a forthcoming volume by
Loewy. There remains the subject of linear groups in a finite field
(including linear congruence groups) having immediate application in
many problems of geometry and function -theory and furnishing a
natural method for the investigation of extensive classes of important
'groups. The present volume is intended as an introduction to this
subject. While the exposition is restricted to groups in a finite field
(endliche Korper), the method of investigation is applicable to groups
in an infinite field; corresponding theorems for continuous and collinea-
tion groups may often be enunciated without modification of the text.
The earlier chapters of the text are devoted to an elementary
exposition of the theory of Galois Fields chiefly in their abstract
form. The conception of an abstract field is introduced by means of
the simplest example, that of the classes of residues with respect to
a prime modulus. For any prime number p and positive integer n y
there exists one and but one Galois Field of order p n . In view of
the theorem of Moore that every finite field may be represented as
a Galois Field, our investigations acquire complete generality when
we take as basis the general Galois Field. It was found to be
impracticable to attempt to indicate the sources of the individual
theorems and conceptions of the theory. Aside from the independent
discovery of theorems by different writers and a general lack of
reference to earlier papers, the later writers have given wide general-
izations of the results of earlier investigators. It will suffice to give
the following list of references on Galois Fields and higher irreducible
congruences:
Galois, "Sur la theorie des nombres", Bulletin des sciences mathema-
tiques de M. Ferussac, 1830; Journ. de mafhematiques , 1846.
Schonemann, Crelle, vol. 31 (1846), pp. 269325.
IV PREFACE.
Dedekind, Crelle, vol. 54 (1857), pp. 126.
Serret, Journ. de math,., 1873, p. 301, p. 437; Algebre superieure.
Jordan, Traite des substitutions, pp. 14 18, pp. 156161.
PeUet, Comptes Eendus, vol. 70, p. 328, vol. 86, p. 1071, vol. 90, p. 1339,
vol. 93, p. 1065; Bull Soc. Math, de France, vol. 17, p. 156.
Moore, Bull. Amer. Math. Soc., Dec., 1893; Congress Mathematical
Papers.
Dickson, Bull. Amer. Math. Soc., vol. 3, pp. 381389 ; vol. 6, pp. 203204.
Annals of Math., vol. 11, pp. 65120 ; Chicago Univ. Record, 1896, p. 318.
Borel et Drach, Theorie des nombres et algebre superieure, 1895.
The second part of the book is intended to give an elementary
exposition of the more important results concerning linear groups in
a Galois Field. The linear groups investigated by Galois, Jordan
and Serret were defined for the field of integers taken modulo p; the
general Galois Field enters only incidentally in their investigations.
The linear fractional group in a general Galois Field was partially
investigated by Mathieu, and exhaustively by Moore, Burnside and
Wiman. The work of Moore first emphasized the importance of
employing in group problems the general Galois Field in place of the
special field of integers, the results being almost as simple and the
investigations no more complicated. In this way the systems of linear
groups studied by Jordan have all be generalized by the author and
in the investigation of new systems the Galois Field has been
employed ab initio.
The method of presentation employed in the text often differs
greatly from that of the original papers; the new proofs are believed
to be much simpler than the old. For example, the structure of all
linear homogeneous groups on six or fewer indices which are defined
by a quadratic invariant is determined by setting up their isomorphism
with groups of known structure. Then the structure of the correspond-
ing groups on m indices, m > 6, follows without the difficult cal-
culations of the published investigations. In view of the importance
thus placed upon the isomorphisms holding between various linear
groups, the theory of the compounds of a linear group has been
developed at length and applied to the question of isomorphisms.
Again, it was found practicable to treat together the two (generalized)
hypoabelian groups. The identity from the group standpoint of the
problem of the trisection of the periods of a hyperelliptic function
of four periods and the problem of the determination of the 27 straight
lines on a general cubic surface is developed in Chapter XIV by an
analysis involving far less calculation than the proof by Jordan.
Chicago, November, 1900.
TABLE OF CONTENTS,
FIE8T PAET.
INTRODUCTION TO THE GALOIS FIELD THEORY.
CHAPTER I.
Definition and properties of finite fields.
Section Page
1 3. Classes of residues with respect to a prime modulus . . . 3 4
4. Fermat's theorem 4
5. Definition of a field 5
6_7. Definition of a Galois Field 68
8 10. Order of a finite field is a power of a prime 9 10
11 17. Period of a mark of a field; primitive roots 11 12
18. Every finite field may be represented as a Galois Field . . 13 14
CHAPTER II.
Proof of the existence of the GF[p~] for every prime p
and integer m.
19 22. Decomposition of functions belonging to the G~F[p n ~] . . . 14 15
71 Yf\>
2325. Irreducible factors of x p x 1516
26 27. Expression for product of all irreducible quantics of degree m
in the GF[p*~\. Their number 1718
Exercices 19
CHAPTER HI.
Classification and determination of irreducible quantics.
29 30. Exponent to which an irreducible quantic belongs .... 19 20
31 32. Roots of an irreducible quantic; their exponents 21
33. When x*-}-xt 1-\- - -\-x-\-l is irreducible 21
34 38. Determination of irreducible quantics in the GF[p n ] whose
degree contains no prime factor other than those of pn 1 . 22 27
3946. Irreducible quantics of degree p* in the GF[pn] 28 32
47 49. Miscellaneous theorems on irreducible quantics 32 34
50 5.8. Primitive roots and primitive irreducible quantics .... 35 42
59. Exercises 42 44
60. Table of primitive irreducible quantics 44
VI TABLE OF CONTENTS.
CHAPTER TV.
Miscellaneous properties of Galois Fields.
Section Page
61 62. Squares and not- squares 44
63. Number of w th powers in a field; extraction of roots . . 45
64 67. Number of sets of. solutions of certain quadratics .... 46 48
68 71. Additive -groups and their multiplier Galois Fields . . . 4951
72. Condition for linear independence of marks with, respect
to an included field 52
73. Conjugacy of marks with respect to an included field . . 52
74. Newton's identities for sums of powers of the roots of an
equation belonging to a Galois Field 53 54
CHAPTER Y.
Analytic representation of substitutions on the marks
of a Galois Field.
76 78. Definitions. Representation of a given substitution . . . 54 55
79 83. Special functions suitable to represent substitutions . . . 56 59
84. Necessary and sufficient conditions for a substitution quantic 59 60
85 89. Applications of preceding theorem. Reduced form . . . 61 63
90. Table of all substitution quantics of degree < 6 . . . . 63 64
91 94. Betti-Mathieu Group. Certain of its subgroups .... 64 68
95. Identity of Betti-Mathieu Group in the GF[pnm~\ with
Jordan's linear homogeneous group in the GF[p n ] on m indices 69 70
96. Exercises 7071
SECOND PAET.
THEORY OF LINEAR GROUPS IN A GALOIS FIELD.
CHAPTER I.
General linear homogeneous group.
97 98. Two definitions of the group 7577
99100. Order and generators 7779
101 102. Transformation of indices. Invariance of characteristic
determinant 80 81
103 107. Factors of composition of the linear homogeneous group . 81 86
108 109. Linear fractional group. Isomorphic permutation group . 87 88
CHAPTER H.
The Abelian linear group.
110 112. Conditions for Abelian substitutions. Inverse substitution . 89 91
114 115. Generators and order of Abelian group 92 94
116 119. Factors of composition of the Abelian linear group . . . 94 100
120121. Conjugacy of operators of period two of the Abelian group 100105
122123. Operators of period two in the quotient -group A(Zm,pn} 105109
TABLE OF CONTENTS, VII
CHAPTER m.
A generalization of the Abelian linear group.
Section Page
124 125. Definition of the substitutions; their inverse 110 111
126128. Structure of the group 111114
CHAPTER IV.
The hyperabelian group.
129 130. Conditions on its substitutions; their inverse 115 116
131. Largest subgroup containing the Abelian group self-
conjugately 117 120
132 133. Corresponding theorems for their quotient -groups . . . 120 121
134 136. Binary linear homogeneous subgroups of the quaternary
hyperabelian group. Application to their quotient -groups 122 125
137. Identity of binary hyperabelian and binary linear group . 125
CHAPTER V.
The hyperorthogonal and related linear groups.
139 142. Definition. Structure in the general case 126 131
143 151. Order, generators and structure in the hyperorthogonal case 131 144
CHAPTER VI.
The compounds of a linear homogeneous group.
153. Isomorphism of linear group with its compounds .... 145 146
154. Multiplicity of isomorphism for general linear group . . 146 147
156. Pfaffian invariant of the second compound 147 148
157 158. Group induced upon certain Pfaffians by the second compound 148 150
159 162. The second compound of the general and special Abelian
groups 151 153
163 165. The second compound of the quaternary linear group . . 153 155
CHAPTER VH.
Linear homogeneous group in the GF[pn], p > 2,
defined by a quadratic invariant.
166 169. Canonical forms of the quadratic invariant 156 158
170 171. Orthogonal substitutions; the first and second orthogonal
groups 159
172 180. Order and generators of the orthogonal groups .... 160 169
178. The ternary first orthogonal group and the linear fractional
group 164
181 198. The structure of the orthogonal groups 169 197
186 188. Senary orthogonal groups isomorphic with quaternary
linear groups 172 179
189. Quinary orthogonal group isomorphic with quaternary
Abelian group 179 182
190. Senary orthogonal groups isomorphic with hyperabelian
groups 183186
195 198. Quaternary orthogonal and linear fractional groups . . . 191 196
VIII TABLE OF CONTENTS.
CHAPTER Yin.
Linear homogeneous group in the GF[Z n \ defined
by a quadratic invariant.
Section Page
199. Canonical forms of the quadratic invariant ...... 197 199
200. Structure of group on an odd number of indices .... 199 200
201 204. Definition, order and generators of the hypoabelian
groups ................... 200206
205. "Invariant defining the subgroup Ji ......... 206 208
206 208. Isomorphism of senary group Ji with certain quaternary
groups ................... 208211
209. Simplicity of Ji on more than six indices ...... 212 216
210. Miscellaneous exercises on chapters I VIE ...... 216 218
CHAPTER IX.
Linear groups with certain invariants of degree q > 2 .
211 213. Definition, generators and structure of group ..... 218 221
CHAPTER X.
Canonical form and classification of linear substitutions.
214 216. Canonical form of linear homogeneous substitutions . . . 221 229
217 220. Substitutions commutative with a given linear substitution 229 236
221 223. Distribution of the substitutions of the general ternary and
quaternary linear groups into sets of conjugate substitutions 236 241
CHAPTER XI.
Operators and cyclic subgroups of the simple group
224 225. Notations. The seven distinct canonical forms .... 242 244
226 237. Conjugate operators and cyclic groups of each type . . . 245 259
238. LF(3, 2 s ) not isomorphic with the alternating group on
8 letters, each group being simple and of equal order . . 259 260
CHAPTER XH.
Subgroups of the linear fractional group LF(2,p n ).
239. Doubly transitive substitution group on p-{-l letters . . 260 261
240 244. Commutative subgroups of order p; cyclic subgroups . . 261 265
245. Concerning dihedron groups and their subgroups .... 265 266
246 248. Subgroups of dihedron and four- group types ..... 267 268
249 255. Subgroups containing operators of period p ...... 268280
256. Subgroups containing no operators of period p . . . . 280282
257 259. Subgroups of tetrahedral, octahedral and icosahedral
types .................... 282285
260 261. Summary of subgroups. Simplicity theorem ..... 285286
262. Galois' theorem on the minimum index of a subgroup . . 286
263. Lowest degree of isomorphic substitution group .... 287
TABLE OF CONTENTS. IX
CHAPTER XIII.
Auxiliary theorems on abstract groups. Abstract forms
of various linear groups.
Section Page
264 267. Abstract groups isomorphic with the symmetric and alter-
nating groups 287 290
268 269. Quaternary linear group modulo 2 isomorphic with the
alternating group on 8 letters 290 292
270 274. Abstract form of quinary orthogonal group modulo 3 . . 292 298
275 276. Its isomorphism with a hyperabelian group 298 299
278282. Abstract group isomorphic with LF(2,p n ) 300303
CHAPTER XIV.
Group of the equation for the 27 straight lines on a general
surface of the third order.
283. Notation for the configuration of the 27 lines 303 305
284 285. Group of the equation. Isomorphism with linear groups . 305 306
286. Subgroups of indices 27, 36, 40, 45 306307
CHAPTER XV.
Summary of the known systems of simple groups.
287. The ten known infinite systems 307
288. Isomorphisms between certain groups of the systems . . 308
289. Two triply infinite systems of non- isomorphic simple groups
of equal order 309
290. Table of simple groups of orders less than a million . . 309 310
INDEX OF SUBJECTS 311312
X DEFINITION OF SYMBOLS.
DEFINITION OF SYMBOLS.
, 12, 19. H0(m, s ), 138.
,p n ), 100. IQ[m, p n ], 16.
FH(2m, 2 W ), 216. J^, 206.
, p n ], 191. L SilJ n, 191.
m, p n ), 89. LF(m, p n ), 87.
GLH(m, p n ), 76. J\r6f(w, p w ), 191, Note 1).
G(m, q, p n \ 110. (Wi ^^ 159 .
r*
^m,p, *, 131. PI^[W, p n ], 21.
G M + (,r 26L 6fJ.(2m, p"), 89.
G 20L 5flT(2m, 2"), 216.
GF[p n ], 14. SLH(m, p n ), 82.
J3(2w, / n ), 115. S0(w, /*), 191.
, / n ), 120. SQ [k, p], 65. ;
FIRST PART,
INTRODUCTION
TO THE GALOIS FIELD THEORY.
DlCK SON, Linear Groups
CHAPTER I,
DEFINITION AND PROPERTIES OF FINITE FIELDS.
1. If the difference of two integers t and r be divisible by a
third integer p, then t and r are said to be congruent modulo p,
or according to the modulus p. This property is expressed by the
following notation due to Gauss:
t = r (mod p).
For example, 7 EE! (mod 3), 1 = 2 (mod 5).
The totality of integers congruent modulo p with a given posi-
tive integer r < p is given by the formula
lp + r (J-0, 1, 2, ...)
This totality, which will be designated C r , is said to form a class
of residues modulo p', it includes every integer which gives the residue r
when divided by p. It follows that the p classes (7 , C 19 C 2 , . . .,
Cp i include every integer, positive or negative. They are therefore
said to form a complete system of classes of residues modulo p.
Example. - The three classes C7 , C 19 C 2 form a complete
system of classes of residues modulo 3; indeed, every integer falls
under one of the three forms 3?, 3Z-J-1, 3Z -f 2.
2. An instructive diagram is furnished by the regular polygon
of p sides inscribed in a circle. Denote the vertices taken in posi-
tive order (counter-clockwise) by C , C 19 . . ., C p \. Regarding <7
to be the origin, we take as the plot of any given integer m that
vertex which is obtained by counting off from the origin m of the
divisions on the circle in the positive or the negative direction accor-
ding to the sign of + m. All integers of the form Ip -f r (I = 0, 1,
2, . . .) are evidently plotted by the one point C r , so that congruent
integers give rise to the same point. The p classes of residues
modulo p are represented unambiguously by the ^vertices of the
polygon.
1*
CHAPTER I.
3. From the numerical identities
s) = (lt)p + (r s),
(lp -f r) (tp + s) = (Up + ls + rt)p + rs,
we obtain the following formulae for the addition, subtraction and
multiplication of classes of residues:
Cy i (-'s == ^r + s) v/r' ^J ~ ^rs-
If two given classes O r and 0,, C s =%= Co, lead uniquely to a
third class O x such that O r = 0,0*, then 0* is said to be the quotient
of C r by C s and the following notation employed
The condition for the quotient is evidently identical with the condition
that there exist a solution x of the equation
1) r = sx -f
In order that a solution x shall exist for r and s arbitrary integers
such that s is not divisible by p, it is necessary and sufficient that p
be a prime number. To prove the condition necessary, let j0=_p 1 p s ,
where p 1 >l, P 2 > 1. Then 1) can not always be satisfied; for
example, when s = p and r is not divisible by p v The condition
that p be a prime is, moreover, a sufficient one by the corollary
of 4. Hence the division of classes of residues, the divisor being
other than the class , is always possible if, and only if, the modulus p
be a prime number.
In particular, these remarks show that the classes of residues
with respect to a prime modulus may be combined by the rational
operations of algebra and that each result is itself one of the classes
of residues. For example, let p = 3. Then
4. Fermat's Theorem. - If an integer a be not divisible by a
prime number p, then a p ~ l = 1 (mod p).
Since the integers a, 2 a, 3 a, . . ., (p l)a are all distinct
modulo p, their residues must be identical, apart from their order,
with the integers 1,2, 3, . . ., p 1.
Forming the product of the integers in each set, we N have
aP ~ l - 1 - 2 - 3 . . . (p - 1) = = 1 2 3 . . . (p - 1) (mod p).
Corollary. - If a be not divisible by the prime number p, there
exists an unique solution of the congruence ax = b (mod p).
Applying the theorem just proven, the solution is evidently
x~a*~ 2 b (mod p).
DEFINITION AND PROPERTIES OF FINITE FIELDS. 5
5. Definition of a field. -- A set of elements u lf u 2 , . . ., w ff , which
may be combined by addition subject to the formal laws
Ui + Uj = Uj + Mi, M f + (Uj -f W*) = (Ui -f %) + U k ,
such that the sum of any two elements is likewise an element of
the set is called an additive -field. If two elements u { and u k are
given , there may or may not exist a third element Uj in the set such
that Ui + Uj = M*. If existent , Uj is said to be determined by sub-
traction, M; EE Mfc MJ. Assume 1 ) that subtraction is always possible
in the given additive -field. The set will contain the differences
MI -- MI, M 2 M2, , u a u a . Each has the additive property of
zero, since M,- -f (M,- M/) = %. From the latter, M,- Ut == Uj Uj
follows by the definition of subtraction. Hence the above differences
all have a common value u. There exists no new zero element M',
since Uj -f u 1 = % requires u f = % - % = *. Two elements are called
equal or distinct according as their difference is or is not the zero
element u. Select from the original set all the distinct elements and
denote them by MO, MI, MS, . . ., M,_I, where M O denotes the unique
zero element.
Assume next that the s elements UQ, u\, . . ., u s \ may be com-
bined by multiplication subject to the formal laws
such that the product of any two elements is itself an element of
the set. Then the element M O will have the multiplicative properties
of zero, viz., for any element M/ of the set,
M./M O = u Q Uj = UQ.
Indeed, since every product UjUt is an element of the set,
Uj(ui U{) = UjUf UjUi = MO, (M/ u t )uj = M O .
Griven two elements M, and u k , MI=)=M O , there may or may not
exist a third element Uj in the set such that UiUj=u k . If existent,
Uj is said to be determined by division, Uj=u k /u,: Assume 2 ) lastly
that division is always possible in the set, and in a single way, the
divisor being other than the zero element. A set of s distinct
elements satisfying the above four conditions is said to form a field
of order s.
To obtain a field of finite order, the assumption concerning
division may be replaced by the postulate that a product of two
1) In the additive - field of all positive integers, not every difference of
two elements belongs to the field.
2) The set of all positive and negative integers satisfies the assumptions
as to addition, subtraction and multiplication, but not that for division.
6 CHAPTER I.
elements shall be the zero element u only when one of the factors
is u . Under the latter hypothesis, the series of products
UoUi, U\Ui, U%Ui, . . ., 'Ug iUi (Ui=%= UQ)
are all distinct and therefore (their number s being finite) are identical
in some order with the series UQ, u\ y w 2 , . . ., u 8 i. Hence if Uj be
any element of the set, the equation
2) xui = Uj (ui =|= MO)
is satisfied by one and but one element x of the given set. Hence
division by any element except U Q is always possible within the set
and gives an unique result.
For a field of infinite order, the assumption that division is not
possible in more than one way may be replaced by the above postu-
late that a product vanishes only when one factor vanishes. Indeed,
if 2) be satisfied by two distinct values % and x 2 of x, then
Ui(xi #2) = %> whereas each factor differs from U Q .
After the above explanations, we make the formal definition:
A set of s distinct elements forms a field of order s if the elements
can be combined by addition, subtraction, multiplication and division,
the divisor not being the element zero (necessarily in the set), these
'operations being subject to the laws of elementary algebra, and if the
resulting sum, difference, product or quotient be uniquely determined as
an element of the set. 1 )
A field may therefore be defined by the property that the rational
operations of algebra can be performed within the field.
The results of 3 may now be stated in the form: The complete
system of classes of residues modulo p forms a field if, and only if,
p be a prime number.
6. Definition of a Galois Field. Let P(x) be a rational integral
function of degree n having integral coefficients not all divisible by
a given integer p. If we divide an arbitrary integral function F(x)
having integral coefficients by the function P(x), we obtain a quotient
Q(x) and a remainder which can be written in the form f(x) + #-*#(#)>
where f(x) is of the form
3) f(x) = a 4- aix + a 2 x 2 + h - 1#" ~~ S
each a t belonging -to the series 0, 1, 2, . . ., p 1. Then
4) F(x) = f(x) + p q(x) + P(x) Q(x).
We say that f(x) is the residue of F(x) moduli p and P(x) and write
4j) F(x)=f(x) [modd p, P(x)~\.
1) Moore, Mathematical Papers, Chicago Congress of 1893, pp. 208 242;
Bull. Amer. Math. Soc., December, 1893.
DEFINITION AND PROPERTIES OF FINITE FIELDS. 7
The totality of functions F(x) obtained by giving to the poly-
nomials Q(x) and q(x) in 4) all possible forms is said to constitute
a class of residues; two functions are called congruent if, and only
if, they belong to the same class of residues. From the form of 3)
there are evidently p n distinct classes.
Consider two integral functions having integral coefficients
F t (x) = /.<*) + J> 2 (*) + P() Qi (*) [ - 1 , 2].
It is evident that the class to which F 1 F 2 or F 1 F 2 belongs
depends merely upon the functions f + f 2 or f f 2 respectively, being
independent of the functions <#, Qi . Hence classes of residues com-
bine unambiguously under addition, subtraction and multiplication.
In order that the division of an arbitrary class by any class (7, not
the class zero G 0f shall lead uniquely to a third class, it is necessary
that the equation dG = Co shall require d = C . Evidently this
will not be the case if p be composite, p=PiP^y or ^ P(x) be
reducible modulo p, viz.,
where the Pi(x) are integral functions having integral coefficients,
the degrees of PI (a?) and P 2 (x) being less than the degree of P(x).
Hence p must be prime and P(x) irreducible modulo p.
Inversely, if p be prime and P(x) irreducible modulo p, it
follows from 7 that to any class Cp l other than the class (7 there
corresponds an unique class G F \ such that GF\GF I is the class unity.
Hence there exists the quotient class
The p n classes of residues therefore form a field called a Galois Field
of order p n . Moreover, the p n classes of residues moduli p and P(x)
form a field if, and only if, p be prime and P(x) be irreducible
modulo p.
As an example, let p = 3 and P(x) = x 2 x 1. The 3 2 resi-
dues are
v/, JL, ~ J. , X , X -f- J. , 3/ J- , $?, X ~T~ -L , i -!
The sum, difference or product of any two of these may evidently
be reduced moduli 3 and x 2 x 1 to one of the nine residues.
Moreover, the quotient of any one by any residue except may be
reduced to one of the set. For example,
1 rf 1 1 /v.2
A w X ^4. tA/ ~~ tX/ o 4
X X\X -~ * 1^ 'T* I 1 >Y* I 1 " ~ /v2 ^ ~
The nine residues thus form a Galois Field of order 3 2 .
8 CHAPTER I.
7. Theorem. - If two integral functions F(x) and P(x) having
integral coefficients admit of no common divisor containing x modulo p,
p being prime, we can determine two integral functions F'(%) and P'(x)
having integral coefficients such that
F\x) - F(x) -P'(x) P(x) = 1 (mod p).
Applying 4, we can set
F(x) 1= a - A(x), P(x) ~ I B(x) (mod p)
the coefficients of the highest power of x in A(x) and B(x) being
unity and the remaining coefficients integers. We perform the usual
process to determine the greatest common divisor of A and B,
neglecting however, multiples of p. Each remainder is congruent
modulo p to a product of an integer r and an integral function R(x)
with integral coefficients, that of the highest power of x being unity.
Supposing for definiteness that the degree of A is not less than that
of Bj we obtain the congruences (mod p) :
-R/n2 =. R ni iQ m 4- r m .
We derive at once the following congruences modulo p
.r.l^ = (r, + ft ftM- (r, Q l + r,
where M and N are integral functions of x having integral coefficients.
None of the integers r 1 . . .,r m are divisible by p-, for, A and B
would then have a common divisor containing x. Hence, by 4,
there exists an integer r such that
r abr r 2 . . . r m = 1 (mod p).
From the last congruence in the above set, we therefore find
1 = rab (MA -- NB) = F(x)-rbM- P(x)-raN (modj).
Corollary. - - If F(x) E|E [modd p, P(x)\, p being prime and P(x)
irreducible modulo p y we can determine an integral function F'(x)
such that ., , , N T/ vi
F\x) F(x) = 1 [moddp, P(x}\.
Note. - By an analogous use of the process for finding the
greatest common divisor, we obtain the following theorem:
DEFINITION AND PROPERTIES OF FINITE FIELDS. 9
//' two integers f and p be relatively prime, we can determine two
integers f and p' such that f'f p'p = 1.
8. The proof of the existence of a function of degree n irreducible
modulo p and hence of the existence of a Galois Field of order p n }
for every prime p and integer n, will be given in 19 27. We
will first prove that no other finite fields exist and that not more
than one Galois Field of a given order p n exists.
9. Consider an abstract field F[s] composed of a finite number
s > 1 of elements or marks u , %, . . ., u s _i. Having every difference
Ui U;., the field contains a mark, denoted by U( Q ), which has the
properties of zero viz., for every Ui,
Ui -f w (0 ) = Ufa}. U( Q )Ui = UiU( ) = U( ).
Having every quotient
Ui/Ui (X=|= %)),
the field contains a mark w (l ) having the properties of unity; viz., for
Ulj
The field thus contains every integral mark
%) = W(D -f w ( i) 4- ---- 1- W(i> (c terms),
Since there exists only a finite number of marks in the F[s\ y
there must arise equalities in the series
If U( r ) = W(,), we have
Denoting by p the least positive integer such that U( P ) = W(oj, the p
marks
are all distinct, while
W( r ) = U( s ) if, and only if, r = s (mod ^)).
This integer p is a prime number. For, if
we have, by hypothesis, U( PI ) =(= w (0 ). Hence, from
we derive U( P J W( ) and hence ^? 2 > i>. Hence the integral marks of the
F\s\ form a field -F[p] which is the abstract form of the field of the
classes of residues with respect to a prime modulus p. When there
is no ambiguity, we denote by c the integral mark U( c ).
10 CHAPTER I.
10. Theorem. The order of F[s\ is a power of p.
If u t be a fixed mark =j= U Q of the F[s], the products
give p distinct marks of the field. If s > p, there exists a mark
not of the form cu r Then
(01,02 = 0> !>>#
gives jp 2 distinct marks. If s > j? 2 , there exists a mark u s not of the
form cu + cu so that
gives j9 3 distinct marks. Proceeding similarly, we must ultimately
obtain all the marks of the F[s] expressed by the formula
0^! + c 2 u 2 H ----- h 0^ n (every c,-= 0, 1, . . ., p 1),
not two of these _p w expressions being equal. Hence s = p n .
Definition. - A set of marks u l9 u 2 , . . ., ut are said to be
linearly independent with respect to the included field F[p], if the
equation
c^i + c 2 u 2 H ----- h c*% = 0,
where the c's are marks of the -FJj)], can be satisfied only when
every c/ = 0.
Definition. - - A rational integral function of any number of
indeterminates X lt X 2 , . . ., X k is said to belong to a field if its
coefficients are marks of that field. It is irreducible in the field if it
is not identically the product of two or more functions belonging
to the field, each function involving some of the indeterminates X/.
An equation between functions belonging to a field is said to belong
to the field.
11. Theorem. Any mark u of the F\s=p n ~\ satisfies an
equation of degree Jc<n,
=
belonging to and irreducible in the F[p].
Indeed, a linear relation with coefficients belonging to the F[p]
certainly holds between any n-\-l marks of the F[p n ] and hence
between
M, M 1 , W 2 , . . ., U n .
If such a relation holds between the first k -f 1 of these powers of u,
u satisfies an equation of degree k.
DEFINITION AND PROPERTIES OF FINITE FIELDS. H
12. Let u be any mark =j= of the F[s = p*\. The marks
u< (* = 0,1,2,...)
belonging to our finite field are not all distinct. From u r = u 8 , we
derive u r ~ s = 1. The least positive integer e for which w e = 1 is
called the period of the mark u, while it is said to belong to the
exponent e. The marks 1, u, u 2 , . . ., w e1 are all distinct.
We may form a rectangular array of the marks =j= of the field
as follows:
1 u u 2 ... u 6 1
Ui UUi U 2 U L . . . U e ~ 1 U l
. . . U
e ~
where w t is any mark =4= not occurring in the first line, u 2 any
mark =j= not in the first or second lines , etc. Evidently the marks
in any line are different from each other and from those in the
preceding lines. Since each new mark Ui gives rise to a set of e
new marks, the number p* -- 1 of the marks =(= in the -F[j? n ] is a
multiple of e.
Theorem. - - The period of any mark =j= of the F\p n ^ is a divisor
of p n 1.
13. Raising u* to the power (jp n !)/, we have
We have thus the following generalization of Fermat's Theorem:
Every mark of the F[p n ] satisfies the equation
We have therefore the following decomposition in the jP|j) n ]:
X"-X=JJ (*-),
i=0
HI running over the p n marks of the F[p n ].
14. Theorem. - - // two marks u^, u 2 belong respectively to ex-
ponents e 1 6 2 which are relatively prime, their product u u 2 belongs to the
exponent e^ and the e l e 2 marks
(d* = 0. 1. .... e.
U= ;i;...;4-
are all distinct.
12 CHAPTER I.
If w x % has the period t, we have
(^Mg)" 1 = V = 1,
whence is divisible by e 2 ; similarly, is divisible by e v But
(tfii5** - 1.
Hence t = e 1 e 2 .
15. We prove as in algebra the theorem:
An equation of degree Jc belonging to a field has in the field at
most k roots, unless it be an identity, when every mark of the field is
a root.
16. Theorem. - For every divisor d of s 1, the equation
X d --l =
has in the F[s = p ri ] exactly d roots.
Setting s 1 = dq, we have the identity
Since the last factor belongs to the -F[s] and does not vanish
for the mark zero, it vanishes for at most d(q 1) marks of the
field. But the left side of the identity vanishes for s 1 marks of
the field. Hence the factor X d 1 must vanish for at least d marks.
17. Decompose p n 1 into its prime factors,
p n l=p h fpl*...p h k k.
For each integer i of the series 1, 2, . . . , Jc, the equation
has by 16 exactly p h .i roots belonging to the f [s = p n ]. Of
these roots p^~ l are also roots of the equation
and thus belong to exponents less than p h .i. The remaining roots w t ,
in number
belong to the exponent p h .i itself. Any product of the form
W = Uj^U 2 ...Uk
will by 14 belong to the exponent p n 1. Forming in every
possible way the product w, we obtain 1 )
1) This number equals ct>(p i), where (t) denotes the number of
integers less than and relatively prime to the positive integer t. See Dirichlet,
Vorlesungen iiber Zahlentheorie, 11.
DEFINITION AND PROPERTIES OF FINITE FIELDS. 13
-such marks. Each mark w belonging to the exponent s 1 is called
a primitive root of the equation
x*- i -i = o
and also a primitive root of the F[s]. Since the powers w 1 , w 2 , . . ., tv s ~ 1
are all distinct, we may state the theorem:
The p n 1 marks =)= of the F[s = p n ~] are the p n 1 successive
powers of a primitive root of that field.
Corollary. If d be any divisor of p n 1, the mark w^ n ~ l ^ d belongs
to the exponent d.
18. We may now recognize in our F[s] the abstract form of a
Galois Field of order s p n . Indeed, by 11, the primitive root w
satisfies an equation of degree k < n.
W k (x) - 0,
belonging to and irreducible in the F\j)]. Every mark =f= of
the F[s], being a power of w, can be reduced by the identity
W*(w) ^ to the form
c 1 w k ~ l + C 2 w k ~ 2 -i ----- h c k iW 4- c k ,
where the c's belong to the -F[p]. The mark zero evidently falls
under this form. Since, inversely, every one of these p k expressions
is a mark of the F[s], we must have k = n. Hence every mark of
the F\s =p n ~\ represents a class of residues moduli p, a prime, and W n (x),
a function with integral coefficients irreducible modulo p. Every
existent field is therefore the abstract form of a Galois Field.
Suppose there could exist a second field F'[p n ~\ of order equal
to that of F[p n \ The field .F[j9 w ] possesses a primitive root w
satisfying an equation W n (x) = 0, of degree n, belonging to and
irreducible in the J^|j>]. The function W n (x) divides x fn x in
the F[pY}- We may, indeed, apply in the F[p~\ Euclid's process
for finding the greatest common divisor of these functions. If there
were no common factor, we would ultimately reach as a remainder
a constant, whereas the process may be interpreted in the GF[p n ~\,
in which field the common factor x w exists. Hence W n and
x pn x have a common factor in the -F(j>]. Moreover, W n is irre-
ducible in that field.
Since F[p\ is contained in F'[p n ~\, the division of x pn x by
W n is, a fortiori, possible in the F'[p n ]. It follows from 13 that
1) Another proof is given in 23.
14 CHAPTER E.
the equation W n (x) --= completely decomposes in the F'[p n ~\. Any
one of its roots w' is a primitive root in the J^'jj) 71 ]. Indeed, by its
definition, W n (x) does not divide x e x in the F[p] for e <p n . The
powers of w 1 therefore give all the marks of the F'[p n ~\. Hence
.F[jp re ] and JF'lj? 71 ] are abstract forms of the same Galois Field. These
results, first proven by Moore (loc. cit.), may be stated as follows:
Theorem. - - Every existent field of finite order s may be represented
as a Galois Field of order s = p n . The GF[p n 1 is defined uniquely
by its order; in particular, it is independent of the special irreducible
congruence used in its construction.
CHAPTER E
PROOF OF THE EXISTENCE OF THE GF[p n ] FOR EVERY
PRIME p AND INTEGER m.
19. The next step is to prove the existence, for every prime
number p and positive integer m, of a congruence of degree m irre-
ducible modulo p, from which will follow the existence of the GF[p m ~\.
We will, however, make a more general investigation, taking as our
basis a fixed 6r.F[j) n ] (in its abstract form), whose existence is supposed
known. We will prove that functions belonging to and irreducible
in the GF[^p ri ] exist for every integer m and will determine their
number. Since the GF\j)] 7 the field of integers taken modulo jp, is
known to exist, we shall have proven (taking n = 1) the existence,
for every value of m, of functions belonging to and irreducible in
the GF[p], i. e., irreducible modulo p.
At the same time, we shall have deduced some important pro-
perties of the GF\j> nm ~] with respect to the included field, the GF[jp n ].
20. Theorem. If two functions F(x) and P(x) belonging to
the GFlp 71 ] have in the field no common divisor containing x, we can
determine two functions F'(x) and P'(x), belonging to the GF[p n ~]
such that ,
The proof is quite analogous to that of 7.
21. Theorem. If, in the GrF[p*\j P\x\ has no factor invol-
ving x in common with F(x) but divides the product E(x) F(x), then P(x)
divides E(x) in the 6r.F[p n ].
Indeed, by multiplying the given equation
E(x) . F(x) = P(x) - S(x)
by JP'(#), determined as in 20, we find
E(x) = P(x)[S(x) - F'(x) - E(x) - P'(x)].
PROOF OF THE EXISTENCE OF THE GF[p*~\, etc. 15
22. Theorem. - A function E(x) belonging to the GF[p n ~] can
be decomposed into factors belonging to and irreducible in the GF\jj n '\
in a single way.
For if E(x)-f l f t ...f ll ~F l F t ...F t ,
where fi(x) and FI(X) are irreducible, F t must by 21 divide one
of the factors f iy and, since the latter are irreducible, be identical
(apart from a factor independent of x) with one of them, say f v
Proceeding similarly with the equality
/2/s A = F 2 F 3 . . . F k ,
we may suppose f 2 = F%, etc. In particular, h = k.
23. Theorem. - Every function F(x) of degree m belonging to
and irreducible in the GrF[p n ~\ divides
xP nm - x.
Upon dividing any function E(x) belonging to the 6r-F[j> n ] by F(x),
we obtain a residue of the form
I I
H ----- \- a m _ l x ,
the a's being marks of the G-F[p n ~]. We denote the p nm distinct
residues of the above form by
5) Z, (*- 0,1,... ,p""-l),
and, in particular, by XQ the residue zero. Consider the products by a
fixed residue Xj=|= -Xo>
6) XjX f (i = 0,l,...,/ m -l).
By the theorem of 21, the products 6) are all distinct and different
from XQ. Hence the residues obtained on dividing them by F(x)
must coincide apart from their order with the residues 5). Forming
the products of the residues not zero in each series,
P nm i
i [mod *(*)]-
Since 77X.EEO, we have by 21,
Xf m ~ l - 1 EE [mod F(xj].
Taking for Xj the particular residue x, the proof of the theorem
follows.
24. Theorem. - If f(x) belongs to the GrF[jp n '], we have, for
every integer t, the following identity in the field:
16 CHAPTER II.
4- . + c k x k ,
where the c's belong to the GF[p*], so that
7) cf=c; (-0, !,...,&)
Raising f(x) to the power p and noting that the multinomial coeffi-
cients of the product terms (viz., those not p ih powers) are multiples
of p, we have the algebraic identity,
ff(x)7 = <$> + clo?+--- + elx'"' + p-Q 1 (x\
We obtain by induction the formula
Wxff - ^ + <S+ + aF+ P Q,(x).
Applying 7), we obtain in the GrF\_p*\ the identity:
Our theorem now follows by a simple induction.
25. Theorem. - A function F(x) of degree m belonging to and
irreducible in the G-F[p n ~\ divides (in the field) the function
x* nt - x
only when the integer t is a multiple of m.
Let t = sm 4- r, where <^ r < m. By the theorem of 23,
xP nt - x =(xP nsm )p nr - x ~ x*> nr - x [mod F(x)].
Hence, if x pnt - x be divisible by F(x) in the GF[p*\, we have
8) xP nr =~x [mod F(xJ\.
Denote by f(x) any one of the p nm expressions
c 4- CiX + c z x* 4- 4 c m -ix m ~ l
in which the c's are marks of the GF[p*\. By 24, we derive
n Q\
[f^]""'' ^ fi^) ~ f(x) [mod F(x)l
Hence the congruence gpB r ^ $ [mod j,^
is satisfied by the jp wm expressions /"(#), which are distinct modulo F(x\
the latter being an irreducible function of degree m. Since r < m,
it follows from 15 that the congruence must be an identity,
whence r = 0.
26. The number N mtf n of functions F(x) of degree m belonging
to and irreducible in the GF^p 1 *] may now be readily determined.
For brevity, such an irreducible quantic will be designated an
IQ[m,p n ].
PROOF OF THE EXISTENCE OF THE GF[pn], etc. 17
It is to be understood throughout the investigation that all our
operations upon quantics are performed in the GF\jn n ~\. We may
therefore state the results of 23 and 25 as follows:
An /(?[%, .p*] is a divisor of x pnr ' - x if, and only if, m^ be a
divisor of m.
It follows that an irreducible factor of x pnri - x will be of degree m
if, and only if, it is a factor of noDe of the functions
9) x* nni1 x (m^ < m, m l a divisor of m).
After showing that the irreducible factors of any such function are
all distinct, it will follow that, if we divide x pnr> - x by the product
of all the distinct irreducible factors of the expressions 9), we obtain
a quotient V m , P n which equals the product of all the IQ[m,p n ].
For example, if m be prime, the irreducible factors of x pUT ' - x
are of degree m or 1. By 13, the product of the distinct linear
factors is x pH x. Hence, if m be prime,
x p nr> x -. pnm pn
We next prove that the irreducible factors of x pnn - x are all
distinct. If such a factor be of degree m, it can be used to define
the GF[jp nm Y). In this field the equation
x pnn> x =
has p nm distinct roots; viz., the marks of the field. Hence no factor
can be a multiple factor in this field and therefore not in the in-
cluded field the GF[p n \. If an irreducible factor f be of degree m x < m,
it cannot be a multiple factor. Indeed, m^ must be a divisor of m,
and f must divide x pnmi x in the GrF[p n ~\. By the former case,
f is a simple factor of the expression just given. It remains to prove
that f cannot divide the quotient
Q = (x pnm - x)/(x* nmi - x).
It suffices to show that Q and x pnntl x have no common factor in
the G-F[p n ]. Setting - .
~^ nnrn /O ***
i rP'^wl) r -^ -t - P
it suffices to prove that y 1 and ^
have no common factor. The condition for a common divisor is
that r be the mark zero in the field. But r ^ 1 (mod p).
i
1) See 28.
DlCKSON, Linear Groups. 2
18 CHAPTER II. PROOF OF THE EXISTENCE OF THE GF[p*], etc.
27. Continuing the investigation, let
m = <ft4^.&,
q\, #2, , q* being the distinct prime factors of m. For brevity,
we use the symbol *
p] =x p x.
We proceed to prove the formula, due to Dedekind for n = 1,
M 77
r m "-
In this expression, the term
f m -I
77* = 77 -rr^ ^r P
|_ 2*1 2' 2 * ' -**J
in which the product extends over the C s ,k combinations q fl , . . ., q, k
of the integers q lf . . ., q s taken k together, occurs in the numerator
or in the denominator according as k is even or odd. Each IQ\m,yP]
occurs once as a factor in 77 = \m\ but divides no other 77*; it is
therefore a simple factor of the fraction. If there be any factor of
the fraction having the degree m t <w, we denote it by F(x). By
25, m i must be a divisor of m. Denote by q lt q%, . . ., q Sl the prime
factors entering in m to a higher power than in m v Then m 1 divides
but not - (j = s^ -\- 1, s i -h 2, . . ., s). It follows that,
if fc > Si, 77^ does not contain -F(^) of degree %; while, for A; < Si,
77^. contains ^(a?) as often as ^ integers can be selected from q lf q%, . . ., q s j
viz., C s ^k times. Hence F(x) occurs in the numerator and denomi-
nator of our fraction to the respective degrees,
1 "h @s 1} 2 ~h C, lt 4 + ', C Sl> i -\- (7 Sl ,3 + t/ f ^6 + ' '
These numbers are equal, since their difference equals (1 I)* 1 = 0.
It follows that every irreducible factor of our expression is an IQ\9H f f^
The number of the latter multiplied by the degree m of must equal
the degree of the fraction, so that
This number cannot be zero; for, upon dividing by the last
term, which is the lowest power of p entering into the expression,
we would then obtain unity expressed as the algebraic sum of a
series of powers of the prime number p with exponents > 1. It
follows that the number of J[m,j? n ] is > 1. [See Ex. 2 below].
28. Let F(x) be an IQ\m,p n ~\. As in 6, the totality of rational
functions of x belonging to the GF\j) n ] can be separated into p nm
CHAPTER III. CLASSIFICATION AND DETERMINATION, etc. 19
distinct classes of residues modulo F(x), each being represented by
one of the p nm residues
a -f- dix + a 2 x* -\ ----- h a m -ix m - 1 (a's in the G-F[jQ n ]).
Proceeding as in 6, we find that these classes of residues form
the GF\j) nm ~\. We can therefore construct the 6r-F[j} r ] in as many
ways as we can express r as the product of two positive integers n, m;
viz. ; by using an IQlm^p^. From the theorem at the beginning of
26 it follows that the GF[p nm >~\ is contained in the GF[p* m ] if,
and only if, m divides m.
EXERCISES.
Ex. 1. Granting the existence of the GF[p n ], the existence of
the G-F[p nq ], q being prime, follows by 26. By induction, the G-F[p r ]
exists for r arbitrary.
Ex. 2. Obtain for the number of IQ[m,p n ] given in 27 the
following limits:
pnm pn = _ $ (m) p nm p n
- > m,p n > ^{ ' ^~
Hint: Expand each power of p n into a series in log p n and apply
1).
Ex. 3. By decomposing modulo 2 the expression (# 2 x)/(x 2 - - a?),
obtain the three IQ [4, 2] given in the left members below. Defining the
6rJP[2 2 ] by means of the irreducible congruence
ft -j- i + 1 EE (mod 2) ,
obtain the six /$[2, 2 2 ] by means of the following decompositions:
T** I /y _l 1 IT" I /y _JL_ 01 [ rf" _!_ /y I 4*1
A> ^^ tX/ ~ JL _ I M ^^ Js ^^ V t \JU ^^ J(j ^^ I/ I ,
/v>4 I ,y.o I 1 " ( /y* I -j /y I -j i \ i /r*2 I a'2/v. i ^'2\
**/ ^^ i*/ ^^ JL It*/ ^^ frtA/ ^^ t I It*/ ^^ t v ^^ t- I.
a; 4 + x 3 + x 2 -f a? + 1 EE (# 2 + ta; 4- 1) (ic 2 4- i 2 x -f 1).
CHAPTER HL
CLASSIFICATION AND DETERMINATION OF IRREDUCIBLE
QUANTICS.
29. Definition. - - An IQ\m,p n ~], as F(x), is said to belong to an
exponent e if e be the least positive integer for which F(x) divides
of 1 in the GF[p n ]. [Compare 32.]
2*
20 CHAPTER m.
The exponent e to which F(x) belongs must divide p nm --1.
For, if 4>>-l = fce + r,
where < r < e, then F(x), dividing of 1, must divide x ke 1
and, by 23, also x ke + r 1. It must therefore divide their difference,
iC* e (iC r 1).
Hence must r be zero.
Furthermore, e must not divide p nt 1, for tf < m; for, if so,
# ? - 1 and hence also F(x) would divide x pHt x, so that the degree
of F(x) would be a divisor of t.
An integer which divides a m 1 but not a* 1, t < m, is said
to be a proper divisor of & m 1. We may state the result:
The exponent to which an IQ[m, p 1 *] belongs is a proper div,isor
of (p n ) m l.
30. Theorem. The number Ni f * of IQ[m, p n ~\ which belong
to an exponent e, a proper divisor of (p n ) m 1 , is O (e)lm.
Let #1, q-2, . . ., q s be the distinct prime factors of e. Proceed-
ing as in 26, we rid x 6 1 of those of its factors which are irre-
ducible in the (r-Fjj)"] and belong to an exponent < e. We obtain
the expression / \
Oe-l) H \x q i q J -l) . . .
L. \ ( e \
^ -i) n U ?t '^ 9 *-V
IT
which is therefore the product of the irreducible factors of x e 1
belonging to the exponent e. Each of them is an irreducible factor of
xP nm - x
and hence of degree m or a divisor of m. Since each belongs to an
exponent which is a proper divisor of (p n ) m 1, the degree must
be m.
The degree of the above function is clearly
?i
Hence m.jyWy. -!>().
31. Theorem. - - If F(x) and cp (x) belong to and are irreducible
in the GF[p n ~] and are of the respective degrees m and t, a divisor
of m, the roots of the congruence
10) y (X) =E [mod F(x)]
are v V" ir 2w -\7- p n(tl)
A 1; Af , Af , . . ., A/
if XL be one root of 10) necessarily belonging to the
CLASSIFICATION AND DETERMINATION, etc. 21
By 24 we have in the 6r-F[j) n ] the identity
Hence, if X^ be a root of 10), so is every Xf r . Since g> (X) is an
IQ\t, P n ]> we have ( 23 ) in the ^T> W ]>
Xf - X t = <p (XJ (X,) = [mod F(a?)].
Hence , m being a multiple of t,
Xf W EE X t [mod JFXa;)].
We next prove that the above t powers of X 1 are distinct modulo
F(x). Indeed, if ^= x ^ [moA F(xj]
for a < 1} < t, we would have, upon raising it to the power p*(m *) f
Xf w ~ X l = Xf (m + b ~ a) [mod F(V)],
so that, by 25, m -\-b a would be divisible by m. Hence b = a.
Corollary. - We have in the G-F[p n \ the decomposition
cp (X) EE (X- X x ) (X- Xf ) . . . (X- Xf ( '~ 1) ).
In particular, F(x) = has in the 6r.F(j> w ] the distinct roots
x, x* n ,...,xP n(m - l \
32. Theorem. If F (x) be an IQ[m, p n ~] belonging to the
exponent e, every root of F(x) = in the G-F[p nm ~] belongs to the
exponent e, and inversely.
We may define the GrF[p nm ~\ by means of F(x). In it, all the
roots of F(x) = satisfy the equation x 6 1 = 0, but do not all
satisfy xf 1 = for f < e. But, p n being relatively prime to e, a
divisor of p nm l, it follows from the corollary of 31 that the
roots of F(x) = in the GrF[p nm ~] all belong te the same exponent.
This common exponent is therefore e.
In particular, for e =p nm 1, the roots of F(x) = are primitive
roots in the G F[p nm ~]. Such a quantic F(x) will be called a primitive
irreducible quantic of degree m in the GF[jp n ~\ and will be referred
to as a PIQ\m, p"].
33. Theorem. - - If e be a prime number, the function
F L r i-4- 4- <r 4- 1
' X-l -
is irreducible with respect to every prime modulus p which is a primi-
tive root of e.
By hypothesis, p belongs to the exponent e 1 modulo e, so
that e is a proper divisor of p e ~ l 1. Hence, by 30 for n = 1,
m = e 1 , the number of irreducible factors of V is = 1.
' a _
CHAPTER IE.
Note. If a be a primitive root of e, then a 4- ke(k = 0, 1, + 2, . . .)
are also primitive roots of e. By the theorem of Dirichlet, this
arithmetical progression contains an infinity of prime numbers. With
respect to any such prime p, V is irreducible modulo p. A fortiori,
V is algebraically irreducible.
Determination of IQ\m,p\ whose degree m contains no prime
factors other than those of p n 1, 34 38.
34. Theorem. - - Let Fi(x),F 2 (x), . . ., F N (x) denote the IQ[m,p n ]
which belong to an exponent
e = (p nm - l)/d,
and let K be an integer relatively prime to d and containing no prime
factors other than those occurring in p nm 1. With the exception of
the case in which k is a multiple of 4 while p nm is of the form 4? 1,
all of the IQ[lm,p"\ which belong to the exponent eh are given by
the N quantics FI(X*), . . ., F N (x*).
By definition, A contains no prime factor other than those
occurring in e. Hence el and e contain exactly the same prime
factors, so that cD(a) _ cp( e )
By 30, we have el ~ e
If we suppose satisfied the conditions (obtained below) under which
shall be a proper divisor of (j) n ) w * 1, we will have
Since e divides^? 11 " 1 1, the irreducible factors of x 6 1 are of degree
< m ( 25). Hence, in the notation of the theorem,
of- 1 - F 1 (x)F 2 (x) . . . F N (x) - Q(x)
where the irreducible factors of Q(x) either belong to an exponent
< e or else are of degree < m. Therefore
where every irreducible factor of Q(x*) is of degree < hm or else
belongs to an exponent < eL Since there are exactly N irreducible
factors of degree mK which belong to the exponent eA, they must
be identical with JFi(#*), . . ., F$(x l \
Calling v the least integer such that p nv 1 is divisible by el,
we seek the conditions under which v = mL Since m is by hypo-
thesis the least integer for which p nm 1 is divisible by e, v must
be a multiple of m. For, if v = qm + r, 0<r<m, then e divides
pn(qm+r)_ i an( j pnmq__ i ^ft hence also their difference p* m 9(p*r ' 1) ?
CLASSIFICATION AND DETERMINATION, etc. 23
which requires r = 0. Having v = qm, we inquire under what con-
ditions does q = A? Since
pnvl d pnmg1
el ~ pnm
it follows that I divides (p*"9 l)/(p nm 1). Raising to the power q
the identity p nm = 1 -f (p nm ~ 1), we find
Let be a prime factor of A and a the highest power of con-
tained in L Since divides p nm 1 and the left member of 12), it
must divide q. Further, if > 2, a divides q. Indeed, the ratio of
the & th term of 12) to the first term q can be written
/ff*-l\
\ )
1-2 ... (&-1)
of which the first two factors are integers, while the third factor
-!) (e-i) _ (*-l)(e-2)
~T~ ~^r
is > 1 if & ^ 2. Hence the irreducible fraction equal to 6 k ~ 1 /Jc has
the factor in its numerator. Hence the terms of 12) beginning
with the second contain to a higher power than the first term q.
Since a divides A, which divides the left member of 12), it follows
that a divides the first term q on the right. Hence, if I be odd
or the double of an odd number, q is divisible by L Inversely, if q
be divisible by A, A being odd or the double of an odd number, the
above argument shows that the right member of 12) will contain
the factor A and therefore that the left member of 11) will be an
integer. In order that v be the least integer for which this can
happen, we must have q = L
If A be a multiple of 4, p nm 1 is even by hypothesis. Then
= 2 will be a factor of q as before. The ratio of the second term
of 12) to the first term will be divisible by 2 if, and only if, p nm 1
be a multiple of 4; the ratio of the & th term to the first will, for k ^ 3,
contain the factor 2. Hence, if p nm be of the form 4? -f- 1, we can
conclude that q = A. [The case p nm = 41 1 leads to the entirely
different theorem of 36.]
35. Let Q be a primitive root in the GF[p n ]. The function x Q*
belongs to the exponent (p n l}/d where d is the greatest common
divisor of t and p n 1. Applying the theorem 34 for m = 1, we
have the result:
If A be any integer containing no prime factor not occurring
in p n l and if t be an integer prime to A, the /[A, p n ] belonging
24 CHAPTER III.
to the exponent A(j n l}/d, f? &em# the greatest common divisor oft
and p n 1, are &e binomials x 1 Q*, the case p n =l 1, A = 4^
&em# excluded.
Inversely, we obtain by this tJieorem every binomial irreducible in
the GF[p*\. In the first place, A and must have no factor in
common, since otherwise x 1 tf would be algebraically reducible.
On the other hand, if I contains a prime factor 0, not a factor of^ n 1,
we can determine ( 7, Note) an integer Q v such that
00! = 1 [mod^-1].
Since p ee '=(>, it follows that Q 6 **= a is a root of
a* $ f = 0.
Hence x a is a factor of X Q g*, so that x*/ e a divides x l tf.
Example. For p n = 1 , we may take Q = 5. Then for A = 2
and t = 1, 3, 5, we obtain the irreducible binomials x 2 5, # 2 -f 1,
# 2 3 belonging to the exponents 12, 4, 12 respectively. For A = 3
and = 1, 2, 4, 5 respectively, we obtain the binomials
/v.3 _ R r 3 _ 4 ,7,8 _ 9 /v.3 _ ft
tt/ t/, t*/ TI, t// / , Jj tj
irreducible modulo 7 and belonging to the respective exponents 18,
9, 9, 18.
36. Theorem. - - Let p n = 2>t - 1, i > 2, J <wW; A = 2^'s, j > 2,
Ze^ fe be the smaller of the integers i and j; finally, let m be
odd. Then if, in the N quantics IQ[m, jp n ] belonging to the exponent
e = m
we replace x by x\ where 'k = 2-^'s is prime to d and contains no prime
factors other than those occurring in p nm 1, we obtain N quantics
of degree ml each decomposing into 2 k ~ 1 quantics irreducible in the
GrF[p n ], so that we obtain all of the 2 t ~ 1 JV r quantics
belonging to the exponent eL
If v denote the least integer such that p nv 1 is divisible by eA,
we find as in 34 that v = qm. In the present case, q is even; for,
if q be odd, v would be odd and p nv 1 the double of an odd
number, whereas A is divisible by 4. By the restrictions on p n
and m,
13) p=2><c--l (rodd).
Raising this identity to the power g, we find
pnm-
CLASSIFICATION AND DETERMINATION, etc. 25
The ratio of the ? th term within the parenthesis to the first term is
+ " 1-2 ... (Z-l) ~ rl ~1 '
where the first and second factors are integers, while the third
factor, being > 1 for I > 2, equals an irreducible fraction with an
even numerator. Hence the first term contains 2 to a lower power
than the remaining terms in the above parenthesis. In order that
pnv_ } shall be divisible by eA, formula 11) requires that A shall
divide the left member of 14). Hence 2 j must divide the first term
of the right member and consequently also 2 l l q. Hence the even
integer q must contain 2 to the power 1 or j i -\- 1 according
as j <J i or j > i. Furthermore, by 34, q must contain every odd
factor of A. Hence, if v be the least possible integer,
or -
according as j < i or j > i, i. e., according as h = j or k = i. Hence
I ml
2* i 2*
As at the beginning of 34, we have
so that the number of IQ[v, p'*\ belonging to the exponent el
is 2*- 1 JV.
By hypothesis,
x e_ i ==F!(X)FI(X) . . . F N (x).Q(x),
where the irreducible factors of Q(x) in the GF[p n ^ belong to
exponents < e or are of degree < m. The irreducible factors of Q(xF)
are therefore of degree < km or else belong to exponents < Ke.
Hence the irreducible factors of degree Am of the expression x e * 1
must, if they belong to the exponent eA, be factors of JFi(#*), . . ., F N (xF).
Since the combined degree of the latter is Nmk = 2*~~ 1 i/JV, and
since there are exactly 2 k ~ 1 N irreducible quantics of degree v
belonging to the exponent eA, it follows that each Fi(aF) is the product
of 2 k ~ l irreducible quantics of degree v.
Corollary. - Since the distinct functions of degree m 1 which
belong to the exponent e = (p n T)/d are given by the formula
x $ ad ,
Q being a fixed primitive root in the (r-Fjj? 71 ] and a being any integer
prime to 6, it follows that x l Q ad decomposes in the G-F[p n ~] into
2* i irreducible factors of degree A/2*" 1 belonging to the exponent
26 CHAPTER IE.
eh, provided p n and A are subject to the conditions given in the
main theorem.
37. Since irreducible binomials are lacking in the case treated
in the last section, we proceed to set up trinomial IQ\l,p n ~\. It is,
however, not necessary to suppose that A is a multiple of 4. We
suppose merely that
p*= 2' l t - 1 (t odd, i ^ 2)
and that I is an even integer containing no prime factor not occurring
w*-l. Set '-
so that v is divisible by 2 1 '. If Q be a primitive root in the
and if s be any integer prime to A and hence also to v, then x Q*
belongs to the exponent (p n l)/^, where d is the greatest common
divisor of s and p n 1, and v is prime to d. Hence ( 36), the
binomial x v Q S decomposes into 2 i ~ 1 irreducible quantics of degree A.
We proceed to determine them.
Since 2* ~ 1 and (p n l)/2 are relatively prime, we can determine
( 7, Note) two integers 1 L and h such that
Multiplying this equation by the even integer s -f (p n 1)/2, we
obtain two integers I and h for which
I2 l - h(p n - 1) = s + (#* l)/2.
Since the (p n I)/ 2 power of the primitive root 9 is - - 1, we
have 2<>i-i , i9
^ u () s = ^ 2 + Q 12 .
In the G-F[p n ~] we have the decomposition
15)
where the % } - a/re marks of the GF[p n ] determined as the roots of the
equation . _ 2
y ' 1 '--+ 2 - o.
In fact, by Waring's formula 1 ), the sum of the (2 z '~ 1 ) 8t powers
of the roots u and : of the quadratic
is found to be E($). Expressed otherwise , if = u -- ? then
1) Serret, Cours d'Algebre Superieure, I, p. 449.
CLASSIFICATION AND DETERMINATION, etc. 27
Hence , if ^ = % r - is a root of JE(|) = 0, we have
M j'+ 1 = 0.
Then, since p n -f 1 = 2% odd, we have
Applying 24, we have modulo
so that every root 1 ) of E() = belongs to the GF[p n ~\. Hence
oi ~ 1 i oi 1
tr + u i
J
Substituting in this identity
pi ' u ~ g l x*ft '
and clearing the equation of fractions, we obtain formula 15).
38. As a simple example, let p n = 1 = 2 3 1, A = 4. The
binomials
/y.16 T^S ( Q 1 Q P^
tX/ ~~" *J I O '" J_ oJ t-J J
\ 7 / /
can be readily decomposed into irreducible quartics. The congruence
E($ HE | 4 + 4| 2 + 2 = (mod 7)
has the roots 1 and + 3. Further
1 ^o 1^1 r^a i Q -fCifoi / 10 ^^7 AS* r\\
/ylD rs /ylu I r\8 -f- o /y>lD_l_ F\4 1 /O_J >i - V / /I l-> W\
*/ ' v " ~ */ |~ c^ tv ' p cJ I o "y" d ~~ u{j - rt, v, O I.
Since 5 J>23 =5 2 ' (mod 7), equation 15) becomes
(mod 7),
holding for I = 4, 2, 3. Taking each in turn, we have modulo 7:
r 16 -J- 4 f/r 4 -r 2 _ /T\ fV 4 J- ^ 2 &\ (^ 9^2 A\ ^ 4 _L 9^ 2 4.^
A/ | a: ___ i ifj x/ ' TX i i iX/ ^^ */ T: I i <AJ o(/ rx i it*/ (^ <iJ i// ~ J ,
2_ //v,4 9^2 O\ //v.4 i 0^2 9\ (~A A~& y\ //y.4_l_ A/y 2 9^
i iX/ . t// ~~~ i i IAS ~l oju j } \J(j ~ ~ rtiA/ -! i i <JU ^^ Jtui/ I )
2 -l).
1) For another proof see Serret, Cours d'Algebre superieure, II, pp. 160 3.
Compare 82 below.
28 CHAPTER III.
Determination and classification 1 ) of the IQ\3?,p n ~\, 39 46.
39. Consider for positive integers fi the auxiliary quantics
16) X ft =x** l '-
where C^j denotes the number of combinations of p things k at a
time. Since C p r , * is a multiple of p, if < k < p r , we have
17) Xf= x*> npr - x (mod p).
Hence , by 26, the product of all the IQ^p^p^ is given by
18) V p -^ p n = XpS/XpS l.
We derive a simple expression for the quotient 18) as follows.
we deduce at once the congruence
19) X^ + 1 -Xf-X M (modjp).
Multiplying together the congruences (for i = 1, 2 ; . . ., v)
Xu + tEEXf+^t-Xi + i-i (modp) ;
and dividing the resulting formula by the product
X U _{_ 1 X u _^_ 2 . X u ^_ y _ 1 ?
we find
M "f" ^ ~" !
20) X B + u =Z B JJ(Zf- 1 -l) (modjp).
i = M
Taking M=jp s ~~ 1 , w-|-t=^ we find from 18) and 20) the result
21) Fp-,p
Further, if v t , i/ 2 , . . ., v^n_i denote the marks =(= of the GF[p n \
we have
i
Since X t - Vj is of degree p wz ' in x, it must decompose in the GrF[p n ]
into p ni ~ s factors each an IQ[p s , p n ].
1) For the case w = l, Serret, Journal de MatTiematiques , 1873, p. 301;
Algebre, II, ch. IV. For general w, Dickson, 5wW. J.mer. Jfa^. /Soc., 1897,
pp. 384 389.
CLASSIFICATION AND DETERMINATION, etc. 29
40. For s = 1, there are p 1 factors in the product 21), given
by i = l, 2, . . , p 1. The irreducible factors of Xf- 1 1 are
then said to form together the i ih class of IQ\_p, p n ]- Consider first
X 1 v = x pn x v,
which is the product of p n ~^ IQ[p,p n ] of the first class. To
decompose it, consider the equation
It follows at once that
c l} c = ri pn - - r] (mod p).
Hence every root t] of 22) belongs to the GF[p l<: \ if, and only if,
c be an integer. Setting in 22)
Ci - - L J / /y 1 ) i .. ._ I J /V \ // __ 2 /V>
~ /w f fi*~ I /V IA/ y /V tv j
where A belongs to the GF\j} n '\, we find
A (x p71 -- x v) = (mod p).
We have therefore in the GF[p n ~] the decomposition 1 )
OO\ 1 (^p n <TJ ^ 2/^ f f ( IP %P /T^J /3 -^
where the ft are the roots of
A or v being determined so that Kv is an integer. We have therefore
the theorem: The quantic l p x p kx ft is an IQ[p, p**] if, and
only tfj T> ap" 1 t ftv n 2 i _j_ np\ ft i A
Corollary. - If b is an integer not divisible by the prime p,
x p x b is irreducible in the GF[p n 1 if, and only if, n is not
divisible by p, in particular, it is always irreducible modulo p.
In fact, the condition becomes in this case
B~nb==Q (mod p).
41. The decomposition 23) may be given a more explicit form
useful below. If ft be one root of 24), then is also
25) Pj = a p -a + p,
for every mark in the GF[p*]. Indeed, we have
1) For the case v 0, this decomposition was given without proof by
Mathieu, Journal de Mathematigues , (2) vol.6, 1861, p. 280.
30 CHAPTER IE.
Further, the formula 25) furnishes all the roots of 24). For, if
a p-a + p = a{-ai+p,
(a aiy= ( ^i) [mod p\ t
so that K = KI + an integer. Hence there are p n /p = p n ~ l distinct
expressions a p a and hence as many roots /fy. Hence
the product extending over p* 1 marks a f of the GF[jp n ~\ no two
of which differ by an integer.
42. Consider an irreducible factor x p x ft of x pn --x-~ 1,
where therefore
PP- l +pp n ' H ----- h/fr-f/j 1.
Denote by I one root of the equation
XP x- p = Q.
Its remaining roots are J+ 1, /+ 2, . . ., 7 + 1? 1.
Then by 23 t ) every root of every IQ[p, p n ] of the first class
is a linear function of J, viz., kx ,-=/ + *, * = integer:
26) x = (1+ i + ;)/A,
the coefficients I/ A and (i 4- /)A being marks of the G-F[p n ~].
Inversely, every such linear function containing I is the root of
an IQ[p, p n ].
43. Consider an IQ[p, p n ] of class fi. Its roots belong to the
and are therefore functions of / of the form
where the u s belong to the QF\_p*\. By B9, f(I) will be a root of
27) X^ = a,
if <? be suitable chosen in the G-F[p*\. But, by 42,
Hence, by 24, we have for any integer m,
[/XI)F m = f(lP nm ) = f(I+ m).
Substituting f(T) in equation 27), X iU being given by 16), we find
The degree of this equation in I being less than j9, it must be an
identity. But its first member is the ^t th difference of the polynomial
f(I) with respect to the constant difference unity attributed to I.
CLASSIFICATION AND DETERMINATION, etc. 31
Since it reduces to the constant <T =4= 0, the degree of f(T) is
exactly ji 1 ). Hence
G
/t=-7> a /i+i= A+2 = " = cc p -i= 0.
ft
We have therefore proved that the roots of every IQ[p,p n ] of class p
are integral functions of I of degree ft.
44. We can readily obtain a formula including all IQ\]p,p n \
In the above expression f(I), let the a, be arbitrary such, however,
that f(I) does not reduce to . To set up the equation of which
f(P) is a root, consider the p equations
P[f(T)-Q-0 (A - 0, 1, . . .,i> - 1).
Reducing the exponents of I below p by using the identity
we obtain the series of equations
(a - g)4- i/4- */ 2 4- - + Op-il*- 1 - 0,
- g + p_i)J+ i/ 2 4 4- ffp-sl*- 1 - 0,
Eliminating 7, J 1 , . . ., J?"" 1 from these ^) equations, we reach the
required irreducible quantic .F(|;),
a
3
Setting a jU _|_ 1 = a^_j_ 2 = = o p _ 1 = and giving to ff , i, . - ., tfy i
all possible values in the GrF[p n ~\ and to a^ every value =j= 0, we
obtain p n f j (j) n 1) irreducible quantics of class ft. Since f(I-\-m)
leads to the same determinant as /"(I), if m be an integer, the number
of distinct IQ[p,p n ] of class ft is ^-'(j)* 1), a result also follow-
ing from 39.
For ft = 1, we find that
cc a
so that we may derive a new proof of formula 25^.
1) Boole, Calculus of Finite Differences, p. 5 and p. 19, formula 3).
32 CHAPTER HI.
An interesting type of IQljp^p^ of class p 1 is given by
setting every ;=0 except GO and p _i; viz.,
Multiplying this by |~ a p i and setting -F() = 0, we find
that p is a linear fractional function of jj. But, by 31, the roots
of _F(Q = may be expressed in the form
Hence its roots are all linear fractional functions of one of them.
This result also follows from the fact that
so that each root is a linear fractional function of I.
45. Formula 19) expresses the fact that X^ becomes X^+i when x
is changed into x pn x. Further, if we set Xo=# 7 19) holds true
for it = 0; viz.,
-
Hence in order to change x into # p "-- a; in any formula involving
the X^ ? we have merely to advance the subscripts of each X M by
unity. Applying this operation to formula 21), we have the theorem:
If F(x), an IQ[jp s ,p n ~\, divides Xf~ - 1 for i <p 1, thm
F(x pn x) decomposes into p n IQ[p\ p n ], each one being a factor of
X P iJ~i l - - 1; but if F(x) divides X^,~ - 1? then F(xv n x) decomposes
into p n ~ l factors each an IQ[p*~*~ l , p n ] which divides X%s~ -1.
46. As an example under the second part of the last theorem,
consider the IQ[p, p n ] of class p -- 1 given at the end of 44.
From it we obtain the IQ[p 2 , # n ],
where or , a^i, ft are arbitrary marks of the GrF[p >l ] such that
,_!+ 0, /^"" 1 + ^"~ 2 + - - 4- 0"+ H= 0.
For an IQ\jP,p\ see Serret, Cours d'Algebre superieure, II, p. 209.
Miscellaneous theorems on irreducible quantics, 47 49.
47. Theorem. - - An IQ\m,p*\ is irreducible in the GrF[p nd ] if n
be prime to m.
CLASSIFICATION AND DETERMINATION, etc. 33
The given quantic being F(x), the roots of F(x) are
x, x*, xP* d ,...,xP d(m ~ l)
all belonging to the GF[p dm ]. If F(x) be reducible in the
the root x will satisfy an IQ\t,p dn ], t<m, of the form
(X - x)(X - x? dn ) (X - x? 2dn ) ... (X- x*> dn(t - 1} ) = 0.
Its constant term must be a mark of the GF[p dn ~\, so that
in virtue of the single relation F(x) = 0. But this requires that tn
shall be a multiple of m, and therefore that t be a multiple of w,
in contradiction with t < w. In fact, by 23, -F(#) divides in the
GF[p d ~] the function x pkd - x if, and only if, k be a multiple of w.
48. Theorem. 1 ) - - An lQ[n,p"\ decomposes in the GF[p nv ] into
8 factors each an IQ N^> p nv , # fte^ngr $e greatest common divisor
of [i and v.
The given quantic being F(x),- the roots of F(x) in the
are
They may be separated into d sets each of p/d roots,
for i = 0, 1, . . , d 1. A symmetric function of the roots in one
set is unaltered upon being raised to the power p nd and therefore
belongs to the GF[p n d ]. The roots of the general set therefore
satisfy an equation
= (X- x* ni ) (X- x* n(S + i) }.- 0,
with coefficients belonging to the GF[p n6 ~\ and a fortiori to the
GF\j> nv ]. If
then
~ J A
We next prove that the JR(X) are irreducible in the 6r-F[jp re(J ]-
Suppose, on the contrary, that in the latter field,
Then
1) For the case % = 1, this theorem and the corollary of 49 were stated
without proof by Pellet, Comptes Rendus, vol. 70 (1870), pp. 328 330.
DlCKSON, Linear Groups. 3
34 CHAPTER IE.
the coefficients of f i + i (X} being the power p n of the corresponding
ones of /i(X), those of f Q being the power p n of those of /j_i. The
coefficients of the product fyfi. . .fsi are consequently unchanged
when we replace the coefficients of f by their (j? n ) th powers and are
therefore unaltered upon being raised to the power p n . Hence that
product belongs to the G~F[p n ~\, so that F(x) would be reducible in
that field, contrary to hypothesis.
Since the degree ft/d of F t (X), an 'iQfjt/d, jp**\ 9 is relatively
prime to v/8, F { (X) is irreducible in the G-F[p nv ~\ by 47.
49. Theorem. If F() be an IQ\m,p n ~\ in which the coefficient
a of I 01 - 1 is such that in the G-F[p n ]
a -f p + **H h a^" 1 ^ 0,
then F(tp |) is an IQ[mp,p n ].
If # be one root of F(g) = 0, its roots are
~2 M (m 1)
X, X p , X p ,. . ., ^ '.
By the hypothesis concerning the coefficient
n n ("* 1)
a = x x p x p j
we have
x + # p -f ic p2 -f + x pnr '4= 0.
Hence, by 40, p g -- x is irreducible in the GF[p nni ]. The
same holds for each of the quantics
~\T I'm 'f nftl f * r\ ^ ^ \
V . CJP__ C rflf I ^ ^^ II ^*9 I 1
Consider the function belonging to the GF[p*],
n 1
By 22, it has in the GF[p nm ] no irreducible factors other than
the X f . Hence if F(|P |) have a factor /"() belonging to and
irreducible in the GFfo*], f(%) must be in the GrF[$ nm ~\ a product
of the Xf,
f(S) = x r x s x t ...,
an identity in virtue of F(x) = 0. Replacing x by x p<n , another root
of F(x) = 0, and therefore X/ by X/ +1 (i < m) and X TO by X , we
obtain from the above identity,
Hence /"(|) contains every factor X,- and therefore coincides with
* |). The latter function is therefore irreducible in the G-F[p n ~].
CLASSIFICATION AND DETERMINATION, etc. 35
Corollary. If F(%) be an IQ\m,p\ in which tine coefficient
of % m ~ l is not zero, F( p ) is an IQ\mp, p].
Examples. The following congruences are irreducible:
(x*- #) 2 + (x 2 - x) + 1 = 0*+ x + 1 == (mod 2),
*-x-l = (mod 3).
Primitive roots and primitive irreducible quantics, 50 58.
50. Theorem. If E be a primitive root of the GrF[p nm ~\ and
m a divisor of m, any IQlwi^p**] belonging to an exponent e may be
exhibited as a product
m\ !
wtore t is a multiple of d = (p nm l)/g such that -T is prime to e.
Inversely, if e be a proper divisor of (#")"* 1 and t be a multiple of d
and -j be prime to e, tlw above product gives an IQ\m v p n ] belonging
to the exponent e.
Suppose first that <p(X) is an IQ[m v p n ~\ belonging to the
exponent e, where m l is a divisor of m. By 23, <p(X) divides
X* nm -X in the GF[p n ], so that any root X l of y(X) = belongs
to the GrF[p nm ], We may therefore set X l = R f . Then, by 31, we
have the decomposition 28). Since <p(~X.) belongs to the exponent e,
X^R* must belong to the exponent e ( 32). Hence t must be a
multiple of d = (p nm l)/c and -j be prime to e.
To establish the inverse, we first prove that R f belongs to the
exponent e. Since et is assumed to be a multiple of p nm 1 ? we
have R et =l. If J?^'= 1, tj is divisible by p nm l. Set t = dd', so
that d' is prime to e. Then must
jd'-(p nm \)/e = (mod p nm 1).
Hence must jd', and therefore, j be divisible by e. Hence
all belong to the exponent e. Upon raising these marks to the
power p n , they are merely permuted. Hence any symmetric function
of them, and consequently <p(X) defined by 28), belongs to the GrF[p n ].
Furthermore, <p(X) is irreducible in the GF[p n ~]; for, if ^(X) be
an irreducible factor of degree m x >l, it belongs to the exponent e.
Then by 29, e would be a proper divisor of (p n ) ml 1, so that
m l = Wj.
3*
36 CHAPTER III.
Corollary. Every PIQ[m, p n ] is given by the formula
F t (x) = (x- JBO (x - B<*> n } ...(x- B*f (m ~\
where t is an integer relatively prime to p nm l.
Evidently F t = F tp n = F tp * EE
51. The determination of a primitive root in the GF[p nn<r \ is
one of the most important as well as most difficult problems in the
theory. Special methods of procedure are illustrated in 54 57.
We may determine simultaneously all the PIQ[m, p n ~\ and therefore
all the primitive roots of the GF[p n \ by the following method of
undetermined coefficients.
The roots of F t (x) = are the th powers of the roots of F^x) = 0.
Hence the equations
are equivalent in the G-F[jp nm ]. Since t is prime to p nm 1 ; we may
determine t 1 by the congruence
tt' = l (modp nm 1).
Hence F^x '/= and F 1 (x t ') = are equivalent equations in virtue
of %p nm = x. By 30, the product of aU the PIQ\m, p n ~\ is given
thus: m
29)
29)
where q lf q 2 , . . . denote the distinct prime factors of p nm 1.
t'
are equivalent if t and t' each run through the integers less than and
relatively prime to p nm 1, which give distinct functions F(x).
Giving F^x) the undetermined form
and forming the product of the -- 0(^ nTO 1) distinct quantics
the result may be identified with the above fractional expression in x 9
giving a series of conditions for the coefficients a, 6, ... The
examples which follow will serve to make clear the method.
52. For p n = 3, m = 2, we have p nm 1 = 2 3 . The integers less
than and prime to 2 3 are 1, 3, 5 7 7. But
- *i(a), F,(x) - F s . ,(*) = F t (x).
CLASSIFICATION AND DETERMINATION, etc. 37
Hence
Since 5-5 = 1 (mod 2 3 ), F & (x) = and F (x b ) = are equivalent in
the GF\&\. Let _ , ,
= x 2 + ax + b.
If # be a primitive root in the 6r.F[3 2 ], # 8 = 1, # 4 = 1. Hence
& z 2 - a + b.
(* 2 + a# + ft) (> 2 - aa? + 6) = x*+ 1,
gmng 2 =2&, 6 2 -l (mod 3).
Hence &E -1, a = l (mod 3), so that the two PIQ[2, 3]
are x 2 + x 1.
53. For p n = 5, w = 2, we have
n
18 8 4 , . .
(a; 12 1) (a; 8 1) or-f 1
The eight integers r less than and prime to 24 are
1, 5; 7, 11-5-7; 13, 17=5-13; 19, 23^5-19 (mod 24).
Each pair of integers furnishes a single F t (x). For each of the
eight values of x, we have t; 2 ^ 1 (mod 24). Hence F t (x) = is
identical with F^x*) = in the 6rjF[5 2 ]. For a primitive root x, we
have x l2 = 1. We have therefore in the field,
F^x) = x*+ ax -f 6, F (x lB ) = x 2 - ax + 6,
^(z 11 ) EE &z 2 - aic + 1, ^ 2 JF;(^ 23 ) EE bx 2 + ax + 1.
The product of these four quadratics is therefore identical modulo 5
with
- x*+ 1).
It follows that & 2 = 1 (mod 5) and, hy subsequent expansion,
2a 2 EE& (mod 5).
Hence the four P/[2, &] are ^ 2 + aa; "t" ^^ 2 ? y i z v
30) rr 2 ic + 2, # 2 2a; - 2.
.
Another method of solving this example is to require that
x 2 + ax + b shall divide x 8 x*-\- 1 modulo 5. We reduce the latter
function by means of the relation
x*= - ax b [a 4 = 6 4 EE 1 (mod 5)],
and find, modulo 5, that
x 8 - x*+ 1 = (- a b b - al*)x - a*b 2 - a*b*+ 2.
Hence
&=_!, 2a^=b (mod 5).
38 CHAPTER III.
54. The eight P/$[4, 3] are the factors of
^-H (a; 80 - 1) (x* - 1) _ 32 _ 24 16 8 .
Or 40 - 1) (a? 16 - 1) ~
It suffices, however, in view of 50, to determine a primitive root p
of the GF[3*]. To get an I$[4,3], we employ the theorem of 37
for A = 4, i = 2, p = 2, Z = 1, giving the decomposition
x s + 1 = n (# 4 x*- 1) (mod 3).
Hence a root * of the irreducible congruence
x*-x 2 1 = (mod 3)
belongs to the exponent 16. If then we find a mark <? belonging to
the exponent 5, Q = ia will, by 14, be a primitive root of # 80 = 1.
We readily verify that the fifth power of fi+ i is congruent to unity
modulo 3. To find the irreducible congruence satisfied by the primi-
tive root Q = i (i?+ f), we form its powers,
^2^ j- ^-3- i __ ^ p3 = qp ^2_ ^ _j_ ^ ^4 = ^ ^_ ^^ j _^_
Eliminating the powers of i, we have
4 (> 3 + ^> 2 + Q -- 1 = (mod 3).
The product of the two P7[4, 3] thus reached is 8 -f- (> 6 4- $*+ 1.
Since the expression 31) contains only exponents which are multiples
of 4, we would expect the new factor Q S p 6 + (> 4 -f-l- In fact, the
product of these two quantics of degree 8 gives p 16 + (> 12 4 + 1?
which divides 31) giving the quotient
e 16 - P IS + e 4 + 1 = (e s + 9 4 + e 2 + 1) ( P 8 + e 4 - > 2 + 1).
We therefore have two new P/$[4 ; 3] given by the decomposition
Since (> 8 + 4 () 2 + 1 is derived from p 8 (> 6 + p 4 -f 1 upon replacing
Q by - - in the latter and multiplying by p 8 , we find
? 8 -h e 4 -h (> 2 + 1 = n (?* Q s - (> 2 + p - 1),
P 8 + ? 4 -(> 2 4-l = //(() 4 4 : ()-l).
Hence the eight PJ[4, 3] are
55. To obtain a primitive root p of the 6rF[5 4 ], we define the
latter by means of a root i of the irreducible congruence
x*=2 (mod 5).
Indeed, by 35, a? 4 3 3 is an J(J)[4,5] belonging to the exponent 16.
Since 5 4 1 =16 3 13, we seek marks belonging to the exponents 3
CLASSIFICATION AND DETERMINATION etc. 39
and 13. We verify at once that 2i 2 -\- 2 belongs to the exponent 3.
To find the most general mark 77 which belongs to the exponent 13,
we simplify the calculations by first determining the marks
??! = ai s -\- bi 2 + ci -f d
of the F[5 4 ] for which if = 1. Then either (+ ^ or (- i?^ 13
equals unity. Now
i 2 ci -f d.
The condition rft 6 1 thus gives
(bi*+ dtf- (ai'+eW=l.
Reducing by $ 4 = 2, we obtain the conditions, modulo 5,
_ 2a 2 - c 2 + 2bd = 0, - 4ac -f 26 2 + d 2 = 1.
For a EE 0, the only solutions are seen to be
6 2 EEl, ^/ 2 E -1, c 2 EEl; b = c = 0, d = L
Hence i 2 -\- ci 2 (c = 1, 2, 3 or 4) ; or else the negative of this
expression, belongs to the exponent 13. We may verify that* 2 -M-f3
belongs to the exponent 13. We may therefore take
p = ;(2^ 2 + 2) (*' 2 -f i + 3)'= 3i 3 + 2i*+ 4.
Then
P 2 E -f*-t-i 1- p s =* 8 -2* 8 +i + l, (> 4 =-* 3 +*-h2.
Hence we obtain the following P/[4, 5] satisfied by the primitive
root ^ ^ p 4 - Q*-Q-2 = (mod 5).
This quartic can be decomposed into the two PIQ[2, S 2 ],
( x
T> J.
^ 2 + 4, (> 25 EE 2^ 3 + 2^ 2 + 4, 9 125 = ^ 3 4- 3* 2 + 4.
Hence (x - ? ) (a; - > 25 ) = a; 2 - (- i 2 + 3) -f 3^ 2 -h 4,
(x - 5 ) (a; - > 125 ) EE aJ 2 - ^(* 2 + 3) - 3^ 2 + 4.
56. The determination of primitive roots in the 6r.F[5 6 ] and in
the 6rF[5 3 ] may be made to depend upon the congruence
32) # 6 4- # 5 -f # 4 + a; 3 + ^+ x 2 + 1 = (mod 5),
which, by 33, is irreducible. The root x belongs to the exponent 7.
The general mark of the (rJPfS 6 ] may be expressed in the form
5
ff E= ^^e t -# ?: (each c/ an integer).
40 CHAPTER HI.
It will belong to the included field GF[&~\ if, and only if, tf 125 = 0.
Applying x 1 " 1, we have (mod 5)
- o.
i = t = 1
Applying 32), this becomes
( C 0~ C l) C 1 X + ( C 5 C l)# 2 + ( C 4~ Cl)# 3 + fe Ci)# 4 -f fe C t )^ 5 .
The conditions that this shall be identical with 6 are
Ct = 0, c 2 = c 5 , c 3 = c 4 (mod 5).
Hence the 5 3 marks of the 6r.F[5 3 ] are given by
33) c, + c. 2 (x* + z 5 ) + c 3 (a; 3 + O [c 0? c 2 , c a = 0, 1, 2, 3, 4].
Since (# 2 + # 5 ) 5 = a; 3 -f # 4 ,
we infer that t = x 2 + x* defines the (rJ^[5 3 ]. In fact, we find
r 5 = x s 4- ic 4 , T 25 = a; -j- a; 6 , r 30 = a; 5 + # 4 -f ^ 3 -I- # 2 7
and finally that T 31 = 1. Hence A = 2(a; 2 4- a; 5 ) belongs to the exponent
4 31 and is therefore a primitive root in the 6r_F[5 3 ]. We derive
at once the P /$[3, 5] satisfied by A, viz.,
2A 3 = A 2 +^ + 1 (mod 5).
We next verify that x 2 belongs to the exponent 2 3 3 2 31,
so that Q = a? (a? 2) belongs to the exponent
5 6 - l = 2 3 -3 2 - 7-31,
so that Q is a primitive root in the 6r-F[5 6 ]. We have
--2)= - 2(x -f # 6 ) = - 2t 25 .
But t 25 belongs to the exponent 31. Hence the exponent of x 2
contains the factor 31 and, moreover, the factor 2 3 , since
(x - 2)^ (56 - 1) =(* - 2) 126 - 81 ' 2 -(- 2) 62 - - 1 (mod 5).
We next prove that the power 2 3 -3 2 -31 of x 2 gives unity. Indeed,
(x - 2) 15 EE (x* - 2) 3 = 2a; 5 - x* + x + 2 (mod 5),
and, by a slight calculation,
(x - 2) 18 = 2a; 5 + x* -f x* + 2# 2 + 4.
This being of the form 33), we have
(x - 2) 2V32 - 81 EE[(tf-2) 18 ] 124 EE 1 (mod 5).
For the same reason,
(x - 2) 2S - 3 - 3 iEE[(a;- 2) 6 ] 124 EE(2# 3 - x*- x* - x 2 + 2x+ 3) 124 =)= 1.
CLASSIFICATION AND DETERMINATION, etc. 41
To determine the PI$[6, 5] satisfied by the primitive root
p = x 2 2x, we form the powers,
We derive at once the required congruence
^6 _ p5 + p4 _ ^3 + 2p 4. 2 = (mod 5).
57. We can set up the PIQ[2, 2 3 ] and P/#[6, 2] by means of
the theorem:
34) A 2 # 2 + kx +
is a PIQ\2, 2 3 ] if, and only if, is a root of
35) J 3 EEJ 2 +1 (mod 2)
aw5 A is any mark except zero and 4 .
By 40, the quadratic 34) is an IQ[2, 2 3 ] for every mark
A 4= in the GF[2*] and for every root of the congruence
04+ 02+ ^ + j EE (0 + l)(/j+ 02+ l) = (mod 2).
Defining the (rJP[2 3 ] by means of the irreducible congruence 35),
we may take = 1, j, j z or j 4 . We first find the exponent ep to
which belongs a root of the congruence
2 EE + (mod 2).
Since g belongs to the GF[2 2 " 3 ], ^ is a divisor of 2 6 - 1 = 3 2 - 7. But
Hence for = 1, ep= 3; for a root of 35), we find
so that 6^ = 2 6 1. The theorem is therefore proven for the case A == 1.
Setting | = A#, it follows that, for =f= 1, a? belongs to the
exponent 2 6 1 unless x 2 = 1, which occurs only when A 2 =0,
i.e., >l = 4 . We therefore reach all |<t>(2 6 - 1) = 18 PIQ[2, 2 3 ].
Half of them are given in the left members of the identities below.
To pick out a set of three whose product gives a PIQ[Q, 2], we
select three which are like functions of respectively j, j 2 , j*, the
latter being the roots of 35). We thus find
(x 2 + x + j)(x 2 + x + j 2 )(# 2 -f x
s x + j0' 5 s*+ PX + J 2 )0' 3 ^ 2 + j*% + j 4 ) = x*+ x* + x* -f
42 CHAPTEE III.
Replacing x by -'- and multiplying by # 6 ? we find
sc
x* + # 4 -f # 3 -f x -f 1, # 6 -|- # 5 + # 2 + # + 1, a? 6 +#4-1,
which with the above three sextics give the six existing PIQ[6, 2].
58. Theorem. The necessary and sufficient conditions that x p X K
shall be a PIQ\jp, p\ are that a be a primitive root modulo p and
that a root of y p = y + 1 (modp) belong to the exponent (# p l)/(p 1).
If a be an integer not divisible by p, the congruence
x p = x + K (mod p)
is irreducible by 40. The product of its roots is
xx p x p2 . . . a?**- 1 ,- xP-
Setting x = ay, we find that
y p =y +1 (mod #).
Hence if a; belong to the exponent p p 1, then a is a primitive root
modulo ^> and y belongs to the exponent (p p l)/(j> 1). The in-
verse is true by 14, since p 1 and (p p l)/(jp 1) are relatively
prime.
59. EXERCISES ON CHAPTER III.
Ex. 1. If Q be a root of one of the P/#[2, 5] of 53, then x 3 $
is an /[3, 5 2 ]. Eliminate Q and derive the following /[6, 5]:
x 6 x+ 2, x* 2x* 2.
Ex. 2. (Moore). If x be a root of the irreducible congruence
x e_ 2a ,3_ 2^0 (mod 5),
a mark c -\- c x -f- c 2 aJ 2 + c 3 iC 3 + c 4 # 4 -j- c 5 ic 5 of the (rjP[5 6 J will belong
to the included field GF[5 S ] if and only if
C 3 = 0, c 4 = 3^-f- 4c 2 , c 5 = 2^-J- 3c 2 (mod 5).
Show that (p = a? -f- o; 2 + 2 a? 4 is a primitive root of the 6r.F[5 3 ] and
that it satisfies the congruence qo 3 = 2<p + 3 (mod 5).
Ex. 3. (Pellet). If y belong to the G-F[p n ] and m be the least
integer for which yP m = y, then x p x y is irreducible in the field
if neither n/m nor y + y p + y p2 H ----- hy* 9 be divisible by p; in the
contrary case it decomposes in the field into linear factors. Prove this
theorem equivalent to 'that of 40 for I = 1.
Ex. 4. (Pellet). If p be a prime number which is a primitive
(y}p _ 3C\n _ 1
root of the prime number n, - - - is irreducible modulo p.
XPX1
Ex. 5. Show that the theorems of 34 and 36 may be combined
into the theorem stated without proof by Pellet:
CLASSIFICATION AND DETERMINATION, etc. 43
If in an IQ[v v p v ] belonging to the exponent , we replace x
by x*; where A contains only the prime factors of w, the resulting quantic
1 Jinv
decomposes into D 2* ' l quantics IQ I ^ o*_ i * -P* belonging to the
exponent Aw, where D is the greatest common divisor of \n and p VVi 1
and where 2*"" 1 is the highest power of 2 dividing the numerators of
each of the fractions J - ^ and ^r when reduced to their simplest
form.
Ex. 6. (Schonemann). If F(x, a) be an IQ[m, p n ] in which the
coefficient of at least one power of x satisfies the equation c**"" 1 ! if,
and only if , v = n or a multiple of w, the product
gives an IQ[mn, p\.
Ex. 7. (Schonemann). Generalize the theorem of 33 as follows:
If p belong to the exponent t modulo e, e being prime, (x e !)/(# l)
decomposes modulo p into (e l)/ quantics irreducible modulo p.
Ex. 8. Prove that x 5 x + 1 is a PI#[5, 3].
Ex. 9. (Pellet). If e be the exponent to which belongs
the product of the roots of an irreducible congruence of degree v, F(cc) EE
(mod j?), and if A be a prime divisor of e, then
1) JP^) is irreducible modulo p if A does not divide (^> l)/c;
2) jF(#*) decomposes into A, irreducible factors of degree i/ if A
divides (p l)/ e - According as A divides or does not divide e,
all of these factors belong or do not belong to the same exponent.
Ex. 10. Using Jordan's irreducible congruence
x*=x + l (mod 2),
show that x belongs to the exponent 73 and x -\- x*-\- x 6 -}- x 1 -}- x s to
the exponent 7. The product y = x(x -\- # 4 -f- X 6 + # 7 -f x 8 ) belongs to the
exponent 2 9 --l and is therefore a primitive root of the G-F[2 9 ]. Verify-
that it satisfies the congruence
2/ 9 +2/ 8 +2/ 4 -h2/ 3 +2/ 2 -f y + l=0 (mod 2).
Ex. 11. If the GF[3 2 ] be defined by i*= i -f 1 (mod 3), the 16
PIQ[2, 3 2 ] are given by the decomposition of the P/^[4, 3] of 54;
for example,
x* x lEE {x 2 (i -f- 1) x i}{x*+ (i -f i)x + i 1},
l)x + i} {x 2 + ix i -f 1}.
44 CHAPTER IV.
Ex. 12. (Mathieu). If H belong to the G-H[p nm ], we have the
decomposition
+ (Hzy n + HZ + d,
where ^ runs through the series of marks of the G-F[jp n ~].
60. Table of primitive irreducible quantics 1 ). When more than
one PIQ[m, p\ is known, we choose that one x m ~ ax r + ^x r ~ l -{-
(mod p) in which the exponent r is as small as possible.
Modulo 2: afe
Modulo 3: x 2 ~2x+l, x*=x + 2, x*=2x*+2x*+ x+1,
Modulo 5: x 2 =2x + 2, x*~2x+3, x*=x 2 +x + 2, x*=
x*= # 5 x*+ x' d - 2x - 2.
/y.5 _ /y.4 _ /y.3 _ /y2 _ /y. _ Q /y> ' rm /y I Q
_ </ "~~^ JU - Jl.' JU JU tjy // - **/ (^ -.
CHAPTER IV.
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS.
Squares, not -squares, m th powers in a Galois Field, 61 63.
61. Every mark of the 6r.F[2 n ] satisfies the equation x* = x, so
that x is the square of the mark x* . Every mark has one and
only one square root, since 1 = -f 1 in the 6r_F[2 w ].
In the GrF[p n ~\j p > 2, a mark may or may not be the square
of a mark belonging to the field, and is called a square or a not-
square respectively. If p be a primitive root of the Q'F\f f \ 9 so that
36) p* n -i=i, 0(/>"-i)/2=--l,
the even powers of g are squares, Q* h = ( p 7 ') 2 ? while the odd powers
are not -squares. In fact, ^ 2/ *+ 1 = x 2 would require
Hence there are (p n l)/2 squares and as many not -squares in the
6r.F[p n ]. Furthermore, the product or quotient of two squares or of
two not- squares is again a square; but the product or quotient of a
square by a not -square, or vice versa, is a not -square.
1) A table of irreducible quantics (not all primitive) is given by Jordan,
Comptes Rendus, 72 (1871), pp. 283 290. His quantic x*-{- X s -{- x* -\-x-\- 1 is
divisible by #*-{- x*-\- 1 modulo 2, while x s -}-x-}-2 is divisible by x 5 mod 11.
2) Serret, Cows d'Algebre superieure, II, pp.181 189.
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 45
62. Theorem. The not-squares of any GF\j)"\j p > 2, are
not -squares or squares in the GF[p nni ] according as m is odd or even.
If 6 be a primitive root of the GF[p nnf \, then Q :-EE <7 U , where
u = (p nm \)l(p n 1), is a primitive root of the GF[p n ~\. Hence
the marks =(= of the GF[p"] are given by the formula
Q V =6 UV (v 1, 2, . . ., jt>* 1).
Let p" be a not -square in the 6r.F[j0 n ], so that v is odd. It will be
a not -square or a square in the GF\j) nnt '] according as uv is odd or
even, i. e., according as u is odd or even. But
m I
u = (jp* m l)/(p n 1) = V**'= sum of m odd terms.
Hence u is odd or even according as m is odd or even.
63. Theorem. - - If d be the greatest common divisor of m andp n \,
there exist exactly (p n \)/d marks =j= in the GF[p n ~\ which are
m^ powers in the field.
If |Lt =(= be the m th power of some mark v of the field, we find,
upon raising ^ = v m to the power (p n i)/d and noting that the
power p n 1 of the mark v m / d =^= is 1, the equation
37) ^ n
Inversely, there are (jp n i)/d roots of 37) in the GF[p n ~\ by
16 and each root is an m th power in the GF[p n ]. To prove the
last statement, we note first that such a root p is a d ih power. In
fact, the roots of 37) may be exhibited as follows:
(
i-0 i
* u, i, . . .,
where Q is a primitive root of the GF\j> n ]. That these roots are
distinct is shown by supposing
- - 1].
Hence i j = 0. We next prove that ^ = Q di is an m th power.
Since m/d is relatively prime to p n 1, we can determine integers I
and t satisfying the equation
Hence
Therefore
Corollary. - Every mark of the GF\p n ~\ will be an m ih power
in the field if, and only if, d = 1. Extraction of the m ih root of an
46 CHAPTER IV.
arbitrary mark of the GF\j) n ~\ is possible if, and only if, m be
relatively prime to p n 1. With this condition satisfied, there exists
but one m th root of each mark.
Number of solutions of certain quadratic equations in a Galois
Field, 6467.
64. Theorem. 1 ) -. If v = + 1 or 1 according as - - a^ is a
square or a not- square in the GF[p*], p > 2, the equation belonging
to the field, _
has p n v or p n + (p n 1) v sets of solutions according as K =4= or K = 0.
Setting K l % 1 =y, the equation becomes
V+ flfjagll = !*
1. If !<> = A 2 , a square =j= in the 6rJP|j) n ], we set
^ + ^I 2 =(>, r} l% 2 =6,
whence
The equation becomes
_
*< 3v.
If % =f= 0, we can give to 6 any one of the p n \ marks == in
the 6rF|j) n ], when the corresponding value of Q is determined by
the equation. There are in this case p n 1 sets of solutions | 1; | 2
in the field of the given equation.
If s = 0, there are evidently \-\-2(p n 1) sets of solutions.
2. If - ff^ be a not -square in the G-F[p n ~\, the equation
<)P 2 = Ofjffg
is irreducible in the field. If one root be i, the other is i pH ~- - i
by the corollary of 31. We therefore have the identity
We are thus led to determine the number of roots in the GrF[p* n ~\
of the equation in the unknown Z = ?] -f- i| 2 ,
38) Z^+^ajX.
If ^ = 0, we have Z = and hence a single set of solutions
& = 0, fe = 0.
If 3c=j=0, let JR be a primitive root of the jF[> 2 *]. We ma J
set (*iK = R k * whence
so that k(p n 1) is divisible by p 2w 1, the exponent to which E
belongs. We may therefore set k = l(p n +\)j I being an integer.
1) The theorems of 64 67 are immediate generalizations ofNos. 197 200
of Jordan's Traite des substitutions.
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 47
Since ^belongs to the G-F[p 2n ], we may set Z=R ( . The equation 38)
beCOmeS
rT PTlPP
t(p*+ 1) = l(p n + 1) [mod jp 2 *- 1].
This congruence has p n + I distinct solutions for t y viz.,
The corresponding values of R'^Z^rj + i^ give^ M -fl distinct
sets of solutions | 1? | g of the given equation.
65. Theorem. The number of sets of solutions (fi> ^j>v >$)
6rJP[p B ], jp > 2, of the eqtiation
wlwre every KJ is a mark =J= in the field, is
_p(8m-l)_ V pn(w-l) ^ x ^ Qj
j,(2m-l)_|_ ^^^^(in-l)) (if K = Q),
%7^re v & -f 1 0r - - 1 according as ( I) m cr 1 2 . 2 t *s a square
or a not -square in the field.
By 64, the theorem is true if m = 1. To prove the theorem
by induction, we suppose it true for equations in 2(m -- 1) variables.
The proposed equation is equivalent to the system of two equations
ii -f aaz = y> 3s H ----- h
1. Let K =f= 0. For each of the ^) n 2 values of ^ different from x
and 0, the first equation has p n I sets of solutions, while by hypo-
thesis the second has p*(* m 3 ) ^p n ( n '^\ where 'k = + 1 according
as - K-^K^ is a square or a not -square, and p = + 1 according as
( l) m ~ 1 a 3 or 4 . . . 3/w is a square or a not -square. For the value y 0,
they have respectively j? K 4-(j? ?l -- 1)A and ^ ra ( 2 s)_ p,pn( 2 ) sets of
solutions. Finally, for >/ = ^, they have respectively j}* A and
^(2m-3)_ h ^^(m-i)_ i) n( w -2)) sets o fsolutions. The total number
of sets of solutions is therefore
(p n 2)(p n
m 3) _|_ ^ Qpn(m 1) _ ^w(m 2)1
By 61, A^Lt = v. Hence the induction is complete.
2. Let % = 0. Separating the two cases ^ =(= and ^ = 0, we
find the total number of solutions to be
-f |> w
48 CHAPTER IV.
66. Theorem. The number of sets of solutions in the G-F[p n ],p>2,
of the equation ,.2 5.2 5.2
where each cij is a mark =|= in the field and K belongs to the field,
isp 2nm -\- wp nm , where o = + 1, 1 or according as ( I) m xia 2 . . .
is a square, a not -square or zero in the field.
Consider the equivalent system of equations
The first equation has one solution if y = 0. If ?/ =|= 0, it has two
or no solutions according as a^ is a square or a not -square. Let
jt = if 3t = 0, and ft = db 1 according as c^ac is a square or a not-
square. We may express the number of solutions of the second
equation by 65, if we set v = + 1 according as ( l) w 2 . . . 03+ 1
is a square or a not -square. Evidently we have \LV = o.
According as ^ = 0, + 1, or - 1, the total number of sets of
solutions of the pair of equations is respectively
i _ -i \
m pn(m 1)1 _j_ 2( - ) [jp(2m 1) v pn(m 1)1 =^2 n m
ml)~\ _j_ 9[j ? n(2m 1)_|_ 1/ ^ ? nm_ pn(m 1)1
(* 8\
K - Jjj ) w(2m 1)_ V pn(m 1)J jj2w_|_ ^nm^
/ w 1\
_ V jpn(?n l)~j _^_ 2! - J|j) n (2 !) v p n (m 1)1 ^^)2w __ ^jpnm,
In each of the three cases, we have enumerated separately the number
of solutions arising when ^ = 0, when 77 = x and when 77 is one of
the values =}= for which the first equation has solutions (viz., two).
67. Theorem. - If 8 denote the number of squares 1 ) G 2 in the
GF\jp n ~\ for which o 2 -f 1 is a square and N the number of square T*
for which T S + 1 is a not- square, we have
S=jO w -5), JV-i(p-l), if-l = square;
S = i(p - 3), J^ = j(^ ra +1), if - - 1 = not-square.
Indeed, the number of sets of solutions |, y in the ff^jp*] of the
equation r 2 = sz i ^
is always p w 1 (by 64). These solutions are of three kinds:
1. =0, ^ = 1;
2. | 2 = - 1, 77 = 0,
occurring when - - 1 is a square;
3. 2 =4=o, 7? 2 = + 14=0,
giving 4^ sets of solutions |, 7^.
1) The mark zero is not reckoned as a square.
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 49
Hence, if - 1 be a square, we have
If - 1 be a not -square, we have
Additive- groups in the GF[p n ~\ and their multiplier Galois Fields 1 ),
68-71.
68. A set of m marks A 1? A 2 , . . ., A w belonging to the GF[p n ]
and linearly independent with respect to the GF[p] give rise to
p m distinct marks of the larger field,
39) c^ + c 2 A 2 H ----- h c m Am (every c,= 0, 1, . . ., p 1).
Indeed, an identity between two of the marks 39) would contradict
the linear independence of A 17 A 2 , . . ., K m . Since the sum of any
two of these p m marks 39) may be expressed as one of the set, they
are said to form an additive -group [A 1? A 2 , . . ., A m ] of rank m with
respect to the GF[p] and the marks A 1? A 2 , . . ., A m are said to form
its basis -system. In particular, the GF[p n ~\ may be exhibited as an
additive -group of rank n ( 10).
These conceptions are capable of the following direct generali-
zation. Any m marks A t , A 2 , . . ., A OT of the GF[p nr ] are called
linearly independent with respect to the GF[p r ] if the equation
in which the y L are marks of the GF |j> r ], can be satisfied only in
case every ^= 0. [See 72]. A system of m linearly independent
marks gives rise to p rm distinct marks of the GF[p nr ]
by letting th y/s run independently through the series of the marks
of the GF[p r ]. These p rm marks are said to form an additive-group
[A 1? A 2 , . . ., Km\ of rank m with respect to the GF[p r ~\, the marks
A 1? . . ., l m forming its basis -system.
If A m _|_i be any mark of GF[p nr ] not in the additive -group
[y^, . . ., AJ of rank m with respect to the GF[p r ], then the m -f- 1
marks A 1? . . ., A m , A m -|-i are linearly independent with respect to
the GF[p r ] and therefore define an additive -group [A 17 . . ., A w , A m _j_i]
of rank m + \ with respect to the GF[p r ].
69. Theorem. Within the GF[p nr ] the number of additive-
groups [A 1? . . ., A m ] of rank m with respect to the GF[p r ~] is
pr} . . . Q> r _ p(m 1) r)
(pmr 1) (pmr pr^ . . . (pmr p(m l)rj
1) Moore, Mathematical Papers, Congress of 1893, p. 214, p. 216; Math.
Ann. vol. 55, 12.
DlCKSON, Linear Groups. ^
50 CHAPTER IV.
We first prove that the numerator expresses the number of sets
of m marks A 17 A 2 , . . ., A m of the GF[p nr ] linearly independent with
respect to the GF[p r ~\. For A t we may take any one of the p nr 1
marks =j= of the GrF[p nr ]; for A 2 any one of the p nr p r marks not
of the form p^, where Q belongs to the GF[p r ]; for A 3 any one
of the p nr p 2r marks not of the form p^-f p 2 A 2 , where Q and p 2
belong to the GrF[p r ]-, etc.
We next show that the denominator expresses the number of
these sets of m independent marks which generate the same additive-
group [Aj, A 2 , . . ., A m ]. In fact, we may use as a basis -system for
the latter any set of m marks A^, Ag, . . ., A' m chosen as follows.
Aj may be chosen in p mr 1 ways:
7/t
i = /,
each yu being arbitrary in the GF[p r ] provided not all are simultan-
eously zero. A 2 may be chosen in p mr p r ways, viz.,
M
^2 = /, fli^i)
t = l
the y 2 i being taken arbitrarily in the GF[p r ] but so as to exclude
the p r sets of values which make A 2 = g^, viz.,
where (> runs through the series of marks of the GF[p r ]] etc.
70. If the p m marks c^-\ ----- h c m h m of the additive - group
[Ai ; .";., A OT ] of rank m with respect to the GF[p] are multiplied by
any particular mark ^ =4= of the G-F[p n ], the resulting p m marks
constitute the additive - group
likewise of rank m with respect to the GF[p]. We will say that
[ftAi, . . ., fjLl m ] is derived from [At, . . ., A m ] by multiplication by ft.
In particular, we seek those multipliers ^ = K which do not alter
[Ai, . . ., A m ], such a mark being called a multiplier of the additive-
group [Ai, . . ., A m ]. If jfj and x 2 be multipliers, then will evidently the
product Xj 3 2 be a multiplier. To prove that ji = x x + K% will also be
a multiplier, we observe first that [fiAi, . . ., ftA^J is an additive-
group included within [Ai, . . ., A TO ], since ^ and x 2 are multipliers of the
latter, and further that it is of rank m if ^ =]= 0. Hence ^ + ^ 2 is
a multiplier unless it be zero. Hence the multipliers K together with
the mark zero constitute an additive, as well as a multiplicative,
group and therefore constitute a Galois Field G-F[p k ] included within
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 51
the fundamental GF[p n ]. It is called the multiplier Galois Field of
the additive -group [A i; . . ., A m ]. Every GF[p k '] included within the
G-F\jF\ is called a multiplier Galois Field of the additive -group.
By 23, k' is a divisor of k and k a divisor of n.
The additive- group [A 1? . . ., A m ] of rank m with respect to the GF\jp\
may be exhibited as an additive- group [Aj, . . ., A'] of rank m' = m/k'
with respect to any multiplier GF [/>*'].
In proof, let yi, yi, . . ., jv run independently through the series
of marks of the GF[p k '^. Taking Aj to he any particular mark A =(=
in [A 1? . . ., A m ], the p k ' marks y^ are all distinct and all belong
to [Ai, . . ., A m ]. Taking A^ any mark in | AI, . . ., A OT ] different from
the ftAj, the p 2k ' marks ftA^-h^^ are ^ distinct and all helong
to [Ai, . . ., A m ]. Proceeding similarly, we obtain ultimately a set
. o f pmk' distinct marks y x A^ + y 2 A' 2 -\ ----- \- y m 'A! m r giving all the marks
of [Ai, . . ., A OT ]. In particular, : jj*"*_p*' m ', so that k f divides m.
Corollary I. Since k is a particular k r , k divides m.
Corollary II. Within the GF^p^ the number A(p f n f m,ti) of
additive -groups of rank m with respect to the GF\ji] which have
the GF[p k ] as a multiplier Galois Field equals the total number of
additive -groups of rank m/k with respect to the GF[p k ~\:
1) (put pk} (pm p2k) . . . (pm pm kj
71. If k be a divisor of m and n and if hi, h%, , . ., h t are the
prime factors occurring in both m and n to a higher power than
in k, there are in the GF[f f \ exactly
* i,iiv
, w, m, ^ fe/) 4-
-f ( iyA(p, n, m } khi . . . hi)
additive -groups of rank m with respect to the GF[p\ which have
the 6rF[#*] as the multiplier Galois Field.
Indeed, from the A(p f n,m f lK) additive -groups having the GF[p k ~\
as a multiplier Galois Field, we must eliminate those having a larger
multiplier Galois Field. It suffices to eliminate those having the
6rF [#**'], for * = 1, 2, . . ., or t, as a multiplier Galois Field. But
the A(p,n,m,khi) additive -groups with the GF[p kh *~\ and the
A(p,n,m,kh2) additive -groups with the GF[p kh *~\ are not distinct but
have in common A(p^n } m } kh l h 2 ) additive - groups each with the
&F[jfi****] as a multiplier Galois Field. By this principle, we readily
determine the number of distinct additive -groups among the sets of
-4.(jp,*e,*,M/) with the GF[p kh i] as multiplier Galois Fields. Sub-
tracting this number from A(p,n,m,k), we obtain the required number.
4*
52
CHAPTER IV.
72. Theorem. The marks AI, A 2 , . . ., A m of the GF[p nm ~\ are
linearly independent with respect to ilie GF[p n ] if and only if the following
determinant 1 ) is not sero in the GF\jp n \:
AI A2 A m
2 P '
AI
"
As
p n(m 1) p n(m 1) p n(m :
First, if AI, A 2 , . . ., A m be linearly dependent, i. e., if a relation
holds, the coefficients % being marks of the GF\p n ] not all zero,
then will the determinant j A vanish.
Secondly, if the determinant vanish, set
m I
i = 1, 2, . . .,
where It is a primitive root of the GF[p nm ] and therefore satisfies
an equation of degree m belonging to and irreducible in the
and where the ft,, belong to the latter field. Then
-p ni T)jp ni ( ' 1 '
where the determinant in E, when written in full, is 2 )
1 1 ...1
E EP H .Rp" v '~ '' o,.. ,m i
n
><
and therefore is not zero in the GF[p nm ]. Hence, if A j = 0, then
must ft =0, so that between the A/ exists a linear relation with
coefficients belonging to the GF[p n ] and not all zero.
73. If A be a mark of the GF[p nm ], the marks
are said to be conjugate with respect to the GF[p n ~\. Any symmetric
function of them is unaltered upon being raised to the power p n and
1) Its decomposition into linear factors is given by Moore, U A two -fold,
generalization of Fermat's theorem", Bull. Amer. Math. Soc., vol. 2 (1896).
2) Baltzer, Determinanten , p. 85.
MISCELLANEOUS PROPERTIES OF GALOIS FIELDS. 53
hence belongs to the GF[p H ]. Hence the m conjugate marks are the
roots of an equation of degree m with coefficients in the GF[p n ].
By 31, the roots of an equation of degree m belonging to and
irreducible in the GF[p n ] are conjugate with respect to the GF[p n ].
In particular, the marks A, A pn of the GF[p 2n ~] are conjugate
with respect to the GF[p n ]._ The conjugate A**" of A will be
designated by A. Evidently A>=A if, and only if, A belongs to
the GF[p n ]. The following relations are proven at once:
74. Newton's identities. If S t denote the sum of the t ni powers
of the roots of the equation
f(x) ~ x>" + ^0?*-*+ a 2 tf'"- 2 -f Y-4 a m _ l x + Ote- 0,
in which the coefficients a,; 'belong to the GF[p n ], tlien
40) =
These identities follow as in algebra upon equating the coefficients
of like powers of x in the following identity, in which tx v . . ., a m
are the roots of f(x) = 0:
41) f(x)~
This identity, evidently true for w = l, may be proven by
simple induction 1 ). Supposing it true for a particular w, we have
proven it true for the value m -f 1. Let
^
Multiplying 41) by x a m +i and adding f(x) to the left member
and x m -\ ----- h a m to te right member, we find
+ ^S^ - (+ i)-+*;(o,-. +1 )*-v
-f (m 1) ( 2 aiflfm+i) % m ~ 2 H ----- h 2 ( w _! a m -. 2 , + i)a;
1) Since equations in the 6r.F[jp] are not algebraic identities, we avoid
the consideration of derivatives. We might, however, employ Weber's definition
(Algebra, I, 13) of the derivatives of a polynomial in x for the derivatives up
to the pth, but not for the higher derivatives on account of the denominators 7T(w).
54 CHAPTER V.
Hence if 41) be true for f(x) with the roots crj, . . ., cc m , a like
formula is true for the equation F(x) = with the roots K{, . . ., W( ,
or m -fi.
Forming the sums
/=! i = l
we derive the new identities,
= S m + (hS m
, = S m + i -
= 5
Corollary. - Iff(x) = have a double root 7 the right member
of 41) must vanish for x = K.
75. Theorem. If t be a positive integer and U Q , u tj . . ., n p i
denote the mwrks of the GF[p n ], then
In fact, the marks u { are the roots in the GF[p H ^ of the
equation n
Applying to the latter the identities 40), we find
CHAPTEK V.
ANALYTIC REPRESENTATION OF SUBSTITUTIONS
ON THE MARKS OF A GALOIS FIELD.
76. Consider the problem to find every quantic <() belonging
to the GF [j> n ] such that the equation O () - ft has a root in the
field whatever mark of the field /3 may be. For example,
3 ^ /3 (mod 5)
is solvable for every integer p, since we have
3 = 0, 1 3 EE 1, 3 3 EE 2, 2 3 EE 3, 4 3 EE 4 (mod 5).
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 55
If we denote the marks of the GF\j?*\ as follows,
44) po, pi, p 2 , ..-, JV-i,
the necessary and sufficient conditions that () = /3 be solvable iu
the field for arbitrary ft are that the marks
45) O (po), < (pi), O (pi), . . . , (py- i)
be identical with the series 44) apart from their order. In fact, the
p n values which 0() takes must all be distinct, since /3 is to have
fp distinct values. When the conditions named are satisfied, the
series 45) forms a permutation of the series 44), and the quantic 0(1)
is said to represent the substitution
(S) > (PO), <t> (PI), . . , d> Ov-i)
0w fce marks of the Qf t [jfy For example, 8 represents the sub-
stitution m _r> i> 2 > 3 > 4
U 3 J"LO, 1, 3, 2, 4
on the marks of the 6rF[5], i. e., the field of integers taken
modulo 5. A quantic of degree k with coefficients belonging to the
GF[p n ~] will be called a substitution quantic SQ[k, p"\ if it satisfy
the above conditions. Its degree k will be supposed <j) n in view
of the equation |j pW = satisfied by every mark of the field.
77. An arbitrary substitution on the marks of the
'?
can be represented by the quantic (jj) given by Lagrange's inter-
polation formula.
t =0
where
and J^'() denotes the function derived from F(g) by formula 41).
Evidently 0() is an integral function of ( of degree
78. Theorem. Two distinct quantics 0(5) and Y(j;) belonging to
the GF[p n ~\ can not represent the same substitution on its marks.
For > if 4>G*) - YG*,) (i- 0,1,.:. ,>-!),
the equation 0(Q Y(Q = would have in the field p" distinct
roots p,- ? whereas its degree is less than p n . By 15, it must be
an identity in jj.
56 CHAPTER V.
79. Theorem. | m is a SQ[m, p"] if, and only if, m be prime
to p n 1.
The theorem follows immediately from the corollary of 63.
However, to illustrate a method of proof used below, we will verify
that, if m be relatively prime to p n 1, | TO takes p n distinct values
in the G-Flp^ when | does. It is sufficient to prove that from
Af\ t m t m
46) 1 = 2
follows | x = $2, provided x and | 2 are marks of the field. This being
evident if either be the mark zero, we suppose j =j= 0, 2 =^ O? so that
47) |f -'_ g"-'_ 1.
Raising the members of equations 46) and 47) to the respective
powers t and x, chosen ( 7, note) so that m + f(.p w 1) = 1, and
forming the product of the resulting equations , we find that ^=^3.
80. Theorem. - - For an arbitrary mark a of the GF[p n ],
<D(g) = 5i 5 +5a| 3 -f 2 |
is a SQ[5, p n ], if p is a prime of the form bm + 2 and n is odd.
To prove that, in the 6r.F[j) n ] ? ^ = 2 is the only solution of
(?i-i 2 ){5fe 4 -f ii| 2 + gg
we set fc 5.
?1 ^ ~T #*> b2 ^ ^^
here 1 ) limiting our proof to the case p > 2. Then 16 times the
quantity within the braces becomes
16{5(5A 4 -f 3A 2 4- KUV-f fi 2 -f ji, 4 ) -f a 2 }
a)
But 2 ) -f 5 is a quadratic residue of no odd number of the form
bm + 2 or 5m 2. Hence ( 62) 5 is a not -square in the 6r.F[5 n ],
w being odd and p = 5w 4; 2. Hence, if the above expression
vanishes, we must have
whence, for p > 2, /Li 2 = 5A 2 , so that I = ^ = 0, ^ = J 2 .
1) An analogous proof for p = 2 is given in Annals of Math., 1897, pp. 84 85.
For an arbitrary prime jo, the theorem is a special case of that of 82.
2) Gauss, Disquisitiones Arithmeticae , Art. 121.
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 57
81. Theorem. The quantic belonging to the
tvill represent a substitution on its marks if, and only if, X = is the
only solution in the field of M'(X) = 0.
Indeed,, the necessary and sufficient condition is that
shall require X l = X 2 , or X 1 X 2 == 0.
Corollary. X pnr - AX pns represents a substitution on the marks
of the GF[p" m ~] if, and only if, either A = or else A is not the
power p nr p ns of a mark of the field.
82. Theorem. - - If k be an odd integer relatively prime to p 2n 1,
and if a be an arbitrary mark of the GrF\j> n ], the quantic
represents a substitution on the marks of the G-F[jp n ~\.
We are to prove that the equation
48) 0,(| 7 ) = /3
has a solution in the 6r_F[j) n ], /J being an arbitrary mark of the
field. By Waring's formula, 3*^, a) is the sum of the k ih powers
of the roots of the quadratic
rf - - %y a = 0.
Hence, in virtue of the equation
g = ,? - a/^,
we have the identity /.. \ / / vi
<t>jfc(i, a) EE ^'- (a/vtf.
The equation 48) thus becomes
rj *k_ pyk_ u*^ 0>
Setting Y 7^*, this becomes
49) Y*-pY-a*=0.
According as 49) is reducible or irreducible in the 6rJF[j) n ], it is
solvable in the 6r.F|j) ra ] or in the GF[p 2n ~\, and therefore always
solvable in the larger field. Call its roots Y i and Y 2 . Since k is
prime to p* n 1, we can determine uniquely ( 79 or 63) two
marks ^ and % belonging to the GF[p* n ~\ such that
k ~\T it -\r
tl = -*! ^2 =12.
58 CHAPTER V.
Likewise, it follows from Y 1 Y 2 = a k that
If 49) be irreducible in the G-F[p H ], we have ( 31, corollary)
and therefore
ft nu nl = %
Hence , , n
(% + ft)* - % + 'fl-
it follows that 48) has the solution in the G-F[p n ]
If 49) be reducible, r t and Y 2 belong to the GrF[p n ], Since &
is prime to p n 1, it follows that ^ and i^ 2 belong to the GF[p n ].
Remark 1. We have shown in 37 that ( ty~ 1 (? 1) com-
pletely decomposes in the CrF\p n ] into linear factors, if p n =2't 1,
odd and > 1.
Remark 2. If ft be a prime number, ^-(J, a) is the only
quantic of degree k suitable to represent a substitution on the marks
of every GrF[p n ] for which p* n 1 is not divisible by Jc (Annals of
Mathematics, 1897, pp. 89 91).
Remark 3. - - The equation 48) is algebraically solvable, having
as r ts Q6 m + 0s k - m (m = 0, 1, . . ., Jc - 1)
where
and s denotes a primitive & th root of unity. This result is a direct
generalization of Cardan's formula for the roots of the reduced cubic
and of Valles' solution of the quintic 1 )
83. Theorem. 2 ) If d be a divisor of p r 1 and v be not a
in the G-F[p n ], the quantic
is a SQ[p r ,p n ].
We are to prove that <t>(|) =-- /3 has a solution in the GrF{jp*\
for /3 chosen arbitrarily in the field. This being evident if |8 = 0,
we will suppose that =|= 0. Writing
1) Formes imaginaires en Algebre, 1869, vol.1, pp. 90 92.
2) For the case r = n, this theorem is included in the theorem of 85.
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 59
we are to prove that
has a solution y in the GF[p n ~\] for, if r} be such a solution
(necessarily =0), then
wiU belong to the GF[p n ] and wiU satisfy <&() = 0.
Setting ^ = 1/w in 50) and multiplying by a pr , we find
This has a solution w in the GrF[p n ] for /3 arbitrary in the field.
Indeed, by 81, corollary,
represents a substitution on the marks of the G-F[p*\, since v/^ d is
not a d th power and hence not a (p r V) 3t power in the field.
Note. - - For p = 3, 5, 7 and partially for p = 11, the author
has shown 1 ) that the only SQ[p,p n '] which exist are reducible to
the form d
where d is a divisor of p 1 and v is not a d ih power in the GrF[p n \.
84. Theorem. 2 ) The necessary and sufficient conditions that 0(5)
shall represent a substitution on the marks of the G-F[p n ] are:
1. Every t ih power of ^(S), for t<p n 2 and prime to p, shall
reduce to a degree <p n 2 on applying the equation % pn *=* jj;
2. There shall be one and only one root in the GT [p n ]
of 0(6) - 0.
After the exponents of are reduced below p n , let
Put for % the p n marks ^ of the 6rjP[p ?i ] and add the resulting
indentities. We find, on applying 75,
If 0(5) represent a substitution, we must have
1) Dissertation, Annals of Mathematics, 1897, pp. 101 108.
2) For the case n==l, this theorem is due to Hermite, Comptes Rendus,
vol. 57 (1863), pp. 750757.
60 CHAPTER V.
Hence a necessary condition is that
_i - (' = l > 2 > ' p a - 2 >
The condition 2 is evidently a necessary condition.
Suppose, inversely, that 1 and 2 are satisfied. Consider the
equation satisfied hy the marks
J7 & - *fo
j = / =
the sum of the m th powers of whose roots is denoted by a m . Then
-erVWuYK- (0 -0 P^ 1 ' 2 '" -i
^- >, L PUWJ - - V-i- U=t= (mod
.7=0
since all but one of the 4>(p/) are = = by 2 and hence have unity
for their (p n 1)*' powers. Applying Newton's identities 40), we
readily find
Ti = \i = 1, 2, . . .,^- 2; =|= (mod #)]
<y p = <?2jo= ' ' ' = Gp n p= 6p n = 0, y p n_ 1 = -
To determine y p , v% p , . . ., we apply the identities 42), viz.,
<?*+ yi^*-i+ ys^-sH ----- h r P n<j k - P =
which here reduce to the form
51) tfjt-hyp^
since by 2,
Furthermore, since any mark equals its (j) n ) th power, we have
<y,4.j,_i= (> s (5 = 1,2,...)
Applying 51) for k =p n +p 1, we find
j^ffp_i= 0.
More generally, for k = p n + Ip 1, Z ^^? w ~ 1 1, we get
yi p <5jP_i= 0.
Hence y^= 0. We have therefore the result
~7l _ j^
so that the marks O(^) form a permutation of the marks \LJ of the
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 61
85. Theorem. 1 ) - - If r be less than and prime to p n 1 and s
be a divisor of p n 1, and if /"(*) be a rational integral function
of % s with coefficients belonging to the GF[p n ~\ such that /"((*) = has
no root in the field, then
represents a substitution on the p n marks of the field.
The conditions of the theorem of 84 for a substitution quantic
are all satisfied by the given quantic. In fact, upon raising it to
any power I, not divisible by s, we obtain a set of terms whose
exponents are of the form ms -\- Ir and therefore not divisible by s
and consequently not by p n 1. If, however, we take the power
I = ts <p n l,
we get the result ' r , since the power p n 1 of f(% s )=^0 is unity in
the field. But Ir is not divisible by p n 1.
Condition 2 is satisfied by our quantic, since it vanishes in the
field only when = 0.
86. As examples under the preceding theorem, we note first | r
if r be prime to p n 1 [Compare 79]. Next, if p > 2,
52) gr (|(*- 1)/2 _ T )2 _2r {& + (p n ~W t -f
represents a substitution on the marks of the GF[p n ] if r be any
mark in the field except +1, - 1, 0. For the remaining p n 3
marks T, the quantics 52) coincide in pairs. We note the following
special substitution quantics 52):
n = l, p = 7: 4 3 and 5 2 2 (Hermite),
For n = \ 9 p = 7, i/ 3 = 1, the theorem of 85 gives the
quantics 2 3 _ 5 g
g(g>- v )= 2 (| 5 + 2v
which together give the following SQ[5, 7] of 80:
5 -|- 3 -h 3 a 2 ^ (a = arbitrary).
1) For n 1, this theorem is due to Rogers, Proc. Lond. Math. Soc.,
vol. 22 (1890), pp. 210 218.
62 CHAPTER V.
87. If O (6) = 006* + ail*" 1 * be a SQ[k, p n ], then will also
0x(), obtained by forming the compound substitution,
- y
if a = 1. We may dispose of y, /3, d to simplify 0i(|). We take
y = Co" 1 , and, in case k is prime to p, we choose /3 so that
7/3a -f- 1 = 0.
Finally, we take $= y0(/3). The quantic 0!(i), in which the
coefficient of (* is unity, the constant term zero, and, when k is not
a multiple of ^), the coefficient of , k ~ l is zero, will be called the
reduced form of 0(1) for the GF[p n '\.
88. To illustrate the use of the theorem of 84, we apply it to
determine all reduced SQ\3, p*]. For p =|= 3, the reduced cubic in
the G-F\j) n ] is | 3 + a J. The sub -case p n = 3m + 1 must be rejected,
since the m ih power of | 3 + contains the power |= gp n i w j^
coefficient unity and hence =4= 0. For the sub- case _p"= 3m-f 2, the
condition given by the power m -\- 1 is (m + 1) a = 0. But if m 4- 1
be divisible by j>, then would also 3m -f 3 =_p*4- 1. Hence must a = 0.
The resulting form | 3 is a SQ[3, p n = 3m + 2] by 79.
There remains the case p n = 3 W , when the reduced cubic is
Raising it to the power S*-^ 3"- 2 H ----- h 3 -f 1 =(3 7 ' l)/2, we
find (mod 3),
The highest exponent of | in this product is <2(3 n 1). The
coefficient of
t8 n 1= |2(3 W ~ 1 + 3 W 2 + +3 + 1)
is evidently a\ -1 . Hence must i = 0. Applying then
the corollary of 81, the resulting form | 8 -f- a 2 is a $$[3, 3 71 ] if,
and only if, either a 2 =0 or else 2 is a not -square in the
89. To treat a more characteristic example, we seek the
when p n is of the form 5m -f 3. The reduced quintic is
The power m-f- 1 gives (m + 1)^3 as the coefficient of
If m -f 1 = (mod p) 9 then 5m -f 5 =p n + 2 = and therefore
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 63
Hence , for p =j= 2, we must have ft = 0. The power m + 2 of
53) |5 +a|8+} ,g
requires C m+ ^f+ C m+2 , 3 .3V + CL+..**- 0.
If ^) is neither 2 nor 7, TO + 2 , 2=|= (mod p) and may be divided out;
for, if m 4- 2 be divisible by p, then is also 5(m + 2) ==p n + 7.
Multiplying the resulting equation by 5 2 and replacing 5m and 5(m 1)
by p n 3 and p n 8, respectively, we have for -p == 2, p =)= 7,.
25y 8 - 15V + 2 4 = (5^ - 2 )(5y - 2 a 2 ) = 0.
The power m -f- 4 of 53) requires, if l ) p n > 13,
If ^) is not 2, 3, 7 or 17, we may divide out the factor
(m -f 4) (m -f 3) (m -f 2) (m -f l)m.
Multiplying afterwards by 5 4 7! and replacing 5(m 1) by 8 (modp),
etc., we find
This equation is an identity for 5y = a 2 , but reduces to - - 10 a 9 for
5 7 = 2 a 2 . In the latter case, a y = ? if p =j= 2. Hence, for
p n =%= 13, 2% 3 W , 7 ra or 17 W , the only possible quintic which represents
a substitution on the marks of the G F[p n = 5m -f- 3] is reducible to
5| 5 +5a 3 + 2 .
We have shown in 80 that this quintic is indeed a SQ[5,p n = 5m -f- 3].
The special cases above excluded require separate treatment.
90. The foregoing methods may be employed 2 ) to show that the
following table gives every reduced SQ[Jc, p n ] for & < 6:
^Reduced quantic Suitable for p n =
I ............... any p"
I 2 ................ 2"
............... 3 n , 3m -f 2
3 -- a | (a = not -square) ....... 3 W
i I 4 3 ............. 7
J 4 + 2 2 + a 3 5 (if it vanishes only for g 0) 2 W
I 5 ............... 5", 5m 2, 5m + 4
| 5 -- ag (a not a fourth power) . . . . 5 W
| 5 2V 2 | ........ 32
| 5 2| 2 ...... ...... 7
S 5 + a! 3 I 2 + 3 2 | (a = not-square) . 7
1) If p= 13, the power m-\-4: = Q brings in terms |24= |2(^n i).
2) Compare the author's Dissertation, I.e. pp. 77 86 and 101102.
64 CHAPTER V.
Reduced quantic Suitable for p" =
5 +a| 3 +y| ( arbitrary) ..... 5m 2
5 +a |3_f_3 a 2 ( a = not- square) ... 13
| 5 2a 3 -h 2 | (a = not -square) . . . 5 W
That in fact these quantics do represent substitutions on the
marks of the corresponding GF\$ n ] follows from 79, 80, 81, 83
and 86, with the exception possibly of the eleventh and thirteenth
forms. To verify 1 ) that the latter two are substitution -quantics, set
Then fi
Since a is a not -square, we can choose an integer k tt so that ft~" 2
shall be a particular not -square v. But
p- 5 Y(f*|) 5 +( f t-2)g 8 + Sftr-'a) 8 ! p-6 2 .
Since ft 3 ^ 1 (mod 7), we can choose the sign of /* = d^a/v) 1 /* to
make the coefficient of 2 unity. It follows, therefore, from 87
that O(|) and ^(jj) will be substitution -quantics modulo 13 and 7,
respectively, for a an arbitrary not -square, if they be such for a a
particular not -square v and for the -f sign in ^(l). We take v = 5,
a non-residue of both 7 and 13. In the notation of 76, these
reduced forms represent the substitutions,
n \ = /0, 1, 2, 3, 4, 5, 6\
U 5 +5 3 +i 2 + 5r-VO, 5, 2, 3, 1, 6, 4/'
/ | \ = /O, 1, 2, 3, 4, 5, 6, -6, -5, -4, -3, -2, -1\
U 5 + 5| 3 + 10fJ ' - VO, 3, 1, 5, 6, 4, -2, 2, -4, -6, -5, -1, -3J'
modulo 7 and 13 respectively.
The Betti-Matliieu Group, 9194.
91. It was shown in 81 that the quantic belonging to the GF[p nm ],
represents a substitution upon the p nm marks of the field if, and only
if, X = is the only solution in the field of the equation
- o.
1) For a verification by means of the theorem of 84, see the author's
paper, American Journal , vol.18, pp.210 218; in particular, 7 and 9.
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 65
Suppose that this condition is satisfied by two functions ^(X) and
and consider the effect of applying first the substitution 1 )
A: X' = YXX) =2* A * Xpn(
/=!
and afterwards the substitution
m
B: X" = V B (X') = V BjX'v n(m - j \
; = i
The result is equivalent to that produced by the single substitution
1. . . , m
(m - j)n( ~ j
After reduction by means of X pnm =X, this equation may be written
C: X" =
each d being a definite function of the A/s and B/s. By hypothesis,
V,[YXX)1 =
requires Y^(X) = 0, which in turn requires X = 0. Hence, X=
is the only solution in the field of ^(X) = 0. It follows that the
transformation C represents a substitution upon the marks of the
GF[p nm ~\. C is called the compound, or product, of A and J5, and
the above relation is expressed in the symbolic form,
C = AR
Giving to the coefficients At every possible combination of values
in the G-Flp"} such that
m
* (
represents a substitution on its marks, we obtain a set of substitutions
having the property that the result of applying first any one of the
set and afterwards any one of the set is identical with the result
of applying a single substitution of the set, called the product of
the two. Such a set of substitutions is said to form a group. In
the present case, the group will be called the Betti-Mathieu Group. 2 )
1) The present notation is used in place of and as equivalent to
A:
2) For n = l, this group was studied by Betti, Annali di Scienze Mat. e
Fisiche, vol. 3 (1852), pp. 49115, vol. 6 (1855), pp. 5 34; for general n, by
Mathieu, Jowrnal de Math., (2) vol. 5 (1860), pp. 9 42, vol. 6, pp. 241323.
The theorems of 92 94 are due to the author, Annals of Math., (1897),
pp. 94 96, 178-183.
DlCKSON, Linear Groups. &
66
92. Theorem.
transformation
54) X'>
CHAPTER V.
The necessary and sufficient condition that the
shall represent a substitution on the marks of the GF[p nm ~\ is
I A ..
0.
p n(m
-"-i
n(m
We seek the condition under which 54) is solvable for X.
Raising 54) to the powers 1, p n , p 2n , . . ., p n (V and reducing the
powers of X by
Xp nm =X,
we obtain the following m equations (written with detached coefficients) :
m 1) -*rn(m 2) -*rn r
A
pn At A p1
y9 . . . ^Lm
n(m l) p n(m
The solution of this system of equations in X, X pn , . . . gives
I As ... ^m_i X'
55)
AX =
A
X'*> n
56) AX?
(m i)
X'
.A
m
n
(m 1)
n (m
_- p
The condition A =(= is necessary, since otherwise there would exist
a relation between the powers of X' with exponents < p nm . To
prove that the condition A =j= is sufficient, we need only verify
that the X given by 55) satisfies the relations 56) for i = 1, 2, . . ., m 1.
Observing that J.^ w = ^ in the field, we find the following relations
upon raising 55) to the power p n ( m ~ ') and moving the first i rows
below the last m i rows:
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc.
67
- (-1)
i (m /)
p
n-
n(m-f 1
p ,.^n i) , n ( TO _f_i)
2 -A-
i(> ) -trf w n(m t)
Moving the last * columns before the m i preceding columns,
which brings in an additional factor ( l) l '( m *^ we obtain the deter-
minant of formula 56).
It follows as a corollary that formula 55) gives the reciprocal
of 54).
A second proof may be given, based on the theorem of 72.
The condition that 54) shall represent a substitution on the marks
of the GF[p nm ~\ is identical with the condition under which
Xij X 2 , . . ., X m shall be linearly independent with respect to the
6r-F[p w ] when it is given that X 1; X 2; . . ., X m are similarly indepen-
dent. We seek the condition under which
. . .X>
x
x
'
ir>
x
)p n(m 1) fp n(m :
0.
Substituting the values of X J7 Xj , . . ., X/ in terms of Xj,
X P J , . . ., X* and the A h as given by the above table, we find that
X
xf
A.
The required condition is therefore that A =)= 0.
93. To illustrate a general method 1 ) of obtaining sub-groups of
the Betti-Mathieu Group, we take m = 3 and consider the totality
of substitutions in the GF[jP n ~] on a variable X of that field,
57) X' = At X^ n +' A, XP" + A s X,
which multiply by a factor Q the function
+ BX (B in the
1) See the author's paper in the American Jowrnal, vol.22, pp. 49 54.
5*
68 CHAPTER V.
The conditions for the identity Z l ^. $Z are readily seen to be
58)
59)
60)
Since the left members of 59) and 60) are the powers p n and p 2n ,
respectively, of the left member of 58), we must have p pM = p. Hence
the totality of substitutions 57) for which the expression
Q = A 5 + ^ M -MLf + B?*-iA$*
is a mark of the G-F[p n ~\ form a group leaving Z relatively invariant.
94. Consider next the substitutions 57) which multiply the
function
by a parameter p, where D is a mark =)= in the G-F[p Sn ].
To form the function 7' into which 57) transforms F, we
note that
.A == *. -"-1 -^- "T" -^-2-^*-
Denoting by W the product of the expression on the right by D and
forming the sum T= W+ WP*+ W^ n , we find that the conditions
for the identity 7' = $Y are the following six relations:
where, for brevity,
61) r =
62) f =
In particular, it follows that 0^ n = Q. Hence those substitutions 57)
whose coefficients A 17 A 2 , A s make r = and give to the function f/D
a value belonging to the GF[j() n ~] form a group with the relative in-
variant Y.
The method may be readily extended to determine for general m
the substitutions 54) which leave relatively invariant the following
function m i
Y ,==^2)P nj X? nj +P n(s + j) (D in the GF[p nm ]),
f
where s may be any integer < m, except perhaps m/2.
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 69
It is found that the number of independent conditions upon
the coefficients A iy in order that 54) shall leave Y s relatively in-
variant, is at most (m -f- 1) or (m -f 2) according as m is odd or
even. One of these conditions merely requires that a certain function
of D and the A- t shall belong to the GF[$ n ~\.
95. We proceed to identify the Betti -Mathieu Group in the
GrF[p n ] with Jordan's linear homogeneous group on m indices with
coefficients in the GF^p 11 ']. Let R be a primitive root of the
6r.F[jp wm ], so that any mark of that field can be expressed in the
form y Q + yijR + r^R 2 -\- -h y-m i-R 7 "" 1 , where each y., is a mark
of the GF[p n ]. Consider the general substitution 54) of the Betti-
Mathieu Group. We may set
m 1 m 1 m I
where each /, ,- and af* belong to the GF[p n ~].
Substituting these values in the identity 54) and reducing the
powers of R to a degree ^ m 1 by means of the equation of
degree m satisfied by the primitive root jR ; we may equate the
coefficients of like powers of E in the resulting identity. Since
we evidently reach a set of m equations of the form
m 1
63) S|-<,t, (*-0, 1, ...,-
in which the coefficients a^- belong to the (rjP[p w ]. By hypothesis,
equation 54) is solvable for X. in terms of X'. Starting from this
solved form, our process evidently yields the ^ as functions of the |J,
so that equations 63) are solvable in the field GrF[p n ~\. Hence a y =(= 0.
According to the definition given in 97, the transformation 63)
belongs to Jordan's linear homogeneous group.
Inversely, every linear substitution 63), with coefficients in the
G-F[p n ~] such that the determinant ^- =4= 0, can be represented in
m 1
the form 54). We note first that 63) transforms Xs^^lfJ? into
x ' =
where
m 1
70 CHAPTER V.
Furthermore, TO, TI, . . ., r m \ are linearly independent with respect
to the G-F[p n ]; for, if x<>, . . ., 3c TO _i be marks of the latter field such
that
KOTO + *I T I + ---- r x TO -ir TO _i= 0,
tnen
*,-,, = (i = 0, 1, . . ., m - 1),
y-O
and therefore, since | a f -y =j= 0, each 39 = 0. Hence, when each |/ runs
independently through the series of p n marks of the 6rJP[p n ], the
expressions X and X' both run through the p nm marks of the GF[p nm ].
Every substitution 63) therefore gives rise to a permutation on
the marks of that field.
But we can always find a set of marks AI, A%, .'. ., Am of the
G-F[p nm ] such that 54) wiU transform the set of marks 1, R, JR 2 , . . .,
H 1 , linearly independent with respect to the GrF[p n ]j into an
arbitrary set of m marks of the GrF[p nm ],
m 1
Bi=^? faR* (* - 0, . . ., w - 1),
j =
linearly independent with respect to the GF[p*\. The conditions are
- ft (* = 0, 1, - - -, m - - 1),
t=l
which can be solved for A\, A%, . . ., ^U, since the determinant in jR
is not zero by 72. The resulting substitution 54) will transform
m 1 m 1
the ^ marks ? kR 1 of the (r-F 71 ] into the marks kBi all
i = / =
distinct; indeed, we have the identity
m t / m \
^mmf ^^J I ^ Jm^ ^
i 1 j = I i ; -= 1
96. EXERCISES ON THE TEXT OF CHAPTER V.
Ex. 1. Verify that 6 4- a 5 a 4 2 (a arbitrary) represents a sub-
stitution on the marks of either the G-F[3 S ] or of the G-F[2 5 ].
Ex. 2. (Hermite). A group of order 168 is generated by the sub-
stitutions . _ ,
x = ax + &, = 0(iC -f &) + c (mod 7),
where 0(#) = x 5 2# 2 and a is a quadratic residue of 7.
Ex. 3. (Rogers.) In applying Hermite's conditions ( 84) for a
substitution quantic, it suffices, when n 1, to test only the first
(p l) powers. This result of Rogers does not generalize immediately
ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 71
to the case w> 1; for $[6, S 2 ] it is necessary to consider, besides
the 2 d and 4 th powers, also the power 5 > (3 2 1).
fl
Ex. 3. By the theorem given by Weber, Algebra, II, p. 299, every
substitution on ^"letters, each affected with n indices #1, #2, . . ., A taken
modulo p, may be represented by the transformation (mod j?),
*S = <M*i, *2, - - ., O (*' 1, 2, . . ., n)
where each O t is a rational integral function with integral coefficients.
Apply the method of 84 and show that, on raising each O/ to the
powers 1, 2, . . ., p 2 and reducing by means of $ = 2; (mod jp), the
coefficient of z\ -^ - # in each power must be congruent to zero.
Ex. 4. The following substitution in the
a) X'-A^X^+AiX (^4=0, Af+
can be reduced to the form Y 1 = EY by introducing a new index
b) T=S X^+S 3 X (JBf+ 1 -.Bf+ 1 4 =0 )
if and only if there exists no root in the G-F^p 1 "-} of the equation
E A 1
? 1 Af-E
JL Z
If A 2 -\- A^=^= 0, it is not possible to reduce a) to the form
Y f KYP n (K in the G-F[p 2n J)
by a transformation of indices of the form b).
[The first result is in marked contrast to that of 214 for m = 2].
Ex. 5. By the method of 95, show that the sub-groups of the
Betti-Mathieu Group defined in 9394 by means of the invariants Z and Y
are identical with certain linear homogeneous groups on m indices in
the G-F[p n ] defined by a linear and a quadratic invariant respectively.
Ex. 6. (Moore.) The multiplier - GF [p k ] of the additive -group
pi, . . ., Aj is the (largest) additive - group common to the additive -groups
OF AI, . . , Af 1 ^] (=!,..., m)
and is contained in the p m 1 additive - groups
[A- 1 A, . . ., A- 1 AJ (A 4= of [A l5 . . ., A m ]).
SECOND PART,
THEORY
OF LINEAR GROUPS IN A GALOIS FIELD.
CHAPTER I.
GENERAL LINEAR HOMOGENEOUS GROUP. 1 )
97. First definition. - - Consider the p nm letters, or symbols,
characterized by m indices, each running through the series of marks
of the G-F[p n ]. The general linear homogeneous substitution A on
the m indices j;,- with coefficients in the GF[p n ] replaces the letter
k l} f 2 , . . ., t m by 1$> V fv . . ., |- m where
A: Si = y 6> 0' =
the coefficients i; - being marks of the field. But A will indeed
permute the p nm letters if, and only if, the determinant of A is not zero,
A = y 4=0.
In fact, there must be one and only one system of m indices which A
replaces by a given system |' and hence an unique set of values fjy
satisfying the equations
tf-"S ( = 1, 2, ..., m).
Let B denote a second substitution with coefficients in the GrF[p"'\,
m
S: li=ft,-i- (fc-1, 2, ..., m)
where
7? I ft, -U n
-L> r^: | IfJJci - \J.
1) Jordan, Traite des substitutions, Nos. 119, 169; author's dissertation,
Part H. Cf. 95 above.
76
CHAPTER I.
The result of applying first the substitution A, which replaces
where
-^fe (i~l,...,m),
by Is* ,
j 1" '
64)
and afterwards the substitution J5, which replaces the general letter
Is'*, ' by L-'', *" where
* " 5 !/ 7?z
=
65)
is identical with the result of applying a single linear substitution,
called their compound or product AB, which replaces ^ i; ..., $ m by
li" v ," i where, by eliminating the [ between 64) and 65), we have
m / m
j = 1 \ t = 1
m
Setting
we may write the product of A and B in the form
By the theorem for the multiplication of determinants
A 4=0.
Moreover, the coefficients y*/ belong to the (r.F[f) n ]. Hence the
compound J^JB is indeed a substitution and has its coefficients in the
same field as those of A and B. If therefore we let the coefficients
of A run through all the sets of values in the GF[p n ] for which the
determinant a {j =j= 0, we obtain a set of substitutions forming a
group called the general linear homogeneous group on m indices with
coefficients in the CrF[p n ] and denoted by the symbol GLH(m,p n ).
Remark. If the substitution A be identical with the substitution
^ = 1, ..., m)
then must ,-/= a^ (*, j = 1, . . ., m). This follows by taking in turn
for j = 1, 2, . . ., m the particular set of values
|,.= 1, ga-O (& = 1, 2, ..., m;
GENERAL LINEAR HOMOGENEOUS GROUP. 77
98. Second definition of GLH(m,p n ). - The essential thing in
the substitution A is the matrix of its coefficients (a^). Taking the
indices 1, . . ., % m to be variable marks of the GF[p n ], we obtained
an immediate interpretation of A as a permutation of certain p nm letters,
so that the linear group was recognized as a permutation -group.
We may, however, let the indices |i, . . ., | OT be arbitrary variables
and consider the linear transformations
A: fS = at& (i = 1, . . ., m), \ a {j \ + 0,
where each coefficient belongs to the GF[p n ]. As in 97, the
compound of two such transformations will be a linear transformation
of determinant not zero and with all its coefficients in the GF[p n ~\.
Since j ccij =|= 0? ^ ne inverse of A exists and has similar properties.
Hence the totality of transformations A form a group, evidently
the GLH(m,p n ). '
Employing this second definition, we may represent the trans-
formation group as a group of permutations on p nm letters. Consider,
indeed, the p nm linear functions AI^I-|- A 2 ?2+ v -h ^mlm where each
A runs through the marks of the GF[p n ]. These functions are merely
permuted by the linear transformations A.
99. Theorem. - - The order GLH[m, #*] of the group GLH(m,p n ') is
(rrtn-m _ ]\ ( finm _ /^n\ f ^nm _ ^n\ ^ ( V) nm _ v) n ( m 1)\
The number of distinct linear functions
by which the substitutions of the group can replace the index ^ is
p nm 1, since the marks ccij may be chosen arbitrarily in the GF[p n ]
provided not all are zero. Let T be one of the substitutions which
replace ^ by a definite linear function f v If then
Ri = I (identity), jR 2 , jRa, . . ., E N
denote all the substitutions of the group which leave | t fixed, the
^products,
will replace t by /i. No other substitution of the group has this
property; for, if U replace 1 by /i, T~ 1 U will leave | x fixed and
hence be a certain E i} so that U= TR;. To each of the p nm 1
distinct functions f there corresponds a set of N substitutions.
GLH\m, p n ~\ = N(p nm 1).
78 CHAPTER I.
The substitutions jR/ are of the form
= 2, . . ., w)
where the m 1 coefficients a^-t are arbitrary and the coefficients
a kj (k,j = 2 7 . . ., &) are such that their determinant =J= in the
field. The latter set of coefficients can be chosen in GrLH[m l,p n ]
ways. Hence
nm -
GrLH[m, p n ~] -=_p(-0(p l)GLH[m -- 1, p"].
This recursion formula gives, since 6?J&JET[1,J?*] = ^/ n 1, the result
100. Theorem. - - Every linear homogeneous substitution A on m
indices with coefficients in the GF\p n ~\ can be expressed as a product BD m ,
where B is derived from the totality of substitutions of the form
Br,sl'- r = r-Mb S , If = fe (* = 1, -, ^5 * 4= r 'l T 4= S ) ,
X &em^r aw arbitrary mark of the GF[p n ~\, and where D m denotes the
substitution altering only the index % m which it multiplies by the deter-
minant of A.
Let the given substitution A be the following:
A: Jf = ^Kjjtj (i = 1, . . ., w).
The product ABi^j. has the form
TO
i =
the matrix of its coefficients being
22
4-
Similarly, the matrix for the product 1,3,1 A is
^12 4-
GENERAL LINEAR HOMOGENEOUS GROUP. 79
To multiply A on the right by Br,s,i, we therefore multiply the
5 th row of the matrix (a,,) by A and add to the r ih row; to multiply A
on the left by the same substitution, we multiply the r ih column by A
and add to the s th column of the matrix (a,-/). We make use of
these operations, which are recognized to be identical with the
elementary operations permissible in reducing a determinant, to sim-
plify the form of the matrix A. It is shown below that, if m > 1,
we can set a n = 1. Then by multiplying A on the right and left
by suitable generators Bi,j,i, we can reach a new matrix A' having
the elements of the first row and first column all zero, except a n
which = 1. After m 1 such steps, we would reach a matrix A^ m ~ ^
having every element zero except those in the main diagonal and
the latter all unity except that lying in the last row. The resulting
substitution would be D m . From the identity thus established,
BiAB-2=D m , where B and 2? 2 are products derived from the Bij,i,
we find
A = #t 1>A - Br l B s D m =BD m .
It remains to be shown that, if m > 1, a matrix can be obtained
from A having an = 1. From the given generators we derive the
substitution
affecting only the indices {;, and ^. In particular, for A = 1, i = 1,
we get . .
T. t ' _ t t. t
5i -- ?y> w - - !
We deterge a substitution K derived from the I?,-,/.* such that the
product A 1 = KA will have the coefficient a2i=t= 0- ^ ^ai^O? we
take K=I, the identity; if 21 =0, but 2j=|= 0, we take K =* J.
The product ,,,_ , r
xA : - A.
i, 2 ,A
has the coefficient n = n+ A 2 i> which can be made equal to unity
by choice of A in the GF[p n ~].
Corollary I. The only linear homogeneous substitutions commuta-
tive with every J5 r>s> * (r, s = 1, . . ., m, r =f= s), where A is a
mark =(= o/" ^e G-F[p n ~\, are those of the form
It follows by inspection of the above two matrices for
and BijtjuA that they are identical only when
of n = or 22 , a n =0 (t 2, 3, .. ., w), 2y =0 (j = 3, ..., m).
Since the indices 1, 2 can be replaced by any pair r, s of distinct
integers ^ m, it follows that every element of the matrix (a,-,-) must
be zero except those in the main diagonal, which must all be equal.
80
CHAPTER I.
Corollary II. The group of binary linear homogeneous substitutions
of determinant unity is generated ~by tine substitutions Bi,2,i and
T t & fc &
61 = ~ 62? ?2 == 5r
Indeed, T transforms _Z?i, 2 ,_;i into -Z? 2 ,i, A.
101. Transformation of indices. We can introduce in place
the m new indices
(t = 1, 2, . . ., m)
Of l, ?2> > 5m
67)
provided the determinant /3/t | =j= 0. In fact, the substitution
will replace ^/ ky0**JMfc> which, by solving 67) ; can be put into
>,*
m
the form'/y^. The substitution A becomes
where B denotes the substitution 67) replacing the | f by the ??,.
In fact
The determinant of the transformed substitution equals that of A,
B--
B
This result is, however, a special case (Q = 0) of the next theorem.
102. Theorem. - - I he characteristic determinant (with parameter Q)
of a linear homogeneous substitution A,
Q
is unchanged under every linear transformation of indices.
It is only necessary to prove the theorem for the following types
of transformations of indices, since by 100 every linear trans-
formation can be derived from them:
GENERAL LINEAR HOMOGENEOUS GROUP.
81
Under the transformation of indices D x , A takes the form
M (i = 2, 3, . . ., m).
The characteristic determinant of the transformed substitution is
22
Q
Under the transformation of indices B^ 2 ,;., -4 becomes
n'i = cc ij%j == anvil + (2 *^i)i?i4->/% (i = 2, . . ., m).
^ = 1 > = 3
The characteristic determinant of this substitution is
ai3+/t23 ... aim-fAa
23 . 2m
#33 .
Multiplying the second row by A and subtracting from the first row,
and afterwards adding the first column multiplied by A to the second
column, we reach the original determinant A(p).
Corollary. - The transformed of A ~by any linear substitution B
has the same characteristic determinant as A. Indeed, by 101, A is
converted into B~ 1 AB by the transformation of indices indicated by
the substitution B.
Factors of composition 1 ) of GrLH(m,p n }, 103 107.
103. Let Q be a primitive root of the GF[p n ~\. If two linear
substitutions have as determinants Q rl and p s/ , their compound has
1) For the case n = 1, Jordan, Traite, pp. 106110; for*general w, author's
dissertation, Annals of Mathematics , vol. 11 (1897), pp. 168 175; also Burnside,
The theory of groups, pp. 340 341.
DlCKSON, Linear Groups. 6
82 CHAPTER I.
the determinant p( r + 5 K Hence the totality of substitutions in the
group G = GLH(m,p n } having as determinants powers of g l forms a
subgroup 6i> Suppose that
. .p k ,
where pi,p%, - -,pk are all primes. Denote by 6r pl , 6> lJ)2 , . . ., Cr p n_ 1 = f
the subgroups of G formed of those of its substitutions whose
determinants are respective^ powers of 0% Q p i p *, . . ., Q p7l ~ l = l. By
63, the orders of these groups are respectively
/Pi> /PiP2> -t /P n 1 (where Q = GLH[m,p n ]).
In fact, by 100, G contains substitutions of every determinant =j=
in the GF[p n ] and contains the same number of one determinant as
of another.
If S and T be linear substitutions, S and T 1 5Thave the same
determinant ( 101). Hence the groups G PI , G PlPz , . . ., f are self-
conjugate under 6r, i. e., each is transformed into itself by any sub-
stitution of 6r. Since p ly . . ., p k are primes, there is no group lying
between 6r and 6r Pl , no one between G fl and G PIP ^ etc. Hence we
may descend from G to f by the composition -series
The group f of all substitutions of determinant unity is called
the special linear homogeneous group SLH(m,p n ). It has a self -con-
jugate subgroup H formed of those of its substitutions which are of
the form
M^i = ft|,. [>"= 1] (i = 1, 2, . . ., w).
The mark p must also satisfy the equation
Hence, if d be the greatest common divisor of m and p n 1, we
find (by the method of proof used in 79) that
68) ^ d = 1.
Inversely, each of the d distinct solutions in the GF[p n ] of 68)
[see 16], leads to a substitution M^ belonging to the group H.
The order of H is therefore d.
If d be a mark of the GF[p n ] which belongs to the exponent d
( 17, Corollary), then p is a power of d. Suppose that
d = q i q 2 . . . q t (each q^ a prime).
Denote by H^, H qi q z> . . ., H d = I the groups formed of those sub-
stitutions of H which multiply every index 3y a like power of tf 5j , by
GENERAL LINEAR HOMOGENEOUS GROUP. 83
a like power of d? 1 ^, . . ., by a like power of d d ^ 1, respectively.
Since we have, for any mark v,
a composition -series of H is given by
H, H qi , -fifcfc., . . ., H Mt . . . q t =I.
In view of the theorem proven in 104 107 , we may state the
complete
Theorem. - - The factors of composition of GLH(m,p n ) are
Pi, P-2, -, Pk, Q/d($ n - 1), q l9 q 2 , . . ;, q h
except in the two cases (m,p n ) = (2, 2) and (2, 3), when the factors of
composition are 2, 3 and 2, 3, 2, 2, 2 respectively.
104. Theorem. - Excluding the above two cases, the group H is
a maximal self -conjugate subgroup of f.
Suppose that f contains a self -conjugate subgroup J which
contains all the substitutions of H and still further substitutions.
We will prove that, aside from the two exceptional cases mentioned,
J coincides with f.
By hypothesis, J contains a substitution
8: ' = / a -| / (i = 1, . . ., m)
/=*!
which is not in H and therefore does not multiply all the indices by
the same factor. Hence, by Corollary I of 100, 8 is not com-
mutative with every B r ,s,i (V, s= 1, 2, . . ., m; r =)= s). Changing the
notation if necessary, we may suppose that 8 is not commutative
with BI,*,*, a substitution of determinant unity and therefore in the
group f. It therefore transforms the substitution 8 of the self- conjugate
subgroup J into a substitution belonging to J. Hence J contains the
product !
-~ b ' JDi )2 ,^oJ5 l5 2,^ ?
which does not reduce to the identity /. In calculating this product,
let <t> be the linear function by which S~ l replaces | 2 . Then T is
seen to have the form, in which the values of the /3iy need not be
determined:
T: -M*5 = 8* " ^i (' = 2, 3, . . .,
Suppose first that the an are not all zero, say ar 21 =j= 0. For
w > 2, we introduce new indices ^ defined by the substitution V of
determinant unity,
f* } *'lj-/'O/< \
1?!= &, rj 2 = |t, ^= If - ^ ( == 3, 4, . . ., m).
a 21
6*
84 CHAPTER I.
The resulting substitution V~~ 1 TV belongs to J and leaves ??,(*> 2)
unaltered:
If, however, every o,-i=0, T itself leaves fixed m--\ indices. In
either case, J contains a substitution =J= I of the form 1 )
3 = 1 3 = 1
Then 7 contains the two substitutions leaving ^ 3 , . . ., vj m fixed:
,.u' t /
1^2 = ^2 ft(ysa
These substitutions are both of the form
U: 1/1 = 171-1- (J^s, ^2 = ^24-^2^3, ^ = ^/ (* = 3, ..., m).
If T 2 and T 3 reduce to the identity, H itself becomes
H ----- h yim^TO, ^2 = ??2 + ^23^3 H ----- h ysm^
If yij= yzj= (j = 4, . . ., w), this substitution =)= J is of the
form U. In the contrary case, we may suppose that y u and y 24 are
not both zero. Then
If
f
is a substitution =J= J of the form f/ and belonging to ?7. Hence,
in every case J contains a substitution U not the identity. For
definiteness, let (Ti =(= and introduce the new indices
1 = > = - , - . -
i
Then U becomes J5i,s, ai . Transforming the latter by the substitution
fc' ; a f _ . i it fc'. __ fc. /y Q w \
5i A ?i? ?2 A '2? 5 5i v.* " > > m )j
where A is an arbitrary mark =f= of the GF[p"-], we reach in J
the substitution S ijS ^ af and therefore every Bi,8,i* The latter is
transformed into JB kt 3j ; i(^ =(= 1 , 3) by the following substitution of l~:
SJ. = it? S* = Si, S5 = li (* = 2, . . ., A; 1, A; + 1, . . ., m).
1) From this point, the proofs by Burnside and Jordan (1. c.) are incomplete.
The specific errors were made in the Traite, p. 108, 1 and in The theory of
groups, p. 316, "This process may now be repeated", etc.
GENERAL LINEAR HOMOGENEOUS GROUP. 85
Finally, for j =j= k, -B*,s,;i is transformed into Bk.j,i by the substitution
Si = - If, % = S 8 , 5J - g f (' = 1, . . ., w; + 3, t 4= j).
It follows from 100 that, if m > 2, J is identical with T.
105. For w = 2, we are given that J contains a substitution
8: $' 1 -li+/JS i , Si-a'Srf^S, (/?'- '/? = !),
which is neither the identity J nor
7?. ' - V - t
-& 5i ~ ~ bl> 5s ~ f'
We proceed to prove that, for p n > 3, J contains a substitution of
the form -Z? 2 ,i,;i in which A 4= 0.
a) Suppose first that /3 = 0, so that 7 contains
.Cf . a' _ _ fyt V . - /y'fc _L ry Ifc
i; 5i " fel 7 feg "~ tt 9i~T B 2?
where ' =4= ^ ^ a == ~ 1 ? since 8^=^=1 or .E.
a A ) If a = ar l 7 whence a = + 1, the group J contains both S
and S^E, one of which has the form
II -fe, ft-S.+ ife (i = ' + o).
85) If =)= ~ J ? e7 contains the substitution =4= I>
S JB%, i, i ^i #2, i, i : li = Si, S'g = Is + (1 ~ B *)ti-
b) Suppose next that /3 =|= 0. The following substitution
X- S-f /3fc fcf l+X 2 a a fc fc
Z: |i xgj+ x/Sgg, Si - x^ > gi~ x S 2
has determinant unity and therefore belongs to I". Hence J contains
viz.
If jp= 4 or if p n > 5, z can be chosen in the GF[p n ] so that
* 4 +l, Jc~ 2 4=% 2 .
Proceeding with & as in case a 2 ) ? we obtain in J a substitution -ft, i,i,
where A =)= 0.
If p" = 5, we take x = 1, when 5 2 -E7 becomes
tt) fc , h
*1 '~ *2*
bl I 2 ~
Our result follows unless fi + = (mod 5). But J contains the
product SB*,}, i S~ 1 5 2 , i, i , viz.,
for which the sum corresponding to the above /3' + a is
(1 + a 0) + (i _ a p + ^) = 02_|_ 2 == (mod 5).
86 CHAPTER I.
We have now proved that, if p n > 3 ; J contains a substitution
^2, i, ;. (A =|= 0). It is transformed into -Z? 2 ,i,;^ by the substitution
Also
By 64, there exist solutions in the GF[p n ] of p 2 -}- G 2 = x/A for sc
arbitrary in the field. Hence J contains B 2 , lt x . Transforming the
latter by (|J_ = 2 , | 2 = ~ ^) we get l?i, 2 , * It follows from 100
that J"= f. By 99 and 103, the order of the group f of binary
linear homogeneous substitutions of determinant unity is p n (jf n 1).
106. For p n = 2, m 2, the group P is of order 6 and is identical
with GLH(2j 2). It contains a subgroup of order 3 generated by
the substitution
fc'__t t' t \ t
fel 5o; b2 *i"r bg-
The index of this subgroup being 2 7 it is self - conjugate. The factors
of composition are therefore 2 and 3.
107. For p w =3, m = 2 7 the group G~GLH(m,p n ) is of
order 48 = (3 2 1) (3 2 3) and contains the following substitutions
A ' V - t V .. t i t .
A-' ?1 - 5lJ 2 ~~ 1 ' ?25
of which ^1 has determinant 1 and the others determinant -f- 1
modulo 3. In virtue of the relations
C 2 =E, CE = EC, CD - ED C;
B*=E, BE = EB, BD=CDB,
A 2 =l, AE=EA, AD^CA, AC=DA,
it results that the groups generated as follows:
(E,D,C}; \E,D,C,B}, {E,D,C,B,A}
have the orders 2, 4, 8, 24, 48 respectively and that each group is
self -conjugate under the following group. The last group is identical
with 6r, whose factors of composition are therefore 2, 3, 2, 2, 2.
GENERAL LINEAR HOMOGENEOUS GROUP. 87
108. From the linear homogeneous substitution A of 98 on the
arbitrary variables 1, %%, . . ., % m) we obtain the linear fractional substitution
, -- - ----- - m-m-- m / _ 1
MI === - i V * !. Ill/
- - ' ' 'T
upon setting #/= 6//6 for i = 1, . . ., m 1. It being only a question
of the ratios of the coefficients a,-j in A', its determinant a^ is
determined only up to a factor |u m , /u, being a mark =^= 0. Also, .A'
is the identity if, and only if, A be one of the p n 1 substitutions
M u : 65 = fife (*~1, ..., w).
The products M/uA and no other linear homogeneous substitutions
correspond to the same linear fractional substitution A'. Hence the
group G = GLH(in,p n ) has (j) w 1, 1) isomorphism with the group L
of the substitutions A'. If Q denote the order of G, the order of L
is Q -j- (p n 1). To the subgroup f formed of the substitutions of G
having determinant unity there corresponds a subgroup A of L com-
posed of those of its substitutions whose determinant is an m ih power
in the field. If d be the greatest common divisor of m and p n 1,
there are exactly ^substitutions of the form M^ in f and they form
the group H ( 103). Hence f has (d, 1) isomorphism with A. The
order of A is therefore Q -~ d(p n 1). Aside from the cases (m,y) = (2, 2)
and (2, 3), H was shown to be the maximal self -conjugate subgroup
of f; hence A has no self - conjugate subgroup other than itself and
the identity and is therefore simple.
The group LF(m, p n ) of all linear fractional substitutions in
the GF [p n ] on m 1 variables and having determinant unity or some
m ih power in the field has ihe order
2)
d being the greatest common divisor of m and p n l. It is a simple
group except in the two cases (m, p n ) = (2, 2) and (2, 3). The group
of all linear fractional substitutions of determinants not zero has d times
the order of LF(m,p n }.
The notation LF(m,p n ) emphasizes the point that the essential
quality of the linear fractional substitution lies in the matrix (or,^)
of degree m and not in the m -- 1 variables x\ y . . ., x m i which play
the ro le of indeterminates. For m = 2, we use the suggestive notation
(A = a - (I? + 0).
v
In virtue of the identity of the two substitutions
(ft any mark + 0)
fill],
\y,<y ?
of determinants A and /i 2 A, we may choose fi so that the substitution
takes its normal form, viz., of determinant unity if p = 2, but of
88 CHAPTER I. GENERAL LINEAR HOMOGENEOUS GROUP.
determinant unity or a particular not - square 'v if p > 2. In fact,
if A is a square, fi 2 A may be made equal to unity by choice of ^
in the field; while for A a not -square, ji 2 A may be made equal to v.
If p n > 3, the group LF {$,$*) of all linear fractional substitutions
in the G-F\p n ] of determinant unity (when in their normal forms) is a
simple 1 ) group of order
M(p n ) ^E P n (P* n - 1} (2 5 l according as p > 2; p = 2).
* i *
There are p n (p^ n 1) linear fractional substitutions of determinant =j= 0.
From the formula of composition of binary linear homogeneous
substitutions ( 97), we derive the product SS of linear fractional
substitutions S=~
r =
' - ' '
Hence if S operate first and S i afterwards, the product SS 1 is 2 )
109. The quotient -group T/iT may be readily represented as a
permutation -group on # E E (j9 nm -- 1) ~ (jp n 1) letters 3 ). Of the
pim_ ^ letters Z^ $ . ..,$ m i n which |i, | 2 ? -, Im denote marks of
the GF[p n ~\ not all zero, we combine into a single system the
p n l letters l^^ ^^ . ..,/&$ in which ^ runs through the series of
marks =|= while |i, J 2 , . . ., % m denotes a set of fixed marks not all
zero. Any linear homogeneous substitution on 1, . . ., m with co-
efficients in the field replaces the letters of any one system by letters
all of some one system and therefore permutes the q systems amongst
themselves. In particular, the substitutions M^ do not displace any
system. Hence the group f of substitutions of determinant unity
corresponds to a permutation -group on the # systems, which represents
concretely the quotient -group V/H.
1) Cf. Moore, Congress Mathematical Papers, pp. 208 242, Bull. Amer.
Math. Soc., Dec. 1893; Burnside, Proc. Lond. Math. Soc., vol.25, pp. 113139
(Feb., 1894); also see 261 below.
2) For the same product of matrices , the notation $ t S is sometimes used,
S operating first.
3) Compare the method of 228, 224; also, for m = 2, that of 239.
CHAPTER II. THE ABELIAN LINEAR GROUP.
89
CHAPTER II.
THE ABELIAN LINEAR GROUP. 1 )
110. A linear homogeneous substitution on 2m indices with coeffi-
cients belonging to the G-F\jp n ~\ is called Abelian if, when operating
simultaneously upon two sets of 2m indices,
5- < / * -4 ^v \
r -i 11-i ' r o VI n ( 1 I s VVt \
*3l 1 ) '(1 1 5 felaj '/(a \ v t/ -*J -* f ' ' ') /)
it leaves formally invariant up to a factor (belonging to the field)
the bilinear function
74) cp
/
/2
The totality of such substitutions constitutes a group called the
general Abelian linear group*) GA(2m, p n \ These of its substitutions
which leave qp absolutely invariant form the special Abelian linear group
SA(2m,p n ). For other definitions of these groups see 160 below
and the author's article, Transactions of the American Mathematical
Society, vol. 1, pp. 30 38.
The conditions that the linear substitution
75)
S:
0'= 1, 2, . . ., m}
shall leave 9 formally 3 ) invariant up to the factor ^ are
> yfy
76)
=i
m
P
it
?ik
o,
(j, fc-l,...,
= 0.
For 0w = 1, the Abelian group GA(2,p n ) is evidently identical
with the general binary linear homogeneous group GLH(2, p n \ In
1) Investigated by Jordan, Traite", pp.171 186, for the case w=l; by
the author, Quar. Jour, of Math., 1897, pp. 169 178, ibid., 1899, pp. 383 4, for
general n.
2) To distinguish these groups from the ordinary Abelian, i. e. commuta-
tive, groups, we prefix the adjective linear. The Abelian linear group is not
commutative in general.
3) The indices | t - and r)i are treated as arbitrary quantities. Formal in-
variance is used in antithesis to numerical invariance.
CHAPTER II.
/=*!
m
determining the structure of the Abelian group, we may therefore
suppose m > 1.
111. We proceed to determine the substitution reciprocal to $,
i-= 1, 2, ..., m).
Supposing 5 to be Abelian, we obtain the same result upon multi-
plying (p by ji that we obtain upon operating the substitution S upon
the two sets of indices. The identity of the two results is not
destroyed by operating the substitution S~ l upon the indices z -i, vjn
(i = 1, . . ., m) of one set. The result obtained upon multiplying cp by p
and then applying the substitution S~ 1 upon the indices ,-i, r]u is
therefore identical with the result obtained by applying the substitution S
upon the indices {,, ^,-2 alone. Equating the two results, we find
!,...,*
From this identity in the indices % t -j, ^-, we find
Hence the reciprocal of the Abelian substitution 75) is
77)
(t'=l, 2, ..., m).
1
p-
When /S" 1 is operated upon the two sets of indices, 9 must be
multiplied by I/ p. Forming the relations expressing this fact, we
obtain the following conditions, together entirely equivalent to the
set of conditions 76):
78) l=
m
V
/ y*
1 = 1
TO
_
= 0,
= 0.
ft !,...,*;
-
THE ABELIAN LINEAR GROUP.
91
112. Since the conditions 76) and 78) will be used repeatedly
in this and the succeeding chapters, it will be found to be of great
assistance to apply the following scheme by which these conditions
can be read off by inspection from the matrix of the coefficients
of S:
"ll
fti
l
nl
12
^22 ^22
F22 "22
Urn 2
ff 1 m
film
2m
P
The 1 st and 2 nd rows of this matrix will be called complementary,
likewise the 3 rd and 4 th rows, . . ., finally the 2m 1 st and the 2w th
rows. Similarly, the 1 st and 2 nd columns will be called complementary,
also the 3 rd and 4 th , . . , finally, the 2m 1 st and 2m th columns.
The left member of each of the relations 78) is a sum of deter-
minants built from the coefficients of two rows, the elements of each
individual determinant belonging to complementary columns. If the
two rows be the s th and tf th , we denote this sum by H si . The
relations 78) may then be written (taking s < t)
79) R2ii, 21= p, R>st= (unless t = s + 1 = even).
Similarly, if we denote by C st the sum of the determinants built
from the coefficients of the s th and t ih columns, the elements of each
individual determinant belonging to complementary rows, we may
write the relations 76) in the compact form
80) Czi i 21= [*>, C s t=Q (unless t = s 4- 1 = even).
113. Theorem. - The factors of composition of GA(2m,p n } are
the prime factors of p n 1 together with the factors of composition
of SA(2m, p n ).
Let Q be a primitive root of the GF [p n ~]. The general Abelian
group contains the substitution
U: g = 9 i ; , ri\ = rii (i = 1, 2, . . ., m)
which multiplies (p by p. Let S be any Abelian substitution and
== Q r the factor by which it multiplies <p. We have
92 CHAPTER II.
S = U r T,
where T is a new Abelian substitution not altering cp and hence in
the special Abelian group. Since r may be any one of the integers
1, 2, . . ., p n 1, the order of GA(2m, p n ) is p* 1 times the order
#J.2f,jp*] of the group J.(2m, #").
Let a, /5, . . . be the prime factors whose product gives _p n 1.
Let A y A a) A a p, . . ., A p n_ l == SA(2m, p n ) be the groups formed by
the combination of the substitutions of SA(%m,p n ) with
respectively. Evidently these groups have the respective orders
,i>],
(jP n -
while each is self - conjugate under A ~ GA(2 m, p*).
114. Theorem. - The group SA(2m, jp w ) ?s generated ~by the
substitutions 1 )
i f j = l, 2, . . ., ?w; i =f= j; W6? t^/tere A s an arbitrary marl' of
the GrF[p*\. Every substitution of the group has determinant unity.
From these substitutions leaving cp absolutely invariant, we
obtain other simple substitutions of SA(2m, p n ) = Gr as follows:
Let /S be any substitution of 6^ and let it replace ^ by
t a i^ y^ not a11 zer ]-
We can set S = VS 1 , where Fis derived from the above substitutions
and S' is a substitution of Gr in which the coefficient corresponding
1) In the expression for each substitution we omit the indices not altered.
For example, Mi alters only the two indices rji and |/.
THE ABELIAN LINEAR GROUP. 93
to a n in S is not zero. Indeed, according as a lj =$=Q or y^ =(= 0,
we may take V = P^j or PijMj. Let /S" replace ^ by
.7 = 1
We can determine a substitution 8 l derived from the above types
which shall replace j^ by co' 1? viz.,
81 = l, /S -Mill, $1, 2, ^ 2 -NJ, 2, y l2 $1, m, 1|n ^i, , xj^
where a and /? are determined by the conditions
Hence $' = $ t S", where S" is a new substitution of 6r which
leaves | A fixed. Let >S rf replace ^ by
"2"-' Ofyfe
> = 1
For ft == 1, flr n = 1, y n = 12 = y 12 = = cci m = yi m =* 0, the relation
jR 12 = k a of 79) gives # n = 1 in the substitution S n . The substitution
S 2 EE ii } r RI, 2 , _ ^ 12 2 , i, _ d 12 . . - JRi, m , _ ^ m Q m , l,-3 lm
will replace ^ by c? 2 if we take
t = Pn Pl2"l2 Pl3"l3 ' ' ' Plm^lm-
Hence /S'' = $ 2 5 f ", where >S'" is a new substitution of 6r which
leaves | x and ^ unaltered and thus has the form
S 1 ":
1 = li ,
+
Applying the following relations of set 79),
Rtt = 0, E 2t = (t = 3, 4, . . ., 2m),
we find ' x. x
a n = p n = yil = d a =0 (* = 2, 3, . . ., m).
The relations between the coefficients a t -j, ytj, fa, d,-j (i,j = 2, . . ., m)
of ^S" ; are seen to be precisely tho^e holding for a special Abelian
substitution on m 1 pairs of indices. Furthermore,
S= VS'=VSiS" =
where V, S 1} S 2 were derived from the types of substitutions given
in the theorem.
After m operations similar to that by which S n ' was derived
from 5, we reach a substitution which leaves fixed all the indices
94
CHAPTER II.
and is therefore the identity. Hence S is a product of substitutions
of the given types. Since the latter are all of determinant unity , so
is also the general substitution S of the group.
115. Theorem. The order SA[2m,
group equals
2) _
of the special Abeljan
There are (j) w ) 2m 1 sets of values of e^-, yi$ (j ~= 1, . . ., m), not all
zero, which give distinct functions ca 1 . In the function o? 2 , d n = 1
while 0H, fa, dij (j = 2, . . ., m) are arbitrary in the field. Hence w 2
may be chosen in (jp} 2n * ways. We have therefore the recursion
formula
SA[2m,
116. Theorem. For p > 2,
- 2
factors of composition of
SA(2m,2) n ) we SA\2m,p n ~\ and 2, the case p n = 3, m = 1
Every substitution of G- ^= SA(2m, p n ) is commutative with
T~T t ,- l T^*...I m ^&-*^$ tt rli^-in (-!,..., m).
The group 7T E E { J, T} of order 2 is therefore self- conjugate under G.
In order to show that K is the maximal self - conjugate subgroup
of 6r, we prove that a self -conjugate subgroup J of 6r, which
contains K without being identical with K, must coincide with G.
Let S, given by 75), be a substitution of J not in K. Then J"
contains the products
where A is a fixed mark =j= 0. Suppose first that all of these
products reduce to the identity. Then, for example, S is commutative
with both LI, * and L\,i, so that, by the proof of Corollary I of
100, S has the form
n
...
n
...
22
7 2 2
. . <K2m
72m
022
#22
02m
$2/n
m2
rm2
<*rom
7mm
{
0m 2
<?m2
0mm
* ,
1) For w = l, /S-4.(2?w,jp) is identical with the group of all binary linear
homogeneous substitutions of determinant unity. Its factors of composition are
therefore given by the theorem of 103.
THE ABELIAN LINEAR, GROUP. 95
But S is to be commutative with every pair L^ and L'l,*. It follows
that S reduces to the form
S: i< = aii, rfi = cca^i (i =1,2,..., m).
By the first type of Abelian conditions given under 79), we have
,-,-= 1. Since S is not in K, the a,-,- are not all + 1 a^d. n t
all - - 1. Transforming $ by a suitable product of the form Pi r P 2s ,
we may suppose that a n = 1, or 2 2 == ~~ 1 i n & Then 7 contains
JNi~i,^-^i,Mj which replaces | t by | x 2ft^ 2 and is therefore (since
p =j= 2) not of the form 5. Taking it in place of our initial sub-
stitution S, we are led to the case next considered.
Suppose that not all of the above products reduce to the iden-
tity J; for example, let
i.
If $ 1 replaces 17, by the linear function ra/A, the product denoted
by 8 has the following form, in which the coefficients of |i have
not been calculated:
S-fc-^i (* = 2, ..., w),
$= ifr 0,-i o> 0' = 1, . . ., m).
From $! we proceed to determine a substitution =f= / belonging
to J and leaving 2m 3 indices unaltered. S l itself is such a sub-
stitution if an= ftn= (i = 2, . . ., m). In the contrary case, the
transformed of S by a suitable P 2> , or P^jM^ will have <v 21 =|= 0.
Consider therefore S when cc 21 =^=Q, and introduce the new indices
61 = 61, % ifi, Ig = la> % = -
2?
an operation equivalent to the transformation of 8 by the following
product T belonging to the group G:
$3,2,-3l/21^ 3 -
where
21
We obtain the substitution S 2 ~^ T~ 1 S i T ) leaving fixed 2m 3 in-
dices, viz.,
1 { .
li (& *i) - L (I 2 - <*2i) = If, i?i = i?,- (* = 3, . . ., m)
96 CHAPTER II.
Writing , : , y { for ,, ^ in S 2 , and applying conditions 79), viz.,
Ru=l, B u =Bu=0, A, -ft, = 1^=0 0'-5, 6, ..., 2w),
we find that 8 2 takes the form
the indices i/, ??, (i = 3, . . ., w), not being altered by S 2 and the
substitutions below, are not written in the formulae.
The group J contains the product
where O is a linear function of | 1?
a) Suppose first that S 3 is not the identity. If 1 cc n =(= 0, we
may define r by the equation
Then J contains $ 4 = Li^S&L'i t i, which has the form
^ = ^1. / , m
f V^12 "T 8 U J
Applying the conditions ^3= ^23= of 79), we find that y 12
21 = 0, so that $ 4 has the following form (with a =(= 0):
If, on the contrary, 1 cf u = 0, t7 will contain M~ S s M ly which
is not the identity and has the form 81). In either case, J contains
a substitution 81) in which a and /3 are not both zero.
If a = 0, ft =)= 0, 81) is of the form L 2 ^ 4= J. If a 4= 0,
J contains the transformed of 81) by ft, 1,2, giving the substitution
Taking A = ~ /3/ 2 , this becomes JVj, 2j a. Then J contains
82) Z 1} _, = JV 1|8itt . Jfr^M^^^^ir 1 ^!^,^^^,!).
Transforming by P 12 , we reach Z 2 , a 2 - In either case, <7 contains
a substitution of the form Z 2) ^ (A =j= 0).
We next prove that J contains all the generators Lf t/l , Mi and
of the group G. Having L*^, J contains the product
T 2 ~t 1 L 2 , zT 2)t = L 2 , i ^ (r any mark 4= 0).
THE ABELIAN LINEAR GROUP. 97
The product of two such substitutions gives L 2 , z. (^ + * a ). But, by
64, marks ^ and r 2 can be found in the G F [p n ] , p > 2, such
that t\ + x\ has an arbitrary value p in the field. Hence J contains
Z 2}/u . Then I contains the product
Hence 7 contains L^ and J/i, the transformed of L^^ and Jf 2
respectively by P 2t . Finally, J" contains 1 )
/, /,/*== -, j, i
it n L jt p Q^ ^
b) Suppose, however, that S 3 = J. Then $ 2 is commutative
with N2 so that
Applying the Abelian conditions I2 13 = J? 23 = 0, we find that <? 12 == 0,
= so ^ na ^ ^ becomes
S 2 is not the identity since S l is not. If y 1]L = 0, S 2 is of the form 81)
considered under case a). If y n =j= 0, J contains $ 2 , the transformed
of 2 by ft,i,i> wnere ^ = "
J *1 ==
ly.-
l
For d = 0, 5 2 = L ltYll . For d =)= 0, J contains the transformed of S' 2
by Ti^l^p, 1 and ft being arbitrary marks =$=Q,' giving the sub-
stitution y __ g , ,2 ' 77
fl .*! ' '11 W* '1 "l*
Forming the product of two such substitutions and noting that,
for p > 2, the equation k\ + #j = ^ has solutions in the 6rjP [p n ] for ^
an arbitrary mark =|= of the field, we find that J contains
In, !*,/: S-=6i+^i, ? 2 = fe + ^%;
where a and /3 are arbitrary marks =)= 0. A suitable product of two
such substitutions gives
Li, a L<2 t p - Z/i, Jkji, *= = ii, 2-
In every case we reach in 7 a substitution Zi, ^, where A =}= 0, and
therefore also L 2j ^- It follows as in case a) that J=G.
117. Theorem. For p = 2, $ J.(2m, p w ) ^s simple except when
m = 2, p n = 2, and when m == 1, p n 2.
1) We might reach ^Ti, 2, by 82) and then obtain Ni,j,p in the group J.
DlCKSON, Linear Groups. 7
98 CHAPTER II.
For p = 2, a substitution S of Gr = SA(2m, p n ) is commutative
with every L,-, i and every L( % only when S is the identity. Proceed-
ing as in 116, we find that a self -conjugate subgroup J of 6r,
which contains a substitution $ =}= 7, will contain either a substitution
of the form 81) with a and ft not both zero or else a substitution S%
of the form 84) in which y u =)= 0.
We next prove that J contains either jC 1? * (A =j= 0) or else
JVi, 2, i 2, i. For d = 0, S' 2 = Li, yu . For <? =4= 0, we transform S{ by
a suitable T^ \ T 2 , ^ and obtain the substitution L^ i jL 2 , i- Hence J
contains 1 ) n _i r r ^ AT r
Vl, 2, 1 -^1, 1 -^2, 1 Vl, 2, 1 = = -Wl, 2, 1 -^2, 1-
For a = 0, 81) becomes J^,/?, so that we reach L^p in J. If
/3 = ; 81) becomes JVi,2, a, so that ; by 82), J contains L^ ^.
Finally, if a. =[= 0, /3 =4= 0, the transformed of 81) by I^T^^ gives
the substitution
In the 6r.F[2 rt ], we may take
p^p-l/2, ^ = a -l^-l ?
when the last substitution becomes N^ 2 , 1^/2,1-
Having a substitution .Li^ (A =|= 0), J will coincide with 6r.
Indeed, Ji^ transforms L it x. into L^m. Since every mark of the
field is a square, we reach Z/ 1? a> (3 arbitrary. Then, as at the end
of case a) of 116, J contains every L^ 0} M iy N^^a and hence
coincides with 6r.
There remains the case in which J contains Ni t 2, i L^ i . Then
J will contain all the products, two at a time, of the substitutions
85) L;, i, Mi, N fjl i (, j = 1, 2, . . ., m; i 4= j).
Indeed, if * and j be any two distinct integers < m, J contains
(Pu ?v )- -ZVi, 2, i 2, i (Pi; P 2 >) = Nij, i i/, i - A- i JV U> i,
Our statement is therefore proved if m = 2. If m > 2, let i, j, & be
any three distinct integers < m. Then J contains
i,^ i Li, ! - L ( - 1 M k = Ni t j, iM k =M k N fjtl .
1) This relation follows from 83), if p = 2, by taking i = 1, j = 2, jit = 1.
THE ABELIAN LINEAR GROUP. 99
We next prove that, for m > 2, J contains L lt i. Since, for p = 2,
L' f , i - Mi Li, ! Ji, JB lW , ! == Jf, Jf, JV,,,, i M t M h
it follows that J will contain the substitution
D = ? JC4, 1 ^2, 1 3, 1 -Rl, 2, 1 J^2, 8, 1 ^3, 1, 1,
the latter being the product of an even number 24 of the sub-
stitutions 85). This product is seen to be
D: iJ-i,-, itf-ifc+ii+is+is (*~1, 2, 3).
But D is transformed into Z lfl by the following Abelian substitution
of period two:
Ii *?i + fe + i s > ^ - ii + i, 4- 1 8 >
*2 = S2> ^2 == 1 ~f~ % ~f~ 2 ~f~ *?2 + is>
ii = 3; ^3 ii + ??i + is + I 3 + ^3-
Hence J contains L it i and therefore also
L,- t i Zi, i Za ? i = i/, i T^ * LI, i T{ t * = LI, &, Mi L^ i L^ i = M i7
Ni, it i L i} i ii, i = N{, y, i , Ti, * N it y f X T,- f = 3/i, jt 1 .
Hence , for p = 2, m > 2, J" is identical with 6r, so that 6r is simple.
For j) = 2, m = 2, 7 contains M M 2 as above , and therefore also
Hence J contains every T^ a . But ^ 1?2 , i transforms T 1}(X into
-Bi, 2, *(i + a) ^i, If w > 1, the G-F\2 n ] contains a mark a neither
zero nor unity , so that 1 -f- a =|= 0, a =)= 0. Hence, for M > 1, the
group J" contains ^1,2, ^(i-f ) = J^i, 2,1, by proper choice of L It
therefore contains N it 2, i . Having the products in pairs of the sub-
stitutions 85), J contains Mi and Z/^1. Thus J=G.
The fact that the case m = 2, # = 2, n = 1 is exceptional is
shown in the following section.
118. Theorem. - The Abelian group SA(4, 2) on four indices
modulo 2 is holoedrically isomorphic with the symmetric group on six
letters. 1 )
By 264 of Chapter XIII, the symmetric group on 6 letters is
holoedrically isomorphic with the abstract group 6r 61 generated by
J5 17 J5 27 B 3 , J5 4? _Z? 5 subject to the generational relations
-I,
1) This theorem was first proved by Jordan by means of the groups of
Steiner, Traite, No. 335. The proof given in the text is due to the author,
Proc. Lond. Math. Soc., vol. 31, pp. 4041.
7*
100
CHAPTER II.
To the operators Bi we make correspond the following substitutions
of SA(4,2):
86) JBi^JMi, Bt^Ii^ty I?3~> S, .Si^Z^, i, JSs^JHj,
where S denotes the Abelian substitution of period two:
0111
1011
1101
1110
We readily verify that the relations corresponding to the above
generational relations are satisfied in virtue of the correspondences 86).
Since $^.(4, 2) has the order
(2 4 -l)2 3 (2 2 -l)2~6!,
the isomorphism between SA(4, 2) and GQ\ is holoedric.
119. In determining the factors of composition of the general
and special Abelian groups on 2m indices with coefficients in the
G~F[p n ], we have been led to a quotient -group, SA(2m, p n )/K,
where K= { 7, T] is of order 1 or 2 according as p = 2 or p > 2.
Owing to the great importance of simple groups, we will designate
this quotient -group as A (2m, p n \ it being a simple group except in
the three cases m = 1, p n = 2; m = 1, p n = 3; m = 2, p n = 2, when
its factors of composition are 2, 3; 2, 2, 3; 2, 6!, respectively. The
order A[2m, p n ] of A (2m, p n ) is
1
V. _t """ / JJ \.A~ ) XT * * * \-t "~ / JL )
Cv
where a = 1 or 2 according as p = 2 or p > 2.
Conjugacy of operators of period two 1 } in SA(2m,p n ) and A(2m,p n ).
120. Theorem. - Within the special Abelian group SA(2m ) p n ')
any substitution S defined by 75) is conjugate with a substitution
which replaces ^ and % by the respective functions
- 1 1?m - 1 +
1 m
+
either #i m _i = or
The theorem is evident if lt - = y lf == /5 lt - = d\j = (* = 2, . . ., m).
In the contrary case, we may suppose that a lOT , y lwi , /3i m; d\ OT are
not all zero, first transforming S by P im where i is a certain one
1) Taken from the author's article, Quarterly Journal, vol. 32, pp.42 63.
THE ABELIAN LINEAR GROUP. 101
of the integers 2, 3, . . ., w. According as 1TO =%= 0, yi m 4= 0,
or dim =j= 0, we transform $ by J, Jfjjj -3fi* or -Mi -Mm respectively
and obtain a substitution /S" in which i m =}=0. Transforming S'
by -L/n,2, we obtain a substitution /S" which replaces | t by
"llll + yil 1 ?! + ---- f" lmlm+ (Pirn ^lro)ty.
Since #1 =|= 0, we can choose A in the field to make the coefficient
of y m vanish. Transforming S" (in which now I TO =4= 0, yi m = 0)
by L'i, Q, we reach a substitution S which replaces | 1? ^ by
respectively ,
Oil
We choose Q to make /3i TO 4- ^im= 0. Hence S l has i m =f= 0,
^lm = film U.
We next determine an Abelian substitution which affects only
the indices %%, 172, % m > tjm and which transforms S into a substitution
$ 2 having lm 4 0, yi m = ft m == y 12 = ft 2 = 0.
a) Let <Y 12 = y 12 = 0. If # 12 = 0, the transformed of S by Jf 2
gives /S 2 . If /3 12 and ^ 12 are both not zero, we transform $ t by L' 2 , Q ,
where /? 12 Q d\ 2 = 7 and obtain $ 2 .
b) Let ff 12 and y 12 be not both zero. Transforming by M 2
when y 12 =|= 0, we may suppose that # 12 =|= in S v Transforming it
by L 2 , $ , we can make y 12 == 0. If then $ 12 =J= 0, we transform by
La 5? and make /3 12 = 0. Suppose, however, that # 12 == 0. If di TO =4=0,
we transform by jR 2 , m, 2, where /3 12 -f- ^^im= 0, and reach $ 2 . But
if ^ lm ^0, we have $ 2 if /3 12 = 0; while for /3 12 =4=0, we transform
by Qm^^Mi, where 12 - Qa lm = 0, and reach S 2 .
In an analogous manner, we can determine an Abelian sub-
stitution which affects only 3, 773, w , ^ m and which transforms S 2
into a substitution $ 3 having
lm 4= > yi2 = ^12 = XlS = ft 3 = y lm = /3 lm = 0.
Repeating the process, we may also make
= 04 === y\ m \ = Plm l = 0.
We therefore reach a substitution S conjugate with S within the
special Abelian group and replacing | 1; rj 1 by respectively
Transforming S by Qm,*, a y where 12 (?i TO =0, we obtain a
substitution of the form S but having 12 = 0. Similarly, we may
make 13 = = a lm _i = 0. If, in the resulting substitution S v
102
CHAPTER II.
d
12
= di m = 0, we have reached Z. If $i m =)= 0, we transform
Q-2,
m,aj
where
=0 ; and reach a substitution of the
2 ,
form 8 but having also d l2 = 0. In a similar manner we make
# 13 = . . . = dim i = and reach Z. Finally, if #i m = but d
$13, . ? #im i are n t all zero, we may suppose that di m _i
first transforming by some P t - m _i. We then transform it by ,
for i = 2, 3, . . ., m 2 in succession, and make
so that we reach Z.
*^ ^13 = ' ' ' = dim 2 = 0,
Corollary. - - If an, y 1? -, /5i t -, <J lf (* = 2, . . ., m) are not all zero
in S, it is conjugate within SA(2m, p n ) with one of the two types of
substitutions:
\mj
Z 2 :
A-fti6i4
Since the conjugate substitution Z then has a\ m ={= 0, we may
transform it by T m , a lm . Then if di m _i=0, we have Z r In the
contrary case, we transform also by I^Ti, dimi and get Z 2 .
121. Theorem. - The special Abelian group SA(%m, p n ~), p> 2,
contains exactly m sets of conjugate substitutions of period 2. I he
r i}1 set includes
' . *i2r(m r)
substitutions all conjugate with T r = l^^T^i . . . T r ,_i.
In order that the special Abelian substitution 75) shall be
identical with its reciprocal 77), for p = 1, it is necessary and
sufficient that
* n /?/'"' 1 nnn \
ij ~ ji) Vij ~ ~ Vji) Pij = ~ Pjh v; J = *?'') *)'
Every substitution of period 2 of SA(2m,p n ),p>2, has therefore
the form
^11 (X-tv Vt9 ... CC-im Vim
'11
im
C 22
Pirn &lm ~ P2m #2m v & mm
For m = l, we have cc^^l, so that 2i 5 _i is the only sub-
stitution $. In order to prove the first part of our theorem by
induction, we assume that every special Abelian substitution in the
THE ABELIAN LINEAR GROUP.
103
GF[p n ], p > 2, on t < m pairs of indices is conjugate within the
group SA(2t,p n ) with one of the substitutions T r (*"<) and proceed
to prove that a like result holds for m pairs of indices. In view of
120, we may suppose that S has one of the three forms Z 1? Z 2
or S , the latter having KH = y lf = fa = 8u = (i = 2, . . ., m).
An $ of the form S is evidently a product TI, + I&, where $ 2 affects
only the m \ sets of indices | 2 , %> > S> ^m- By hypothesis,
S 2 is conjugate with one of the products, J, J 2 , 1> ^2, i -^s, 1> ?
T 2 ,_i T 3) _i . . . T Wi _i. Hence an 8 of the form ^ is conjugate with
some l r (r = 1, 2, . . ., m). We proceed to consider Z t and Z 2 in
the following three cases.
Case a), d ==(= in Z r Then $ has the form
11
...
1
]
11
...
d
22
. - 2m-l
y2m 1 #2m
^2m
o""
22
ftm 1
ff..-i A.
to
*
"2m
~y2n
i "m 1m
/ m 1 m "m m
1
-ft m
2jj
^ - Pm 1m
m 1m
"mm.
The Abelian conditions 79) give at once
_ O ( ' f) ., -\\ -y I O
Wim == Pirn == ^ir/i === Oj TO ^ ^, . . ., //t- - ly, WH -}- Ct mm U.
Hence Zj = Z' t Z!/, where Z^ f affects only the indices
t' (i V w> - 1^1
,:, ^ \i -- &, . . ., m ij,
while Zj affects only |i, ^i, | m , ^ m , viz.,
fcf
rm
a ll
1
Ojj
d
d
-11
1
11
By hypothesis Z^ is conjugate with some product of the
In order to make the induction from one to two pairs of indices,
we must prove that
T m? _ i. Transforming
O 11 a u 4-
d
VO 1-
".^ is conjugate with a product of ^\, i and
'i by Gi, m, a; we obtain the substitution
5d d
-(?(
104
CHAPTER II.
Taking or 11 +(?d = and transforming the resulting substitution
by T f ^d j we obtain P im = (Jilm) (^i^m)- The latter is transformed
into Ji,_i by the following Abelian substitutions (and by no others):
2 (auan-
L,]
It follows that, if d =4= 0, Zj is conjugate with some T r .
Case b). d = in Z r The Abelian conditions 79) now give
= ^im = (i = 2, . . ., m
= U, a
2 __
11
1.
Transforming Z x by
TT-
where 1 2Aa n =0, we obtain
' a ll
...
'
11
...
^22
A
A
a
A
Hence W= T^ + iW, where W affects only ,-, ^ (i = 2,..., m)
and may therefore, by hypothesis, be transformed into a product of
the Tj t i by an Abelian substitution on the same indices. It will
transform W into a product of the T^ _i (j = 1, . . ., m), which is
conjugate with some T r .
Case c). In virtue of the Abelian conditions, Zg becomes
' a u
1
}
11
1
22
frm-l
2
o
o"
*22
"2m 1
0,
o
1
"2m 1
-**-!...
-11
r
1 1 TO ^
-a n
1
1
1
02m
tf 2m
^m 1m
Transforming Z 2 by the product Qi, m , iQ m i,i,a> where 1 2^a n = 0,
1 + 2tfff n = 0, we get a similar substitution but having zeros in place
of the four elements 1. Since it is of the form W, we may proceed
as in case b).
THE ABELIAN LINEAR GROUP. 105
To complete the proof of our theorem, we note that
rri _ m m m m rri m m rri
1 - . J-1,1, -is - : -Li, i J-t, !, ., J-m - - -M, 1 J-2,1 - - -t/n, 1
have the respective characteristic determinants (with parameter K)
Hence no two of them are conjugate under linear transformation.
The most general substitution of S A (2m, p n ) commutative with T r
is seen to be A = A r A m -. r , where A r is an arbitrary special Abelian
substitution on the indices | f , rj { (i = 1, . . ., r) and Am r an arbitrary
one on the indices |/, rjt (i=*r -\-l, . . ., m). By 115 the number
of substitutions A r and A m r is respectively SA[2r, p n ] and
S A [2m 2r, p n ]. Dividing SA[2m, p n ] by the product of the
foregoing numbers, we obtain the number of substitutions of
SA(2m,p*) conjugate with T r within the group.
Operators of period 2 of A(2m, p n ), 122123.
122. By 119, we obtain the quotient -group A (2m, p n ) by
considering as identical S and S T = TS, where S' is an arbitrary
substitution of SA(2m, p") and T is the self -conjugate substitution
Tj 5 _iT 2 ,_i . . . T m ,_i. In particular, T r and T r T become identical
in the quotient -group. But the latter is conjugate with T rn r .
Furthermore, if s = m/2 or (m 1)/2 according as m is even or odd,
no two of the operators Ji, T 2 , . . ., T s are conjugate within the
quotient -group. The special Abelian substitutions of period 2 lead
therefore to just s distinct sets of conjugate operators of A(2m, p n \
p > 2. To complete the study of the operators of period 2 of
A(2m, p n }, it remains to determine the conjugacy of the special
Abelian substitutions S for which S*=I. Being of period 4, such
an $ is not conjugate to any T r . Moreover, no two of the cor-
responding operators of the quotient -group are conjugate, since that
would require one of the four relations
- T nv TT A~ l (RT\A T nr TT
J. r OP _/. J-r<) -L I O . J-ti -L r OI J. JL r .
A being Abelian. But any of these would require that S be conju-
gate with some T t within the special Abelian group, whereas their
periods are different. Making use of the result of 123, we may
state the theorem:
According as m is even or odd, the group A (2m, p n \ p > 2, has
exactly (m + 2) or (m -f 1) distinct sets of conjugate operators of
period 2.
106
CHAPTER II.
123. Theorem. Within the special Abelian group on 2m indices
in the G-F[p n ], p > 2, every substitution S, such that S 2 = T, is conju-
gate with M EE Mi M a . . . M m . 1 )
Taking as S the general substitution 75), whose reciprocal is
given by 77) for f* = 1, the condition S = S~ 1 T is seen to require
= - d ) i> ?H = Vi h fai = fa ft J = 1, ' ; ).
The matrix of coefficients of the general S is therefore
8 =
Via
^
011 - 11
12
P22 K 22 ' ' '
im
Vim -02m Vim- - - <*>,
ft Q
K lm P2m -<X2m - - P*
subject to the special Abelian conditions.
Take first m = 1. Then S has the form
ll
It is conjugate with a similar substitution in which a n = 0. In fact,
if /3 11 =j= 0, the transformed of S by L^ * replaces ^ 1 by
in which the coefficient of % may be made zero by choice of L If
/3 n = 0, r n ={= fy we first transform S by M t and then proceed as
before. If /3 n = y u = 0, we first transform S by L[, % and obtain a
substitution which replaces % by 2A n ^ a u ^ 17 so that the new
ft. 4= o-
With ff u = 0, S takes the form
/ <y\
(-V-* o)
and is the transformed of M by the special Abelian substitution
(
Indeed, by 64, there exist solutions in the G-F[jf f ], p > 2, of
To prove the theorem by induction for m pairs of indices, we
assume it true for t pairs of indices t < m.
1) For the number of conjugates see Ex. 8, end of Ch. VIII.
THE ABELIAN LINEAR GROUP.
107
If ccu= /3i,-= yi,-= 8u= (* = 2, . . ., m), then $ = ^5', where
S l affects only | 1? % and is therefore conjugate with M ly and where
>S^ affects only J/, ^,- (*' = 2, . . ., m) and is, by assumption, conjugate
with M*M S . . . M m . Hence S is conjugate with M 1 M 2 M S ... Jf m
within SA(2m,p*).
In the contrary case, 8 is conjugate (by 120) with one of the
two substitutions Z 1; Z 2 . We consider the following three cases.
Case a). If Z 1? with d =|= 0, be of the form 8 above, the Abelian
conditions give
= Pim= 7im = dim= (i = 2, . . ., m 1),
a,-
Hence Z t = Z' t Zj, where ZJ_ has the form
fern -
.J
1
d
-1
while Zj f affects only | f , ^ f (* = 2, . . ., m 1) and is, by assumption,
transformed into M^M^ . . . M m \ by some special Abelian substitution
affecting only the same indices. We proceed to prove that Z^ may
be transformed into MiM m by a special Abelian substitution on the
indices |i, rji, | TO , y m . The proposition that Zj is conjugate with
.MiJfa . . . Jf TO under SA(2m,p n ) wiU then follow.
^ fti == 7ii == ^ i n ^-i? we transform it by N^ m ^i and get
-2A
-d
-
1
d
2A<5
n
This is of the form ZJ, but has y u =)= 0. Next, if y n = 0, /3 n =|= 0,
we transform ZJ_ by MiM m l m , and get a substitution of the form
Zj in which, however, y 1]L =4= 0, j8 n = 0. We may therefore assume
that y n =(= in Z' r Translbrming it by L'i t ^L' m ^/^ where A, = cc ll /'y u ,
we get a substitution of the form
F~
'
yu
i
'
ft
d
-d
dy n
-i
- Bit
x
108 CHAPTER II.
If /3 = 0, then d = 1 and the transformed of F by R^ m ,
f y u
gives
If /3 =)= 0, the transformed of F by N^ TOI ^ gives a substitution of
the form W. Since TF is the product of a substitution on the
indices 1? % and a substitution on the indices | m , ^ m , it is conjugate
with MM m .
Case b). If Z 17 with d = 0, be of the form S, the Abelian
conditions give
Ai = ym = 0, a n + TOWZ = 0, y im = d im = (* = 2, . . ., m -- 1).
Transforming Z t by .Li, 0? where y u 2<7a u = 0, we get
a ll
L
...
1
'
...
K 22
722
. . . a 2 -i
J>2m 1 #2m
A,
~22
- - - fci
"2?n 1 P2m
...
a u
-1
&2m
2
. &n-i
#m 1m Pm j/i
^11 ,
Transforming 1L[ by l7 TO) ff; we obtain a substitution Z" which differs
from Z^ only in having the coefficients 1 replaced by (1 20 n ).
By choice of (?, the latter may be made zero. Hence Z" = S^S',
where S 1 affects only J 1? ^ and is therefore conjugate with M v
while S' affects only | /? ^ f (* = 2, . . ., m) and is, by assumption,
conjugate with M^Mz . . . M m . Hence Z" is conjugate with
becomes
c). If Zg be an Abelian substitution of the form S, it
K ll
<y n
...
1
'
ftl
*11
...
1
#22
722
...
72m 1 ^2m
A. -
22
...
*,.-! ft
-1
-
- ^2 TO 1
72m I
-11
7m 1m 1 &m 1m
Vii
...
a n - ft n
...
7n ~ ii
v
-1
ftm -
ff2m
...-/5 n
C^m 1 m Pm m
11,
THE ABELIAN LINEAR GROUP. 109
Suppose first that n = 0, so that --ft 1 y n = l. Transforming
Z 2 by Ri,m,i> where 1 + A7 11 =0, we reach a substitution equal to
a product $j$', where S 1 affects | 1? % only and $' affects
only io W (f-2,...,)
Suppose, however, that a n =f= 0. Transforming Z 2 by J& OT _I I? ,
= 0,
we obtain a substitution Z 2 of the form Z 2 and having y OT _ lni _i= 0.
Transforming Z g by li, ? , where /3 m?M 2 p n = 0, we obtain a sub-
stitution Z 2 of the form Z 2 , but having (l mm = y m \ m \= 0.
If ft 1 =y 11 =0, we transform Z' 2 f by ftn 1,1,*, where 1 2Aa n =0,
and afterwards by Ci,m,?> where 1 2pa 11 = 0, and obtain a product
$!$', where S t affects only 1? % and S 1 affects only |,-, t? t - (i > 1).
If y u = 0, fti + 0, we transform Z' 2 ' by P m -. lm M 1 M m _ l M m
and get
~ K ll
~~ Ai
...
1
'
n
...
1
_1
fa
s
- m -i
Ai
...
-u .
...
Ai n
1
#2)
1 72m 1
. . .
#m 1
ff u ,
which has the form of Z 2 with y u =^= 0. We therefore treat the
latter case only. Transforming Z 2 by L' m , Q , where
m 1m
we obtain a substitution C7 of the form Z 2? but having a m _ lwl and
y m imi both zero. Transforming U by ^ OT _i, W ,A J^i, ^, where
a n -f- A^ n == 0, we get a substitution of the form
7n
...
1
'
1/7
'll
...
1
-1
$2m 1
72,
.-i ...
-7n
...
l/7n
...
-7n
-1
ftm
-2r
... 1/yj
i
Pmm
o ;
Transforming this by Ri, m ,i, where 1 -f 4y n = 0, we get a similar
substitution with the elements + 1 replaced by zeros, and therefore
the product of a substitution on | 1? ^ by a substitution on the
indices ,-, ^ t - (i = 2, . . ., m). It is therefore conjugate with
MiMz . . M m .
110
CHAPTER IE.
CHAPTER HI.
A GENERALIZATION OF THE ABELIAN LINEAR GROUP. 1 )
124. Those linear homogeneous substitutions in the CrF[j[> n ]
on ma indices.
* 7 m /
87) S'. Xij = ^ (CCjciXkl ~f" a k2%k2 ~f" ' ' ' ~f~ &kq%kq)>
^_J
k = l f -, -j \
\* = " L y > m 'i J = ~ L i - - -> Q.)
which, if operating simultaneously upon q independent sets of mq
variables, the j ih set of which is given the notation
U) U) (fi
leave formally invariant the function
(i = 1, 2, . . ., m),
iq
form a group (r(m ; q,p n ), which for g = 2 is the Abelian group
SA(m, p n ).
The conditions upon S for the absolute invariance of are seen
to be those given by formulae 88) and 89), viz.,
i=i
31
= 1
89)
il
a J Q l
/each
I each
(j = 1, . . ., m)
> 1, 2, . . ., q] x
1, 2, . . ., m, }.
' ; . /
j q not all equal/
125. The inverse of the general substitution 87) of G(m, q, p") is
90)
.s
'r., -.
-rs*'i'L\* a -rs
(r 1, ..., m; s = 1, .. .,
1) Taken from the author's paper, U A class of linear groups including the
Abelian group", Quarterly Journal, July, 1899. The group is mentioned, but
not investigated, by Jordan, Traite, p. 219, No. 301.
A GENERALIZATION OF THE ABELIAN LINEAR GROUP.
where A' r { denotes the adjoint of a J / 5 in the determinant
111
1 1
i
i q i q
Ctrl . . .
In fact, the product 87) 90) replaces x rs by
k = 1, . . ., m
I = 1, . . ., q
Here the coefficient of x k i is
1, . . . , m
!,...,$
i = 1
&
il
. . . CC rs i
a
il
il
I ... &rq
iq
cc iq
iq
CCjel
iq
a iq
and therefore, by 88) and 89), equals unity if (k, I) = (r, s), but
equals zero if (&, T) =(= (r, s). Hence the above product replaces x rs
by x rs . The reciprocal of 8 is therefore obtained by replacing
by ^L*j for i, k = 1, . . ., w; i, .; = 1, . . ., g.
Writing relations 88) for $~ L given by 90), we find
^
.
zg
i =1
a il ' ' ' K iq
'jq a jq
'il ' iq
2-1
holding for j = 1, 2, . . ., w.
Note. - - For substitutions 87) which multiply O by a constant 9,
the reciprocal is evidently obtained by replacing a* t by ---^*y-
126. The structure of the group G(m, g, p n ) is essentially different
in the two cases q = 2 and # > 2. The case g = 2 has been investi-
gated at length in Chapter II. In the following investigation we
assume that q > 2, a restriction necessary for the treatment given.
Let J 2 > Js; > Jq nave nxe( l values not all equal chosen arbitrarily
from 1, 2, . . ., m, and let k s ,k 9 ,...,k q have fixed values chosen
from 1, 2, . . ., q. Then for j = 1, . . ., m; & t = 1, . . ., q, we obtain
mg equations 89). In fact, since q > 2, jj, j 2 , . . ., j q are not all
equal and hence do not lead to conditions of the type 88). Expanding
the determinants of 89) according to the elements in the first columns,
our mq equations may be written
112
CHAPTER III.
JS" = o
1, . . . , m
!,...,<?
where
92)
in which & 2 , & 3 , . . ., i
Since the determinant
a'! 6 *
Jz "2
* b n i b n
a. I . . . K. I
3* "2 3q K q
denote the integers 1, . . ., Z 1, Z -f- 1, . i .,
a
ii
'it,
=|= 0, being the determinant of $, we have
(*1 \
= 1, ..., m\
l = l,...,qJ
Hence the determinant 92) vanishes for i = 1, . . ., m and for
& 2 , & 37 ? &? an arbitrary combination of g -- 1 distinct integers < q.
If # = 3, we have reached the relations 95) below. If q > 3,
we denote by (/!*! the adjoint of a! 6 * in the determinant 92) and
consider the following expansions:
93)
= 0.
Of these consider the mq equations in which i, j 3 , . . ., j q have fixed
values chosen arbitrarily from 1, 2, . . ., m, but such that j 3 , j, . . ., j q
are not all equal, and Jc 3 , . . ., ^ ? fixed values chosen arbitrarily from
1 ? 2, . . ., q, while lastly j 2 takes the values 1, 2 7 . . ., m and & 2 the
values 1, 2, . . ., q. Since the matrix
comprises q 1 rows of the matrix of $, not all of its determinants
of order q 1 are zero. Hence the q -- 1 determinants 0, which are
the same in each of the mq equations 93) , must be zero, viz.,
94)
ic n
. . . a. q ,
where c 3 , . . ., c q denote any q 2 distinct integers < q.
If q = 4, we have reached the relations 95) below. If q > 4,
we proceed as before. After q 2 such steps, we reach the set of
relations
a*r a*, r ,, / ? -i m . -I
95) = ( ' ^ ^, , ; ".
/ rolS rtilS \ nf O If L* 1 I /
\ / o n/ A/ J. . u
S / S * f -*
a*;
A GENERALIZATION OF THE ABELIAN LINEAR GROUP.
113
In virtue of the relations 95), the conditions 89) all reduce to
identities. In fact, in each relation 89), at least two of the j's are
distinct, say j =$=j z , and therefore all minors formed from the first,
and second columns vanish in virtue of 95).
A substitution S belongs to the group G(m, q, p n ), q > 2, if and
only if its coefficients satisfy the conditions 88) and 95).
127. Theorem. - - Every substitution S leaving O invariant can
be derived from the totality of linear substitutions of determinant unity
on q indices
q
(j = l, ..., q),
together with the linear substitutions, each on 2q indices,
/ Y* ^1T i ( Q ^ ^^ I OT7 i
We can evidently derive from these generators a substitution T
which belongs to G(m, q, p n ) and replaces an arbitrary index Xki by
any particular index as x lv We may therefore suppose that in the
product S'~TS, S being defined by 87), the coefficient JJ 4= 0.
If then we set
)* = (7jF*ag (j = 2, . . ., m; A; = 1, . . ., q)
it follows from 95), for * 1, r 1, j' 1, # 1, j > 1, that
96) J- = }f ( j = 2, . . ., m; fc, s = 1, . . ., g).
Substituting these values in the relation 91) for j = 1, we find
2-1
,11
c.
/ 7
9-1
= 1
JL
It follows that
11
Hence the following substitution is of determinant D =f= 0.
.' = a ii x -f . . . -f a{ 1 XL
^lq
DlCKSON, Linear Groups.
114 CHAPTER III. A GENERALIZATION OF THE ABELIAN LINEAR GROUP.
If we denote the determinants of <J> by D t so that O E
we readily see that E multiplies Dj by the factor D but leaves
unaltered D t - (i = 2, . . ., m). Hence, if W denote the substitution
"FT: a?[-i = Dxu (i = 2, . . ., w)
the product TF.R multiplies by the factor D. The product
^^(TF-R)" S' multiplies by D~ and therefore satisfies the
relations 89) and consequently also relations 95), derived from them.
But S l ajffects the indices # u , x 12 , . . ., x\, q as follows:
where J{ denotes D times the earlier a*^, for fc = 2, . . ., m. For
the substitution S t we have ajj = (s = 2, . . ., q). Hence by 96),
Also eel] = (s = 2, . . ., q), J* = 1. Hence, by the following cases
of 95), a ii a ii /? = 2 w& = l q\
f\
we find ajk = 0. Hence every a** = 0, for j > 1, so that ^ leaves
~h "v*o/i /Y* /Y* *Y*+
I 1 -A. Vy Vl. tA/-| I A/-| O 5 * * * 9 *^1 Q *
Applying the Note of 125 to form the reciprocal of S ly we
find that the matrix of ST has zeros throughout the first q columns,
except the diagonal terms D in the first q rows. By the above
argument, the remaining elements of the first q rows must be zeros.
Reciprocating this matrix by the same rule, we find that D == 1 and
that S 1 reduces to a substitution on the indices
/ * C\ \
/y /yi /y . I /| --- - T/ /^M I
v/lj ^}3) ; *^^? \</ ^ * *l "^y*
Since T7 is the identity, S = T~ l S' = T~ 1 ES^ where 1 and 12
are derived from the generators given in the theorem. Proceeding
with $! as we did with S, we reach a substitution $ 2 on the indices
. . ., Xj q . Finally, we reach the identity.
128. It follows from 127 that the group G(m, q,p n *)> q > 2,
has an invariant subgroup f composed of the substitutions
Q
f -j a -j \
ik > ? ' '? ? J
where, for i = 1, 2, . . ., m, the determinant
= 1 ( j, fc = 1, . . ., 3).
CHAPTER IV. THE HYPERABELIAN GROUP.
115
The quotient -group is generated by the substitutions P,j and is thus
holoedrically isomorphic with the symmetric group on m letters. The
group f is the direct product of m groups each the special linear
homogeneous group in the GF[p n ] on q indices ( 103). The sub-
stitutions of the * th group are given as follows
<i
III C i t * 1 "I fl\
1, ...,!,*',/- A, ..., (I).
The structure of the group G (m, q, p n ) is therefore completely de-
termined.
CHAPTER IV.
THE HYPERABELIAN GROUP.
129. The totality of linear homogeneous substitutions in the
8: SS = >'M* ( = !,..., 2m)
which leave absolutely invariant the function
v
~
i
2i 1
n
: P
1
forms the hyper abelian group 1 ) H(2m,fP*). Its name is derived from
the fact that the totality of its substitutions whose coefficients belong
to the included field GFlfP] constitutes the Abelian group
which is therefore a subgroup of the hyperabelian group.
A general substitution 8 transforms Y into
j, *=!,... ,2?n
--
J-\-
Ik tl*
/
The conditions upon 5 for the absolute invariance of Y are thus
97)
2
(j, *-l, ..-,
where ^ t = 7 unless j and k differ by unity, when
1) Introduced by the author, Proc. Lond. Math. Soc., vol. 31, pp. 30 68.
It will hardly be confused with Picard's hyperabelian group of infinite order.
8*
116 CHAPTER IV.
The reciprocal of the hyperabelian substitution S is
m
< ^. 1 / *)W }- - $
9; 1 = > (cc p a p
/ ! \ 2j 2 1 *2j 1 2j 1 2 1 ~
1
S~ L :
(Z - 1, . . ., m).
Indeed, the product SS replaces | 2 z i
1, . . ., m
^
/,*
'2J21
52*:
/,*
'S.? 1 2/fc t*2;2/fc
f 2; 1 2 i 2/ 2 I
~
Similarly, $$~ replaces
m
"
The relations 97) in which j > k are derived from those in
which j < & by raising the latter to the power p n . We may there-
fore express the hyperabelian conditions in the convenient form
98)
Ik
11 (if fc = j + 1 = even)
(unless ft = j + 1 = even)
( j, It = 1, . . ., 2m; j
The corresponding relations for 8~~ are found by replacing
by respectively
2.;
/ Hi;
Writing out the four sets of relations 98) according to the evenness
or oddness of j and fc, and making the replacement just indicated,
we obtain four sets of relations for the invariance of Y by the sub-
stitution S~ L and therefore together equivalent to the relations 98).
We may combine the four sets into the single formula
1 (if ~k == j -f- 1 = even)
(unless k = j + 1 = even)
(j, &-!,..., 2m; j ^ fc).
5 hyperabelian substitution S must
99)
130. 2Ae determinant A o/"
satisfy the relation
100) A^
THE HYPERABELIAN GROUP. 117
For proof, we reflect on its main diagonal the determinant of
$~ 1 , then change the signs of the 21 -- I 8t row and column for
I = 1, . . ., m, and finally interchange the 21 I 8t row with the 2Z th
row for I = 1, . . ., m, and likewise interchange the corresponding
columns. We obtain the determinant
<J
1
Hence AA* W = 1, being the determinant of the product SS
131. Theorem. - The maximal subgroup M of the hyperdbelian
group H(2m, p 2n ) which transforms into itself the Abelian group
SA(2m, p n } is given by the extension of the latter by the substitution
V Q ' i2/-i piai-i, 61. .fT^lii (Z = l, ..., m),
where Q is a primitive root in the GF[p 2n ]. The index of SA(2m, p n )
under M is p n 4- 1.
We determine all hyperabelian substitutions
2m
ntl ('-I,--., 2m)
which transform the Abelian group into itself. Now 8 transforms
the Abelian substitution , affecting a single index,
52r 1=
into the substitution
I'/ = !,-+ ilr-
(*-l,..
whose coefficients must therefore belong to the &F[j>*], viz.,
2r-i a ^2r-i ft j = 1, . . ., 2w; r = 1, . . ,, m).
Likewise, /S must transform the Abelian substitution
?2r = ?2r 4" ?2r 1
into a substitution belonging to the 6rjP[p n ]. Hence the products
/2r} r ft j = 1, ., 2m; r = 1, . . ., m)
must belong to the GF[p n ]. The reciprocal S~ must transform
the Abelian group into itself. From the above results, it follows
therefore that the products
i.S"_ lt (,< !,.. ., 2m; r = l, . . ., m)
must belong to the GF[p n ]. Combining our results, every product
118
CHAPTER IV.
j /\ H \ /i- ^ A W /
101) CtirCC p . CC r ,-CC P . (l. 1, V
/ } r y T i \ 7 t/ /
must belong to the GF[p n ].
But, if /3, y be marks of the GF[p**] such that
^ w =^ - = mark of
then, if ^4=0, p/y = pir-p*-i is a mark of the G^F^]. Hence
by 101), the ratios of the non- vanishing coefficients in any row or
any column of the matrix of S must all belong to the GF[p*].
Suppose first that m = 1. If cc n =f= 0, we have
21 = A u , 12 = ft u (A, ft in the GF[p n f).
Then if A and p be not both zero, 22 = v n , v being in the GF[p*\.
For A = ft = 0, the hyperabelian condition gives Oui"~l, whence
22 = va 11 . If, however, a n = 0, both 12 and 21 are not zero.
Hence 22 = pa 12 , (> in the ^JPf^? 71 ]. By the hyperabelian condition,
- 12 of|"=l, whence # 21 = <? 12 , ^ in the 6r.F[j;> n ]. In either case,
we have reached a substitution of the form 103) below.
For m > 1, S transforms the Abelian substitution
into
=(=
Hence the sums
l ^2r-l ft # V' . .,
must all belong to the (rjFjj) 71 ]. In like manner, if S transform
each of the following three Abelian substitutions (in which r =|= $)>
,2rl
** I *"
T>* /* ^~" 1 II fei 5
52s 2* ~ 2r5
:' fc fc
i2s 1 = b2 1 b2r :
; 2s
into substitutions belonging to the GF[p n ], then must the respective
sums
tf/2s
./ * / j.
t,j-l, ..., 2m\
belong to the 6rjP[^> n j. Combining our results, every sum
102) ^ a*f s + ^ s ccf r (i, j,r,s = l,...,2m; r 4= s)
belongs to the G-F[p n ].
THE HYPERABELIAN GROUP. 119
Of the coefficients in the i ih row of the matrix of S, we may
suppose that ec,- r =J=0, for example. If, then, oy r -f0, the ratios of
the coefficients in the i ih and j ih rows must all belong to the GF [p n ]
[by the result following from 101)]. If, however, y r =0, we may
suppose that, for example, oy, =)= (s =j= ") Then, by 102), the
products a t> ?" belong to the 6rjP[^ w ]. We have in either case the
result that the ratios of the coefficients in the ^ th and j ih rows belong
to the GF[p n ]. Hence the ratios of all the coefficients in S to any
one non- vanishing coefficient belong to the GrF[p n ], so that 8 may
be written 2m
103) i} = a^ In 6, (i 1, . . ., 2m),
.7 = 1
where the A,,- belong to the G-F[p*].
Inversely, every hyperabelian substitution of the form 103)
transforms into itself the Abelian group defined for the GF[p n ].
The conditions that 103) shall be hyperabelian are
wn
104) V
J 1*
(if A; = j + 1 = even)
(unless Jc = j + 1 = even)
(ij ^==1, . . ., 2m; i <.;).
The substitution (A^-), or 103) with the factor a deleted, therefore
belongs to the general Abelian group GA(2m, p n ) and multiplies V
by the mark cc~ pn ~~ 1 of the GrF\_p*\. If then we set
105) a _ == A i
we find that S = / F a , where
2m
U: l}-^^ (^ = 1,..., 2m),
/Si
F a : S^-i=ia/-i, i f 2=~ pn i2z (? = 1, ..., m),
so that V ay and therefore also Z7, is a hyperabelian substitution.
Moreover, in virtue of the relations 104) and 105), U belongs to
the special Abelian group SA(2m, p n } and is therefore of determinant
unity. The first part of our theorem is therefore proven.
If we form a rectangular array of the marks =f= of the GF[p 2n ]
with those belonging to the GF[p n ] as first row, the
p n + l~(p* n I)/ f (p n 1)
"multipliers" form a set of marks 1? # 2 > > V+i such that none
of their ratios belong to the GF[p n ], while every mark of the
GF[p 2n ] not of this set has with some mark of the set a ratio
belonging to the GF[p n ]. Furthermore, the product
120 CHAPTER IV.
belongs to SA(2m,p n ) if and only if a 1 a' belongs to the GF[p n ].
It follows that the substitutions V a . (i = \, 2, . . .,jp w -fl) give the
totality of substitutions V a such that F'F7 does not belong to
SA(2m, p n \ Hence an identity of the form
UV a . = U' V a . (i and j ^p + 1; i + j)
is impossible when U and V both belong to SA(2m, p*\ Every
hyperabelian substitution 103) is therefore of the form 7F., i being
chosen from the series 1, 2, . . .,^? ra + 1, while an identity 7F. = U'V a .
requires *=,;', 7= U'. Hence the number of distinct substitutions 103)
is (p n -\-~L)SA[2m, p n ]. The second part of our theorem is there-
fore proven.
132. Those substitutions of the hyperabelian group H(2m,p 2n }
which have determinant unity form a self - conjugate subgroup H' of
index p n + 1. In fact, for G any mark =)= of the G-F[p 2n ], the
substitution
SI- ok, Si-*-*"!,, SS-S< (* - 3, . . ., 2m)
belongs to H(2m,p 2n ). Its determinant a"^""^ can, by choice
of 0, be made equal to any one of the p n -\-\ roots of A pn + x = 1.
Hence there exist hyperabelian substitutions whose determinant A is
any root of this equation. By 130, there are no other values of A.
The group H' contains a self- conjugate subgroup formed by the
substitutions
106) T x : |5 = x& (i = 1, . . ., 2m) \v? m = 1, x^+i == 1J.
The quotient -group will be denoted by the symbol HA(%m, p 2 ").
It will be proven simple except in the special cases m = 1, p n = 2 or 3
( 138, 145, 148). By the same references its order HA[2m,p 2n ] is
L(^2m_ l)r)(2w 1) fpn(2m 1) _j_ l)^w(2m 2) ^ ^ < ^2 M _ ^^
where q denotes the greatest common divisor of 2m andp w + 1- The
order of H(2m,p* n ) is
The Abelian group SA(2m, p n ) has an invariant subgroup formed
by the identity and T_I. The quotient -group A(2m,p n ) is simple
except in the three cases m = 1, p n = 2; m = 1, p n = 3; m = 2, _p n = 2
( 119). But H(2m,p 2n } contains SA(2m, p n ) as a subgroup. In
order that T x shall belong to the latter, the coefficient H must belong
to the GrF[p n ]. But j^ w =% and xP w + 1 = l require K 2 =l. Hence
THE HYPERABELIAN GROUP. 121
would T x be the identity or T_!. It follows that A(2m, p n ) is a
subgroup of HA (2m, p n ). We proceed to determine the number of
conjugates to the former group within the latter group, using the
result of 131.
133. Theorem. - - The largest subgroup M' ofHA(2m, p 2 ") which
transforms A (2m, p n ) into itself is identical with A(2m, p n ) ifp = 2
or if p > 2 and p n -f- 1 contains a higher power of 2 than m contains;
in the remaining case, the order of M' is double tlie order of A(2m, p n \
The determinant of S = UV a being supposed to be unity and
that of U being unity, it follows that V a has determinant
-|A7\ /v m(p n 1) 1
iu<; a i.
Now V a and T x V a correspond in the quotient -group HA(2m, p 2n )
to the same operator. We investigate the conditions under which
T x V a has its coefficients in the GF[p n ]. The necessary and sufficient
condition is seen to be
Hence must x 2 =a pn ~ 1 and therefore
i
or a must be a square in the GrF[p 2n ]. The remaining condition
K 2m = 1 becomes an identity in virtue of 107). Hence, if the solutions
of 107) are all squares in the GrF[p 2n ], the substitution 8 = UV a
will correspond in the quotient -group to an operator belonging to
A(2m, p n }. But, if there occur not-squares as solutions of 107), the
resulting substitutions V a may be expressed as products V V V^, v being
a particular not -square. Then Vp. corresponds in the quotient -group
to an operator of A(2m, p n ), while V* does not. In this case the
group A (2m, p n ) is transformed into itself by a subgroup of
HA(2m, p 2n } of double the order of A(2m, p n ).
For p = 2, the theorem follows at once since every mark of the
G-F[2 2n ~] is a square. For p > 2, we are to determine in what
cases 107) has as its solutions in the CrF[p 2n ] only squares. A
common solution of the pair of equations
108) (p w -i) = l, a ^-i = i
is required to be a solution of a 2 (p 1} = 1. A common solution
of 108) satisfies a c?(pW ~ 1) = 1, where d is the greatest common divisor
of m and p n -f 1. The condition is therefore that d shall divide
^
y (p n + 1). It is satisfied if, and only if, p n -f 1 contains 2 to a
higher power than m does.
122
CHAPTER IV.
Corollary. If g = 1 or 2 according as the order of M' is
equal or is double the order of A(2m, p n ), the number of subgroups
of HA (2m, p 2n ) conjugate with A(2m, p n ) is
HA[2m, p 2 n ] ~ g A[2m, p n ] =
where a = 1 if p = 2, a = 2 if p > 2, and g denotes the greatest
common divisor of 2m and ^* -f- 1.
134. The conditions that the quaternary substitution in the
f h | <-
> *2 == a 22?2 T ^24 ^4?
1 AQ\
)
' fc I fc f
bl == a ilbl T a l3b3> *2 == a 22
' t I t tl
5 3 a 3l*l T W 33b3? ?4 a 42
shall be hyperabelian include the following:
Setting A^agga^ ^34^42? we
A n n ,si A , ?i , w
-MM. P=== "U> 31 AP =-f4^ 1
2l + U 5l K ll = 0,
these conditions that
The above substitution then takes the form
/
(
a 44 P
A
ll +
Inversely , the substitution T is seen to leave absolutely invariant
if 22? a 24? a^, a 44 belong to the GF[p 2n ], so that T belongs to
H(4, p 2 n ). The totality of the substitutions T forms a group G
holoedrically isomorphic with the general binary linear group
GLH(2, p 2n ). Among the substitutions T occur the simple ones of
the form
'37
i i
where J. and 5 are arbitrary marks of the GF [p 2 "] such that J. =4= 0.
We proceed to determine every hyperabelian substitution
8:
(' -I,--., 4)
which transforms the subgroup G into itself. The product S
i
THE HYPERABELIAN GKOUP. 123
must belong to 6r. Hence the coefficient of 2^ 1 must vanish if i
be even and that of | 2 ^ if * be odd. Taking first 5 = 0, we find,
after dropping the common factor ( 1)*,
^-"* - > A + a., f 4 -4*" - 0,4 < .i- 1 = 0,
where i and A; are both even or both odd.
If p n > 2, this leads to an equation in A of degree 2p n <p 2n 1.
Being true for every A=$=Q, it is therefore an identity, so that
110) ?ia*2 = 0, /sfc4 = (i, fc both even or both odd).
Taking next the terms in 5, which can have two values =J= 0,
we find
111) 2 3^2 = (i, Jc both even or both odd).
Similarly, if S transform the following substitution of the form T,
T 2 : _/ " 3
into a substitution of 6r, we find from the terms in C that
112) /i a*4= (i y k both even or both odd).
If any a/,- =f= 0, i and j being both even or both odd, the sub-
stitution S reduces to the form 109) and must therefore belong to 6r.
In fact, the relations 110), 111) and 112), holding if p n > 2, may
be combined as follows:
113) a'2/_i2/-ia2;fc-i2,i=0, astst-ias*** "P (^, ?, A;, Z = 1, 2).
Hence, if c^*-, 12*, i H=0, we get 2it-i2^=0 (&, A = 1, 2). Then,
for fixed A, a^j^i is not zero for both k = 1 and ~k 2, since other-
wise all the coefficients in the 2A th column would be zero and
therefore the determinant of S would vanish. It follows therefore
from the second set of relations 113) that cczi^i 1 = (*, I = 1, 2).
Hence $ has the form 109). Similarly, the hypothesis 2 ^2^=1=0
requires, successively,
s 2*-i = (i, 1 = 1, 2); a a *-i2;.= ft A = 1, 2).
124 CHAPTER IV.
I
If every a,-,- = 0, when i and j are both even or both odd, for
p n > 2, 5 reduces at once to the form
** ' < j. <. f iti **
fel == a l2t2 ' ^14^4? fe 2 == ^21 1 ' ^23 ^3 7
SS^ a 32*2 ~^~ a 34fe4> 4 = ^41 fel H~ <*43 fe3'
This is of the form Vg, where g is of the form 109) and V denotes
the hyperabelian substitution not in G,
* ' 1 = ~t>2> b2 = l; $3 = '4J fe4 == te3'
The theorem stated below has thus been proven for p n > 2.
For # w > 2, we consider the reciprocal of $ and find the
conditions corresponding to 111) and 112) that S~ l shall transform
T! and T 2 into substitutions belonging to 6r, viz.,
114) i*ff4*=0, 2i3i=0 (i, k both even or both odd).
By 111), 112), 114), S must be of the form g or Vg, g being of
the type 109). To illustrate the method of proof, let cr 13 =J= 0. Then
# 41 = #43=0 by 114). Since o^ and 44 can therefore not both
vanish, 12 =a 14 =0 by 114). Likewise from 111) ^ 12 = 32 =0,
# 23 = #43 = 0. The hyperabelian condition involving the coefficients
of the first and third rows then gives ffis<*? = 0, whence 34 =0.
Then 31 and 33 can not both vanish, so that a 21 =0 by 114).
Hence S has the form 109).
The order of G is (p* n - 1) (> 4 *-^ 2 *) by 99. The order
of If (4, j) 2 *) is O 4 - l)^ 3 *(y w + l)^ 2w (j) 2w - l)_p"(_p+l) by 132.
Theorem. The quaternary hyperabdian substitutions T with
coefficients in the G-F[p 2n ] form a group G Jioloedrically isomorphic
with GLH(2, p 2n ). The only substitutions of H (4, p 2 n ) which trans-
form the subgroup G into itself are of the form T or VT. H(4:, p n ~)
contains exactly N~ : -~-(p* n + l)p Sn (p n -\- l)p n subgroups conjugate
with G.
135. Consider the subgroup H' formed of the substitutions of
if(4, p 2n ~) of determinant unity. By 132, its index is.p*+l. The
determinant of the substitution T is seen to equal A~^ -1 . Those
substitutions T in the GF[p 2 "] whose determinant is unity form a
group G 1 of order (p* n T)p 2n (p n 1). Since T^ and T 2 are of
determinant unity, the proof in 134 leads to the following theorem:
Within the group H' of quaternary hyperabelian substitutions in
the GF[p 2n ] of determinant unity, the subgroup G' of the substitutions T
of determinant unity forms one of a complete set of N conjugate sub-
groups, each being Jioloedrically isomorphic with the group of binary
THE HYPERABELIAN GROUP. 125
linear substitutions in the G-F[p* n ] with determinant in the G-F]p n ].
The only substitutions of H ( which transform 6r f into itself are the
substitutions g' of G 1 and the products Vg*.
136. The substitutions T for which A = 1 form a group G l
holoedrically isomorphic with the group of binary linear homogeneous
substitutions of determinant unity in the CrF[p 2n ]. Since 6r x con-
tains T! and T 2 , it follows from 134 that g f and Vg' (g' in G- 1 )
are the only substitutions of H' which transform Cr 1 into itself.
Hence 6^ is one of a complete set of N conjugate subgroups of H'.
For p = 2, H' is the simple group HA(4, 2 2w ) and G is the
simple group LF(2, 2 2w ). For p > 2, we pass from H' to the simple
quotient -group HA(4c, p 2n ~) by making the substitutions T x 106)
correspond to the identity. In particular, T_i corresponds to the
identity, so that 6r becomes Z.F(2, jp 2w ). The only T x belonging
to G are T_i and the identity. We have therefore proven the
Theorem. The simple group HA(4:, p 2n ) contains a complete
set of - - (p Sn -\- T)p 3n (p n -j- \}p n simple conjugate subgroups LF(2, p^ n \
4
137. Theorem. - The group of hyperdbelian substitutions S of
determinant unity on 2 indices with coefficients in the G-F\j> 2n ~\ is
identical with the group of binary linear substitutions of determinant
unity with coefficients in the G-F[p n '].
For m = 1, the conditions 98) and 99) that S shall be hyper-
abelian are
- = 0-
Hence the products u ^, a n a fi> a ii a fa
being equal to their own (j? n ) th powers. Hence if n =)= 0, the ratios
of or 22 , 21 , 12 to n all belong to the G-F[p n ~\. Similarly, the
products a 2 2 a ii? ^22^12' ^22^21 a ^ ^ e l n g to tne ^rl^lj)"] and there-
fore, if 22 =[=0, the ratios of # 21 , 12 , 11 to 22 all belong to the
G-F[p n ]. Finally, if cc^ = 22 = 0, we have a^ a^ = 1, so that
the ratio of cr 21 to a 12 belongs to the 6rJP[jp w ]. In every case,
S has the form
fc' / fc i fc N fcf f fc i fc N
j == a (a^ ^ -f- a 12 c 2 j, 2 = (^ 21 bi T ^22 ?2/
where the a-,j belong to the GF[p n ~\. Since it is to be hyperabelian
and since it is to have determinant unity, we have the respective
relations
Hence, by division, a* n 1 = 1, or a belongs to the
126 CHAPTER V.
Corollary I. - - HA(2, p 2n ) = A(2, p) = LF(2, p n ).
Corollary II. The group of all binary hyperabelian sub-
stitutions in the GF[p 2n -] taken fractionally is the group of all
linear fractional substitutions in the GF[p n ~].
138. In virtue of the transformation of indices,
% = J%i + %*> % == ?J pn %i 4- P? 2 ,
where J and Q are primitive roots of the respective equations
we have the following identity
tf+ . + tf+ 1 = ( 3 - J
Hence the hyperabelian group on 2m indices with coefficients in. the
6r_F[^ 2n ] is holoedrically isomorphic with the group on 2m indices
in the GrF[p 2n ~\ defined by the invariant
CHAPTER V.
THE HYPERORTHOGONAL AND RELATED
LINEAR GROUPS. 1 )
139. We first investigate the linear homogeneous group in the
CrF[p n '\ defined by an absolute invariant of the general type
<t> r EE Aili + A 2 |2 H ----- h lmn,
where each A is a mark =|= of the GF\j) n ~\.
If rpQr v we have in the 6rF[j) n ] the identity
Hence a substitution which leaves <t> r absolutely invariant will at
most multiply the function
by a mark ^ which satisfies the equations
1) Dickson, Mathematische Annalen, vol. 52, pp. 561581.
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 127
from which ^ = 1. We may therefore limit our discussion to the
case in which r is prime to p.
In order that the linear substitution on m > 1 indices
8:
shall leave O r formally invariant, the following conditions upon its
coefficients must be satisfied 1 ):
115) *' = ^ 0' = 1, .,
;=i
m
116) - - r V A.ar! r . . . . r ? = 0,
J r^ \ r~ ! _ r ! / / Ui o* u.
holding for every partition of r into s integral parts
r = r -f- r 2 + ---- h ^ 5? m > s > 1,
while for each partition j l9 j 2 . . ., j, may take every combination
of s distinct integers chosen from 1, 2, . . ., m.
If r be not divisible by p, the inverse of /S is
k =
1
1
k
* t=
Indeed, the product $$ replaces . t by
j = l \i = l /
upon applying 115) and 116) for r 1 = r 1, r%= 1.
140. Theorem. //" r > 2, */" r &e wo^ a multiple of p, and if
r 1 ~be not a power of p, the only linear homogeneous substitutions in
the GF[p n ] which leave O r invariant are those which merely permute
the terms AII ? . . ., l m %m amongst themselves.
Consider for r > 2 the following equations of the set 116), in
which fa and J 2 denote two arbitrarily fixed distinct integers < m:
1) If, as in 97, the indices are to belong to the G-F\_p n ~\ so that the
invariance of cj> r is numerical and not formal, we must take r < pn in order
that our results shall still hold true. Cf. 152.
128 CHAPTER V.
!& 'O'fta = 0,
,-'.)*- 0,
t = 1
If neither r nor r 1 is divisible by p, we may drop the numerical
factors from these m equations. 1 ) But
a fi 4=0 ft j = l, . . ., m)
being the determinant of S. Hence we have
ki&iji <*ij t = ft j v j 2 = 1, . . ., m; fa =%= j 2 ).
Hence only one element of each row of the matrix for S is not
zero. The determinant of S being not zero, the non- vanishing
coefficients lie in different columns as well as in different rows.
Hence S merely permutes the terms of the sum O r .
Suppose next that r 1 is divisible by p and set
where g is not divisible by p. We now consider the case g > 1.
We make use of the following equations of the set 116):
m
fant^lil ^ i / o \
W* >* ( 1 ryti 1 )P S fyP Mxy
i- / . s / I/U.IA. . C* . . I l/l . . \/
\(Q 1)>*+1 ! s ! _! V ' ? A */*^ 7i
L\t/ / JT I J JJ ^^B
(nV)S -j- 1 ^ ? X. 1 /" e \
\y-f ' / ' ^y '/ 1 |y(p 1)^3 ffP ]fV ==; Q
[(a 1)^*] ! (p s ~|~ 1) ^^J ^ z ^i </i' O'a
of which the first two alone occur when m = 2. We may verify
that the numerical factors are not divisible byj9. 2 ) Then, since ,-, 4=0
It follows as before that S at most permutes the terms of $> r .
1) If m==2, only the first two equations occur. The same conclusion
follows in this case that was derived for m ^> 2.
2) This result follows by inspection from a general theorem on the residue
of a multinomial coefficient taken modulo p given in the author's Dissertation,
Annals of Mathematics, 1897, 14, p. 75.
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 129
141. If r is not divisible by p and if r=%=p s -\-\, the structure
of the largest linear homogeneous group leaving O r (r > 2) invariant
is now evident. Indeed, the group has as a self - conjugate subgroup
the commutative group of the substitutions
iS = & (* = 1, ..., w) [<& = !],
the quotient -group being the symmetric group on the m letters |,-.
142. Theorem. - Ike structure of the linear group in the GF[p n ~\
which is defined by the absolute invariant O r , r^p*4-l>2, results
immediately from the structures of the groups in the GF[p**] defined
by absolute invariants of the type
= 1
i
| I
~r
For the case r = p*-\- 1, the conditions that S shall leave O r in-
variant may be derived as special cases of 115) and 116), but are
given by inspection from the identity,
p* + 1
By either method, the conditions in question are seen to be:
117)
= A. (j ==!,..., m),
3 \ J ' /"
118)
4 =1
By 139, the inverse of S has the form
By the same rule, the inverse of the latter substitution is
m
..
j ;
Hence this substitution must be identical with S. Hence
119)
The determinant of S~ is
DlCKSON, Linear Groups.
130 CHAPTER V.
Hence, since the product SS~ = 1 has the determinant unity, we
have
1 9/Y\ \ iy.. P S + 1 1
L6\JJ I CCjj J 1.
From the form of the reciprocal $ 1 , it follows that
I i A
- c\ -4 \ J m s "V * / ' < \
121) -j-aP = -^ (t, j = 1, . . ., m)
where Ay/ denotes the adjoint of ay t - in the determinant
XJ __ / -J \
The value of n, defining the G~F\_p n ] to which the coefficients of our
substitution S and the quantities Aj were assumed to belong, has
played no part in the above formulae. We proceed to prove that
our problem can be reduced to a series of similar ones in which
n = 2s. Consider the G-F[p* n >], which includes the G-F[p n ] and
the 6rF[^? 2 *]. Raising 119) to the power 2s _ i> we have
(*L\ " +1 =1
w
it
if a-ij =J= 0. Hence - would be the power p s -f 1 of some quantity
in the G^ 2 *'. The substitution T
.,;
transforms qp r into
*==> *=i
in which the coefficients A[ and A} are equal. Evidently 1} transforms
8 into a substitution with coefficients in the GF[p 2nt ~].
Suppose that the coefficients a ]2; ]3 , . . ., i mi do not vanish,
while iy = for ^ > m lf in all of the substitutions leaving tp r
invariant. Then the group is isomorphic with a group of substitutions
in the GF[p 2ns ] leaving invariant
772
<p'r = ^> i ^6 t (ti = & = ' ' ' = ^mj-
*s=l
In the latter substitutions the coefficients aiy (j > mi) are all zero.
If, among the coefficients 2 y (/>l^j), any one as a 8 /,. =f= 0, we
transform the invariant qpj. by T A , giving the function
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 131
But this function is invariant under the transposition (ii^,) and hence
(p r must have been invariant under a substitution in which i^ =)= 0.
It follows that /x /.
cdj = (i = I, . . ., m^ j = m + 1, . . ., m)
in every substitution leaving cp r invariant. Considering the form of
the reciprocal , we have
ctji = (i = 1, . . ., m^ j = m l -f I, . . ., m).
Hence every substitution leaving cp r invariant is the product of two
commutative substitutions, the one affecting the indices | 1? . . ., | OTl
only and leaving invariant
and the other affecting only ,+ 1, . . ., | m an( l leaving invariant
m
f fcj>*+l
Proceeding with the latter substitutions in the same manner, it
follows that the structure of the group in the 6rJF[^ n ] leaving O r
invariant results immediately from the structures of various linear
groups in the GF\_p^ nx } denned by invariants of the type O. But
the relations 119) for substitutions of the latter groups become
a? 2 * = a. . (i, 7=1..... m).
ij ij ^ 7 > '
Hence there is no limitation imposed in assuming that the field to
which the substitutions belong is the GrF\_p**].
143. We designate by G- m , p , s the group of all linear homogeneous
m-ary substitutions in the 6r.F[jp 2 *] which leave O invariant. For
p > 2, those of its substitutions whose coefficients belong to the
G-F[p s ] constitute the first orthogonal group 1 ) in the 6rjF[jp*] on m
indices. Indeed, relations 117) and 118), for Aj=l, then become
The group Gr m)Pi ,, having the orthogonal group as a subgroup, will
be called the hyperorthogonal group in the GF[p 2$ ] on m indices.
We proceed to determine its structure, treating first the case m = 2.
144. Theorem. If p* > 3, the group of the substitutions of 6r 2 , Pi s
of determinant unity has a maximal invariant subgroup of order 1 or 2
according as p = 2 or p > 2; the quotient -group is LF(2, p*).
1) See Chapter VJI, 171. For p = 2, see Ex. 4 of 210.
9*
132 CHAPTER V.
For m = 2, we have by 117) and 121), when A 1 =A 2 = 1,
Inversely, every substitution satisfying these relations is seen to leave
If L + 2 ' absolutely invariant. Every such substitution is the
product of a substitution
122)
by one of the p* + 1 distinct substitutions
t' __ t tt . . nt
1 1? $2 --^$ .
The number of distinct substitutions 122) is (p 2s 1) j>*. Indeed, for
the p s -\-\ values of a 12 for which f 2 =1, we must have n = 0;
while for each of the remaining (p 2s p* 1) values of 12 in the
6rF[jp 2 *], there exist j/ + l solutions in the field of
for, the second member belongs to the GF[p<] and is therefore the
p s + 1 power of some mark in the GF[p 2s ~\. But
The group of the substitutions 122) has an invariant subgroup
of order 1 or 2, according as p = 2 or p > 2, generated by the
substitution
Cn v - t t r - t
1^2* 'i - ~ bl? 5o ~ ~ '2*
The quotient group (obtained concretely by taking the substitutions 122)
fractionally) is, by 137 138, simply isomorphic with the group of
linear fractional substitutions of determinant unity in the G-F[pP].
By 109, it is a simple group when p s > 3.
Corollary. Every binary hyperorthogonal substitution in the
GF[p 2s ] taken fractionally may be given the form
V
of determinant a mark of the G-F[p*], where A, B belong to the
Indeed, since D p l = 1, we may set D = R p ~ 1 , 12 belonging to
the GF[p* s ']. The fractional binary hyperabelian substitution becomes
Rc&lz V
***/
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 133
The group may be transformed into the group of all linear fractional
substitutions in the G-F[p s ] (see 138, 137, coroUary II).
145. For m general, let S be an arbitrary substitution of G-, n , piS ,
m
S: | = **& (* = 1, . . ., w).
By 139, its inverse is obtained by replacing a,-^ by *>*. Hence the
relations 117) and 118), for A/ = l, when written for the inverse S~\
give the equivalent set of conditions for the invariance of ^^|f "
m i = l
123)
124) = (* * ~ !' *; J
7=1
By 146, the number of distinct linear functions
by which the substitutions of G- m , ptS can replace ,_ is the number
POT,*,* of distinct sets of solutions in the G-F[p^ s ~\ of the equation
125)
.7=1
Let T be a substitution of the group which replaces | x by a
definite function . Then, if Z, Z', . . . denote all of the Q m , p , s sub-
stitutions of the group which leave ^ fixed, the products TZ, TZ', . . .
and no other substitutions of the group will replace |j_ by . Hence
the order Q m>1>l of the group (r m ,^,, is
But the substitutions Z, Z f , . . ., have
cc n = 1, ftu = (i = 2, . . ., m).
Hence by 124), for j = 1, we have
tence Z, Z f , . . . are substitutions of the group G m i tP ,s on the
indices | 2 , . . ., | m , so that Q,^^ s = Q m _ 1;ft ,. Hence, since
134
CHAPTER V.
is the number of substitutions affecting one index only, we have
Q- P P P
m,p, s ~ ' m,p,s m l,p, * r J,.p, s-
To evaluate P w ,j,s, the number of sets of solutions of
we note that, for the P n i,p, s sets of values of 7/ 2 > ., ?? which make
= l^ the corresponding value of ^ is zero; while, for each
of the j) 2 *(w 1)_ P M _i ?J9)S sets of values in the GF[p 2s ~] for which
that sum =|= 1, there exist p s -j- 1 values in the GF[p 2s ~\ for ^. Indeed,
belongs to the CrF[p s ] and is therefore the power
in the GF[p 2s ]. Hence we have
of a mark
Since Pi,_p, s = j9*-i- 1, we find by mathematical induction that
For another proof of this result, we consider only the case p > 2.
Then if v be a not -square in the G-F[p s ], the GF[p 2s ] may be
defined by means of the irreducible equation
Setting
we have
Hence
= !,.. ., w)
By 65, this quadratic equation has j/( 2 -i) (_ iyp*(n-i) se t s o f
solutions <*},...,, (t lf . . ., fi n in the (rjF[j9 s ]. Hence ^ wl ,_p )S equals
146. Theorem. - - If n , 12 , . . ., I OT be any system of solutions
in the GF[p 2s ] of the equation 125), there exists a substitution S in
the group G nhp , s which replaces ^ ~by
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 135
and which is generated ~by ilie following substitutions [in which only the
indices altered are written]:
an additional generator being necessary if p* = 2, m 2> 3, viz.,
W: li = ^ 4- 1 I, + / 2 i 3 [^ = J+ 1 (mod 2)].
If m = 1, we may take S = T^ ttu - If w = 2, we take
$ = 0JJJ 12 .
If m > 2, we prove the proposition by induction. Suppose first that
the f* +1 (i = 1? ? w) are not all unity, for example,
The left member belongs to the GF[p s ~\. Hence we may write
1 O\ rfp* + 1 _J_ ., .p* + 1 1
L\J) 1% ' r^ ~~ *>
|it being a mark =(= in the 6rF[j? 2 *]. The group therefore contains
a substitution of the form Offi*. By 125) and 126), we have
Assuming our theorem to be true for m \ indices, the group contains
a substitution 8' replacing ^ by
<* 1 1 # 1 * 1 ,
11 o I fc I I
^i"* ~^3 -r * ' + ~.r
Hence the product 8 EE S 1 Of^ 1 - will replace ^ by /^ .
Suppose on the contrary that
If the group contains a substitution S 1 replacing | x by |j+ | 2 H ----- h lm ?
the product __ Q
~ -^ 1, ii -^ 2, 22 J-m,a lm >i t '
will replace | x by /^ But the group will contain a substitution of
the form S l if it contains $ 2 ~ Oil$S lf which replaces ^ by
136 CHAPTER V.
If p =)= 2, we can take a = ft p \ since the condition
can be satisfied by a mark in the G-F[p* 3 ]. In this case, $ 2 0'}
replaces 2 by the function
^ r 5s ~r 53 ~r * * ' ~r im ?
and therefore belongs to the group by our assumption on m I
indices. If p = 2, s > 1, we can choose and ft among the sets of
solutions in the GrF[p* s ] of
127)
in such a manner that
Indeed, the condition is (since ^) = 2),
*X+/j.
Since jp*> 2, we may take for a a mark neither zero nor unity in
the GF[p'~\ and then determine a solution ft of 127) such that
/?4=/3* S . Then will a*V=|= aft. To prove that such a choice for ft
is possible, we note first that
K P S = a , 2 4= a; hence a** + 1 =J= 1, ft 4= 0.
Further, if a', /3' be one set of solutions of 127), then is also a', r/3',
where T is any root of
Not every root % belongs to the 6rJP[_p*], and therefore not every
solution ft corresponding to a given a belongs to the GrF\_fP]. Hence,
if p = 2, p s > 2, we may suppose that in the substitution $ 2 the
coefficient a n is such that ^ + 1 =f=l> when the proposition follows
as above.
For p s = 2, an additional generator W, for example, is necessary
since the only substitutions of the form #2 are the products
T^T^-i and (g^^T^-i 3 =1).
Indeed, there exists in the 6r.F[2 2 ] only six sets of solutions of
viz., a = Q, ft = and a = 0, ft = p, where p 3 = 1. Hence the
substitutions T^ t and O"'/ can not combine to give a substitution
replacing ^ by | t + 2 + Is? for example. It follows readily that the
additional generator W is sufficient, together with the substitutions
T and 0, to generate the group 6^2,1-
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 137
147. Lemma. - - If a substitution S of the group G^p^ be commu-
tative with 0%f, for certain values of a, then the following coefficients
of S must be zero, , . __ . , ,
Among the conditions for the identity S0%f = 0"lt S occur
,- = o,
O' = l, ...,w;j=J=r, t).
Hence the theorem follows if the determinant
(a -- 1) (a* 8 - 1) + P^ +1 EE 2 - a - ^ s =(= 0.
The equation 2 a a pS = has p* solutions in the 6rF[j0 2s ]; indeed,
a* 8 '=(2-X=2- ^=.
But for arbitrary there exists a mark /3 in the GF\_p 2s ~\ such that
Hence there are sets of solutions a, ft for which the above determinant
does not vanish, as well as sets for which it vanishes.
Note. Another statement of our result is that S breaks up into
the product of a substitution affecting only | r and % t by a substitution
affecting only (-, (j = 1, . . ., w; j + r >
148. We proceed to determine the structure of the group G m , pi3
of order Q TOj _p, s . For m = l, the group is a commutative (cyclic)
group of order p s -f- 1. For m = 2, its structure was determined
in 144.
The substitutions of 6r OT) ^ ?s of determinant D = 1 form an in-
variant subgroup H,,,,^, of order Q m , P ,s/(p" '+ 1). Indeed, we have
shown that D must be a root of
120) D^+^l.
Inversely, substitutions do exist in the group G- mi p,* having as deter-
minants every root of 120); for example, T^ t and its powers, where x
is a primitive root of 120). Hence the factors of composition of
G~m,p, s are those of H TO? ^ S together with the prime factors of p s -f 1.
Supposing m>3, let / be an invariant subgroup of \-\ m , P ,s con-
taining a substitution
Si g =^ ctij^j (i = 1, . . ., m)
not of the form
T: g = T& (i = 1, . . ., m) [r^+! = 1, r- = 1].
138 CHAPTER V.
With the single exception m = 3, p* = 2, when H 3 , 2 , i is of order 72,
we shall prove that I coincides with H. Therefore the substitutions
T form a cyclic group of order d, the greatest common divisor of m
andj>*-f 1, which is the maximal invariant subgroup of H^,^. Hence
the quotient -group gives a simple group of order ^^ s . We shall
designate it by the symbol H0(m, p 2y ).
149. Theorem. There exists in the group I a substitution replacing
t ~by K^-}- (?| 2 and not reducing to the identity.
Suppose that 13 =j= 0, for example. Transforming S by 0^ y we
obtain a substitution S' replacing ^ by
.7=4
To make the coefficient of 3 zero, we have the conditions
The condition for t is therefore
Unless a^+i-f- a|-M= 0, there exists a solution ^ in the
of this relation; indeed, the value of |u-^ 9 + 1 belongs to the GrF[p*~\
and is therefore the (p s -\-l) st power of a quantity t a in the G-F[p 2s ~].
It follows that we can assume that the only coefficients i ; - (j > 1)
which do not vanish are c* 12 , . . ., ai mi and that, if m > 2, they have
the property that
128)
If % == 2, the theorem is proven. If m > 3, the terms in 128)
must all be equal and therefore zero unless p = 2. Supposing first
that p =f= 2, our theorem is proven unless m 1 = 3, when we have
129) 1 <+ 1 = 1, + !+ +i-0, -0 y-4,..., m).
In the latter case we may assume that not both
+ 1 +f l +1 -0 ('-2, 3);
for, if so, f2+ 1= "fs"^ 1 an( i hence each is zero by 129), sincej0=j=2-
For definiteness. let
f;+> + <+^H=o.
If the left member be unity , then a 12 = by 129) and the theorem
is proven. Suppose therefore that the left member is neither zero
nor unity and consider the substitution
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 139
~a $~*(~j n $ri n S C C
where S tt = S~ C,C Z S is seen to be the substitution
I!- - fc- 2u<S,-2^;;S, (t = 1, . . ., m).
.7 = 1 > = 1
The coefficient c? u in S is therefore
12
Hence
+i= r; n = (i- 2af 1 +'-a+') i ,
which =)=! since afi +1 4- af^ 1 is neither zero nor 1. Applying the
above process to S in which a^ s + 1 =j= 1, we reach a substitution in
the group I in which all but one of the aj (j = 2, . . ., m) are zero.
Suppose next that p = 2. We have by 1 28)
The ratios of 12 , 13 , . . ., i mi therefore satisfy the equation
130) xP s + 1 = l.
Hence by transforming S by suitable products of the form
T^r-iT,,*, (i = 3,...,m\
where the r { are roots of 130) , we reach a substitution S' belonging
to Jin which 12 = 13 = = cc imi . Transforming S' by the reciprocal
of 2 |s, we obtain in / a substitution S n which replaces ^ by
ii8i + i2 { (^ - ^) ? 2 + (f* + A*") ^3 + S 4 + ' ' + Ul-
If p* ={= 2, we can choose A and ^ [see 146] such that
;u>'+i+ p*'+i= i, (A - ^Jp'+i^ i.
Hence in /S" f the sum of the ( p s + I) 5 ' powers of the coefficients ^ 2
and e^' 4 is not zero in the 6r.F[2 2 *]. As above we can therefore
make " 4 = 0. If p s = 2, we reach at once the same result by
transforming S' by (^^W^t,^), W being defined at the beginning
of 146.
Repeating the process , we reach finally a substitution in J, not
the identity, in which either
i> (j = 3, .. ., m)
or else
+1 =g +1 "H>, w-0 (j - 4, ...,>
140 CHAPTER V.
In the latter case, the substitution S thus obtained has (since p = 2)
a p s +i= 1
**ii
Transforming it by TT^rT^i, we obtain in I the substitution
where S 1 denotes the substitution
Hence for S'^S^^T the coefficient of | t in |{ is
Setting for brevity
since T**-M = 1, that
= a, a mark =j= in the GF[p <r \, we find,
' - r - 1).
Since the theorem follows as above if c^ 1 s + 1 =f= 1 ? we seek to prove
that a value x can be found for which
But a root of
only when
131)
==l will satisfy
x^--r --1 = a
1
= ar.
The desired value of t certainly exists if p* -f- 1 > 3. But if p s = 2,
we have a = 1, whence the equation 131) has the single root T = 1
in the 6rF[2 2 ]. The theorem has therefore been proven for all cases.
150. Theorem. - - Excluding the case m = 3, p s = 2, the group I
contains a substitution leaving one index fixed and not reducing to the
identity.
By 149, I contains a substitution S =4= 1 which replaces | t
by a function of the form K^-{- c? | 2 . Hence
where S 1 is a substitution of H TO7iVS of the form
= 2, . . ., w).
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 141
Consider the substitution belonging to H,
where i > 2. The group I will contain the product
s 1 EE S^T^ST^S-^T-^T,
since T and 0J are commutative. Since S' leaves | x fixed, our
theorem is proven unless S' reduces to the identity. In the latter
case, we find by comparing the values by which S^T und TS l
replace | 2 that
2/=0 (j = 3, . . ., w; j =H)> r2/=t
If w > 3, $ has at least two values and therefore
If w = 3, the same result holds if p* > 2. For then a value of r
exists satisfying T**+ I = 1 but not T S = 1. Hence must a 2t - = 0.
Excluding the case w = 3, j>* = 2, it follows that S 1 (which was seen
to leave x fixed) alters | 2 at most by a constant factor L Hence
s = o^r M z,
where Z leaves | t and 2 fixed. Hence I contains
which leaves | 3 , . . ., | m fixed. If S' 4= 1> the theorem is proven. If
S' = l, we find by comparing the values by which STnTwi and
Ti t Ti t iS replace L that
ra = r~ 1 0.
Hence, taking for x a value for which x 2 =|=l, we have ^ = 0. The
only case left for consideration is therefore that in which
S= Ti, y .T^. / ,iT^j L lL.
If S be not commutative with every Oi,g, we obtain at once a sub-
stitution =j= 1 in I which leaves | 8 , . . ., | TO fixed. In the contrary
case, A = 3t 2 , and therefore
^ 4i,*J*a f jZ*
If m = 3, Z = T 3)X 2 ? the determinant of S being unity. Trans
forming S by (liis)^, we obtain the substitution
$ 2 = l\ x T^y,Tl, % 2,
belonging to I. Then I contains
leaving | 2 fixed and not reducing to the identity. For that requires
x 3 = 1 , when we should have
S = T^ y .T^ x T^ K
contrary to the hypothesis made in 148.
142 CHAPTER V.
Let m > 4. If Z be not commutative with every
= Of/ (i, >-,.;., m; *+! + /*+! - 1)
then I contains the substitutions leaving | x and | 2 fixed,
not all of which reduce to the identity. In the contrary case, Z must,
by 147, have the form
! = o>& (* = 3, . . ., w).
Hence I contains the product
'_ '- t 4
* *
which does not reduce to the identity; for, if so, x = CD and S would,
contrary to the hypothesis made in 148, have the form
S = w& (i = 1, 2, . . ., m).
151. Theorem. Except in the case m = 3, #' = 2,
coincides with the group hL,^.
The proofs of the theorems of 149 150 hold for any value of
m ;> 3. Hence by a repeated application of these theorems, we finally
reach in the group I a substitution S =|= 1 leaving m 2 indices
fixed and therefore of the form O^z, we may assume. If it reduce
to OjOg, when ^ =j= 2, its transformed by Oi,s gives the substitution
^aP'+l-^ + l.-aa/J^ r
tfi, 3 ^i ^ 2 9
so that I will contain an Oi, 3 neither the identity nor C t C s . Indeed,
by 144, there exist solutions a =)= 0, )3 4= in the GF[p 2s ], p>2,
of the equation ^ ?s + 1 +/3^+ 1 = 1. Hence I contains a substitution
Oijg neither the identity nor C C 2 . It follows then from 144 that,
for p s > 3, I contains every substitution O"^. Transforming by sub-
stitutions of the form (|/^)(7/, we obtain in I every OJJ/.
These substitutions suffice, except when m > 3, p* = 2, to
generate the group H/^,. Indeed, by applying the formula
l T f}"^ T~^ n a '.P
^ t Uii j J. /, t - - f /, j
where
cc'= *'+' -j- r- 1 ^'+!, ^ f = ccptt- 1 -- 1); T^'H- 1 = 1,
it follows from 146 that every substitution of Gr m , p , s has the form
Ji or hT m , y . where h is generated from the Of/ and has determinant
unity. Hence the substitutions of H w? ^ s (of determinant unity) are
of the form h.
THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 143
For the case p s = 3, we first prove that I contains the sub-
stitution CjOg. We have shown that I contains an Ojf not the
identity and therefore 0'/ given by 132). If /3 =|= > we can
'=0; indeed, if a be not itself zero, we have in the 6r.F[3 2 ]
(\ R f
and we need only take t = -- 1. But the square of O\fr gives C^C*
since /3' 4 = 1 when ' = 0. If, however, /3 = 0, then =(=!. If
= --!, we have at once O^z = C C 2 . If =(= 1, then the square
/ /^or, /i<* 2 , rt n
of Oi,2 gives 0i,2 = (7 1 t; 2 .
Having 1 C 2 , I contains (as above) the substitution
0S, A EE !'+!- jS^+S /i EE - 2/3 EE aft (mod 3).
Taking for a and /3 an arbitrary set of solutions of
a 4 = 1, /3 4 = 1, whence a 4 + 4 = 1,
we have Oi' 3 where ^ = a/3 is an arbitrary solution of ft 4 == 1.
Hence I contains
Transforming the latter by Ojja, we obtain by 132),
Hence I contains every such Oi/ 2 For a = 0, /3 4 = 1, we have
a' = ft ? ft' = 0; for a 4 = - 1, /3 4 = 1, we have ' = 1 ^. We
have therefore reached in the group I every 2 in which = ^u,
0, + 1 + p, where p is an arbitrary one of the four roots of ^ 4 == 1.
Defining the 6r.F[3 2 ] by the irreducible quadratic congruence,
i^ = -l (mod 3),
we have % = 0, + 1, + i> i 1 i *' Hence sc takes every value in
the GF\3*\. We thus reach aU 24 substitutions OJJ. It foUows
that I coincides with H^^I.
For the case p s = 2, we have in I a substitution OJ? =|= 1. By
the result at the end of 146, it must be one of the six substitutions
The transformed of the latter by T ljr T s ^i gives
(Sila)^!,?^,?" 1 ' ^,V|j, i.
Hence, in every case, I contains a substitution of the form
144
CHAPTER V. THE HYPERORTHOGONAL etc.
Its reciprocal gives T^ r iTg f *. If m > 3, I contains
r + 1 r+l
T X+l
+ l r
. 1
where
W =
11
1 r
x r
Hence I contains
and therefore "FT. Hence I contains TF^
with H m , 2 , j if m > 3.
Hence I coincides
152. Theorem. TAe grrtmp ^X*>,* ^ isomorphic with a subgroup of
the linear group 1 ) on 2m indices in the GF[p'~\ defined by a quadratic
invariant
Indeed, we may define the GF[p 2s ~\ by an equation of the form
I 2 - 01+ 1 = 0,
belonging to and irreducible in the 6rF[j>']. Its roots land I p = 7~
belong to the GF[^ 2s ]. Set
& = a?i H- Iy i9 an = a fj + Ic tj (i, j = 1, . . ., m).
Then
The invariant ^ |f '' L becomes the quadratic form #. The general
substitution of Gr m ,p,s,
m
Si $-&&$' (f=l, ...,)
takes the following form
(t 1, . . . f m).
y\ = \CijXj
1) Cf. Chapters VII and VHI. See also the note to 139.
CHAPTER VI. THE COMPOUNDS OF A LINEAR etc.
145
CHAPTER VI.
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 1 )
153. It was shown in 98 that the linear substitutions
A. '-
-a, 5
combine according to the law
where
(t = 1, ...,
(-!,...,
i, j 1, . . ., w).
In Sylvester's umbral notation, the general q ih minor of the
determinant
Ci;
is as follows:
P| * 2
A A
- . In
a
The formula expressing the q ih minors of /J in terms of the
minors of
133)
and of \a'ij is the following 2 ):
1*1 ... 1q l/i . . . t/q
t . '. L i
the summation extending over the C m , q combinations
the m integers 1, 2, . . ., m taken q at a time.
Consider the linear homogeneous substitutions on
. . ., l q of
variables
[a], : I? t - ,- = V
'i,...,^
where the sets (i it i 2 , . . ., ^ 3 ) and
%'.'
' . *
take independently
1) This chapter gives a new exposition of results published by the author
in the following journals: Bulletin of the Amer. Math. Soc., vol. 5 (1898),
pp. 120135; Proceed. Lond. Math. Soc., vol. 30 (1898), pp. 70 98; Trans-
actions of the Amer. Math. Soc., vol. 1 (1900), pp. 91 96.
2) Scott, Theory of determinants, p. 53.
DlCKSON, Linear Groups. 10
146
CHAPTER VI.
the C m , q combinations q together of the integers 1, 2, . . ., w and
where we suppose
The determinant of the substitution [a] 2 is called the q ih compound
1 2. . m
of the determinant
1 2. . m
a
and equals 1 ) the latter raised to
the power C m i, q i. In virtue of formula 133), we have the
following formula of composition:
rvn ^ [>-] r a -\
IK j 2 L J 2 L a _lr
Hence if the substitutions A = (a,,) form a group G m , the substitu-
tions \K\ form a group G m , q called "the q ih compound of the m-ary
group 6r m ". We may therefore state the theorem:
Any linear homogeneous group is isomorphic with each of its
compounds.
154. Theorem. The general linear homogeneous group GLH(m,p n }
has (d, 1) isomorphism with its q ih compound, if d be the greatest common
divisor of q and p n \.
We verify first that at least d substitutions of G m = GLH(m, p n )
correspond to the identical substitution in its q th compound G m , q .
In fact, there exist in the 6rF[^ n ] exactly d marks d for which
$<*= 1 ( 16). For every such mark d, the substitution belonging
to G m ,
(d 0...(M
d. .0
I
..*
gives rise to the substitution [a] 2 ~Jin G- m , q .
To prove the inverse, consider the matrix J formed of certain
coefficients of the substitution [a] ?; in which
(-i)
1 2.. .212
2 3 ... q j
1 2... 2-1 2
1 2
2 3 ... 2 j
1 2 ... q - Ij
1 2.
2 3.
1 2.
2 3.
. . q Iq
' q J
q-ij
2 J
2 3.
2 3.
' q J
-q o
1) Muir, Theory of determinants, 174.
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 147
Consider also the matrix A of determinant A,
>
CC
qq . . .
A =
The composition of the matrices J and A gives the result
(A 0. .
JA =
A. .0
O...A
We seek those substitutions of Gr m which correspond to the
identity in Gr m ^. Suppose, therefore, that [a], reduces to the identical
substitution, so that the matrix J is the identity. In this case we have
) IK J II \ ) *) ; J J 5 I /*
Taking in turn j = q + 1, q -f 2, . . ., m, we have the result
A 0... 0}
A...
O...A
Hence A =)= and therefore A 3 = 1.
155. Theorem. The special linear homogeneous group SLH(m,j)*)
has (#, 1) isomorphism with its q ih compound, if g denotes the greatest
common divisor of m, q, p n 1.
The proof is quite similar to that of the last section. The
following m-ary substitution of determinant unity in the G-F[p n ~\,
< d 0. . (
o:.. c
will give [a] 3 = Jonly when d 3 = l. Hence must d 9 =l. The inverse
is proven as above.
156. Theorem. The second compound of the general linear
homogeneous group GLH(m, p n ) leaves invariant the Pfaffian
Y Y
-*- "i Q * -M~ -1 Q
[1 2 ...m]
Z% '
1m
10*
148
The square of [1 2 .
CHAPTER VI.
. m] is the skew -symmetric determinant
where Y f j I},-.
By 100, G = GLH(m, p n ) is generated by the substitutions
-#/-,, a and D m . The corresponding substitutions of the second com-
pound G m ^ will therefore generate the latter group. To BI^I and D
there correspond the respective substitutions of G^^'
y =
t -
(i - 3, . . ., m)
.. ., w; <j).
But A is unaltered by an interchange of any two subscripts as 1
with 3; for, the resulting determinant may be derived from A by
interchanging the first and third rows and the first and third columns.
It therefore suffices to prove that A remains invariant, up to a
multiplicative constant, upon applying the substitutions ^1,2,2 and D.
By inspection, D t multiplies A by Z) 2 . Also ^1,2, A transforms A
into the determinant
T
~f~ ^ *
32
r 82 o
23
TT i i y y" y y Q
This reduces at once to A since Y 12 + Y 21 = 0.
157. Theorem. For m odd, the substitution [a] 2 of the second
compound gives rise to the substitution
(i = 1, . . ., m)
An denotes the
-' = * A F
7. ^-tj-f-j
y i
upon the Pfaffians F$ E^ [1 2 . . . j -- 1 j + 1 . w],
minor 1 ) complementary to a^ in the determinant a,-,-
1) Or the adjoint of u., without its prefixed sign.
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 149
Consider the Pfaffian Fj, j being a fixed integer <J m. By the
last section, it is unchanged by the substitutions
3 r ,; t i (r, s = 1, 2, . . ., j - 1, j + 1, . . ., m).
Furthermore, B Jttt i alters no element of Fj and hence leaves Fj
unchanged. Finally, we prove that B r ,j,i replaces Fj by
Indeed, BIJ,I replaces Fj by
I ,
;
23 ;
, . .., a.j1
Y YI--I
v v
-*-23> ' * > */ 1
. .,
tn 1m
m 1m
> *f
= [1 2 3 . . . j - 1 j + 1 . . . m] + A [j 2 3 . . . j - 1 j + 1 . . . m]
Interchanging 1 with r, we see that B r j,i replaces
[r 2 3 . . . j - 1 j + 1 . . . r - 1 1 r + 1 . . . m] z -
b
Hence B r j,jL induces upon the Pfaffians FI the substitution
By inspection, Dj gives rise to the substitution
Our theorem is therefore true for the particular substitutions -/-,,, A,
which generate the group Gr m .
To complete the proof of the theorem, we show that, if $EE
induces upon the F t the substitution
150
CHAPTER VI.
m
I:
!,..., w),
where
is the minor of
complementary to a,-y, the products
D S and B r ,s,t8 will induce upon the J^ the substitutions called for
by the theorem. First, the product Z^/S will induce upon the F {
the substitution
8,
The matrices of the two products ^S and
4,1 DA
22
12
(i = 1, . . ., m).
are respectively
.DJ.1,
DA 22 . . . DA^
} . . . 1) A. mii
Here the second matrix is derived from the first by the law expressed
by our theorem.
Next, the product B r ,,,iS induces upon the F { the substitution
+ (-
^ - If v )
The matrices of the two products E^,^S and
i n An ...A lr
JU>1 -^J-oo . -A-lr
are
J^
..A<i m
\
A A A _l_ ( 1 V+ r +l 1 A A
*L m l j?L m 2 . . . ^l.ffi r |- ^ Lj ' A^nL mx ... ^1.
The second matrix is seen to be derived from the first according to
the law expressed in the theorem.
Corollary. The second compound of any linear homogeneous
group G m gives rise to a linear group on the m Pfaffians F lf . . ., F m
which is identical with the m -- 1 st compound of G m .
158. We can establish in an analogous manner the theorem: The
linear substitution [a] 2 of the second compound of any m-ary linear
homogeneous group G m , which corresponds to the substitution (%) of G m ,
effects upon the C m ,2 Pfaffians . .
\ "9 ^^ ^ ^****^ ^^ ft I
\ t <^ ^ <^ ' - <^ ^m 2
a linear homogeneous substitution identical with the substitution [a] m _ 2
of the (m 2) nd compound of G m .
The group induced by the second compound of G m upon these
Pfaffians is therefore the (m 2) nd compound of G m .
mm j
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 151
159. Theorem. The q ih and m q ih compounds of the special
linear group SLH(m, p n ) are holoedrically isomorphic.
The theorem follows from 155 since the greatest common
divisor of m, q, p n 1 equals that of m, m -- q, p n 1.
We proceed to set up the correspondence between the individual
substitutions of the two groups. We may express the q ih minors of
the determinant
, adjungate to D = <r t -y , in terms of the
m q ih minors of the latter determinant by the formula,
It) . . . In
Jl fa ' 'fa
A
. . . m
. . . m
Hence, if we write (for every ^ < 2 < < i q < m)
Yl 2 ti 1 !! + !... fj 1 2 + l...m == Z^ ,- a ...,' 3 ,
the general substitution [a] OT _ 3 of the m q ih compound of the
general m-ary linear group takes the form
7 1 - DI-
^j'i i t . . . iq * /
A
Ji fa
A
If we take D = 1 , this substitution belongs to the q ih compound,
being derived from the substitution (A^ of determinant
Hence to [] m _ ? , the m g th compound of (a^) of determinant unity,
corresponds [yi] 3 , the g th compound of (-^7).
160. Theorem. The general Abelian group GrA(2m, p n ) is the
largest 2m-ary linear homogeneous group in the G-F\j) n ] whose second
compound has as a relative invariant the linear function of its C m ^
variables Y-
It will be convenient to employ a notation for the general sub
stitution S of GrA(2#i, p n ) more compact than that of 110, viz.,
S:
(i 1, . . ., 2m).
The Abelian conditions 76) then take the form (see 112)
139)
m
2
f Ijfc
= |P (if*
even)
(unless Jc =j + 1 = even)
These conditions may also be obtained by the method of 129.
152
CHAPTER VI.
The corresponding substitution of the second compound is
Y- ' f'^> '** = 1> ' > ^ m
a
2^2
In virtue of 139), [a] 2 transforms Z into
r,< -v.
A,
Inversely, if [a] 2 transforms Z into t u^, the relations 139, follow.
161. Since the Abelian group G-A(2m,p n ) contains the substitution
T: I!---!* (i = 1, . ., 2m),
it is (by 154) holoedrically or hemiedrically isomorphic with its
second compound according as p = 2 or p > 2.
If $ belong to the special Abelian group SA(2m, p n ), so that
ft = l, the corresponding substitution [] 2 of the second compound
will leave Z absolutely invariant. Since S then has determinant
unity ( 114), [cc] 2 will leave absolutely invariant the Pfaffian
[1 2 ... 2m] ( 156). If in SA(2m, p n ) we consider S and TS to
be identical, we obtain the quotient -group A(2m, p n \ The latter is
therefore simply isomorphic with the second compound of SA(2m, p n ).
Applying 119, we may state the theorem:
Except for (2m, p n } = (2, 2), (2, 3) and (4, 2) the second compound
of SA(2m,p n ) is a simple group which leaves absolutely invariant the
Pfaffian [1 2 ... 2m] and the linear function Z.
162. For 2m = 4, p > 2, we introduce as new variables
'- ~2~ (-M2 ~~ ^M)J ^i E E Y ( -MS "i" ^M)'
The general substitution [a] 2 of the second compound of SA(4, p n )
takes the form, in which the unaltered index Z does not appear 1 ),
13
14
23
24
Y' =
f-14
V
- L ~
2
12
1
12
12
12
12
12
13
14
23
24
13
13
13
13
1 3
12
13
14
23
24
14
14
14
14
14
12
13
14
23
24
23
23
23
23
23
2
12
13
14
23
24
24
24
24
24
24
12
13
14
23
24
1) In 164 below, the second compound [a] 2 of an arbitrary quaternary
linear homogeneous substitution is written in matrix form.
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP.
153
For example , [a] 2 replaces Y n by the function
13
12
13
13
13
14
13
23
13
24
13
34
which becomes Y^ 9 of the table if we apply the Abelian relation
13
12
13
34
11
a
12
31
a
32
Similarly, it replaces Y by the function
By means of the Abelian relations
12
34
12
= 1-
12
12
34
1-
12
12
12
13
12
34
34
34
34
13
34
34
12
12
0.
34
13
12
13
Hence Y is replaced by the function Y f given by the above table.
It is therefore a substitution on five indices leaving absolutely
invariant the function
= Z\ - [1 2 3 4] = F 2 + r u Z M - Y U Y
U
For p > 2, the simple group A(4, p n ) is holoedrically isomorpMc with
a subgroup of the quinary linear group leaving the quadratic function O
absolutely invariant.
This theorem and the results of 163 165 find application in
Chapters VII and VIII.
163. By 155, the quaternary linear group of determinant unity
SLIf(4:, jp n ) = 6r^ is holoedrically or hemiedrically isomorphic with
its second compound 6ri,2 according as p = 2 or p > 2. By
103 104, 6rJ has as maximal invariant subgroup the group
generated by the substitution
M,*: & = f*l< (i = 1, 2, 3, 4),
where ^ is a primitive root of [i d = 1, d being the greatest common
divisor of 4 and ^ w 1. The quotient -group LF(4c, p n ) is a simple
group of order
JL (^j4 _ l)^8n (^8 _ 1)^2 ^2n _ ]_) ^n
To Mp there corresponds in Cri^ the substitution which multiplies
every index by ^ and therefore the identity if p = 2 orp n =4Z + 3;
while, for jt?n= 4? + 1, it is the substitution T multiplying each of
the six indices by 1. We may state the theorem:
154
CHAPTER VI.
For p n = 2 n or p n = 4Z -f 3, 6rl i2 is a simple group holoedrically
isomorphic with L _F(4, p n ). For ^ n =4Z + l, 6rl )2 has a maximal
self -conjugate subgroup (I, T} of order tivo, the quotient -group being
holoedrically isomorphic with LF(4:, p n ). If e = 1 or 2 according as
$> = 2 or p > 2, the order of 6ri )2 is
164. Theorem. - - The second compound 6r 4)2 of the general linear
homogeneous group 6r 4 in the GF[p n '\ contains the substitution
Y f
J.
V Y V Y Y !
^- X J- i -*-~
' Y
J "
^/J v be a square in the field.
To the substitution (a,-^) of 6r 4 corresponds in 6r 4)2 the sub
stitution a:
12
13
14
23
24
34
T 1 =
*
12
12
12
12
12
12
12
13
14
23
24
34
13
13
13
13
13
13
12
13
14
23
24
34
14
14
14
14
14
14
12
13
14
23
24
34
23
23
23
23
23
23
12
13
14
23
24
34
24
24
24
24
24
24
12
13
.
14
23
24
34
34
34
34
34
34
34
12
13
14
23
24
34
Consider the "partial substitution", possibly of determinant zero,
23
24
34
141)
Y' =
23
23
23
24
23
34
Y'
24
24
24
24
23
24
34
Y'
^-34"
34
23
34
24
34
34
Its determinant is readily seen to equal
a A
THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 155
v must be a
If [a] 2 be the particular substitution 140), the u partial substitution"
141) becomes f
010
.0 v-
of determinant v~ 1 . Hence if 140) belong to
square in the field.
Inversely, if v be a square, 140) is the second compound of
the following substitution of determinant unity:
i/A 000^
i; 1 /.
v-'A
000 v-'A ,
Note. The second compound contains the substitution
Y'=vY Y' =vY Y' =Y Y 1 =Y Y r =v~ l Y Y'=v
12 v -*-\%1 -*-i3 ^-*-13; -*-l4 X 14J X 28 -^23? ..4 -*-24? -*-34 *
In fact, the latter is the second compound of the substitution
i/OOO
0100
0010
v~ ]
165. Theorem. For p = 2,
the relation
substitution of 6r{ j2 satisfies
12
34
12
34
12
34
13
24
13
24
12
34
+
13
24
14
23
+
12
34
T
13
24
13
24
14
23
14
23
14
23
14
23
12
34
+
13
24
"
14
23
= 1 (mod2),
formed ~by multiplying each coefficient of the partial substitution 141)
by that coefficient of the matrix [a] 2 which lies symmetrical to it.
Gi, 2 does not contain the substitution M ~ ( F 12 Y 34 ).
The left member of our relation is seen, to be the expansion of
the expression
"
12
23
24
32
34
43
44
C 44
l ll
1 21
C 31
C 12
C 22
1 32
^13
C 23
C 33
and is therefore = 1 (mod 2), since a^ = 1. The substitution M l
does not satisfy the relation and so does not belong to the group 6ri j2 .
156
CHAPTER VII.
CHAPTER YE
LINEAR HOMOGENEOUS GROUP IN THE GF[p*~\, p>2,
DEFINED BY A QUADRATIC INVARIANT. 1 )
166. Any quadratic form with coefficients in the 6r.F(j) n ], p > 2,
f= ff u (;} + 2ff 12 |J 2 + ff 22 f + 2a 18 | L ! 8 H ----- h ftinmim
may, by using the notation o^- = a^- 7 be written in the form
/. _
/ =
*.;
By the determinant (or discriminant) of f we mean
l ll U 12 ' ' '
^21 #22 '
A =
167. Theorem. Upon applying to f a linear m-ary trans-
formation of determinant D, the determinant A of f is multiplied by D 2 .
In view of 100, it suffices to prove the theorem for the types
of transformations considered in the cases 1 and 2 following.
1. Upon applying to f the transformation
we obtain the function
Its determinant is
flf si~r* a ii #2
^
33
Multiply the first row by ^ and subtract from the second row; after-
wards multiply the first column by A and subtract from the second.
We obtain the original determinant A =
K;
1) The results of this chapter were given by the author in the American
Journal of Mathematics, vol. 21 (1899), pp. 193 256, and partially in earlier
papers there cited. For the case n = 1 , the order of the first orthogonal group
was determined by Jordan, Traite, pp. 161 170.
LINEAR GROUP WITH QUADRATIC INVARIANT. 157
2. Upon applying to f the transformation
! = I; (* = 2, . . ., m)
2,..., TO
we obtain the function
Its determinant is
D 2
*
. . . #7
168. Theorem. A quadratic form f tvith coefficients in the
6rjF[p w ], p>%, and of determinant A =4= can ~be reduced % a linear
homogeneous substitution belonging to the field to the form
142)
(each Ki 4= 0).
Since A =j= 0, the coefficients n , 12 , . . ., cci m are not all zero.
If ff n = 0, we may suppose that a 12 =j= 0, for example. Applying
to f the substitution of determinant 2/1 =}= ?
2Aof
12 .
we obtain a form in which the coefficient of |J is cc 22
Taking for A any one of the ^? ra 2 marks different from zero and
from --#22/2^12; the coefficient of |^ will be not zero. Whether a n
be zero or not, we thus obtain a form
i + 0)
whose determinant A f is not zero by 167.
Applying to f 1 the substitution
ri
52;
'. (4 9 <wA
bi (* *9 ? m>)
ni
we obtain a form in which the coefficient of |j_ | 2 is zero, while
that |J remains /3 n =4= 0. In a similar manner, we can make the
coefficients of & S<Sm all zero. In the resulting form
. .,
2, . . . , m
the coefficients
of the transformed form is not zero by 167.
are not all zero, since the determinant
158 CHAPTEE VII.
Proceeding with this form as we did with /) we reach a form
s, . . . ,
of determinant =)= 0. After m 1 such steps we reach the form 142).
169. Certain of the ,- in 142) are squares and the others are
not -squares in the GF[p r ']. By applying a suitable substitution
which interchanges the (/, we may suppose that in the resulting form
!, . . ., or, are squares, say a 2 , . . ., a*, while ff s _|_i, . . ., cc m are not-
squares, say rojfi, . . ., vc&, v being a particular not -square. Apply-
ing the substitution
i; = -'!. (i -!,...,)
our form is transformed into
Furthermore, we can transform f s into + 2 and wee versa. In
fact, the substitution of determinant a 2 + /3 2
transforms |?+ g into ( 2 + /3 2 )(|f + g). By 64, a and may be
chosen in the GrF[p n ~\, p > 2, such that
We have therefore only two canonical forms, f m and f m \. The
latter form may be dropped if m be odd. Indeed, f m i can, for m
odd, be transformed into
^=,,(11 + %+...+ !).
But the linear group leaving f Q invariant leaves also
/ * 4- _L . . 4- 2
/m bi T^ 3 i T^ 5i
invariant. We may therefore state the theorem:
The group of all linear homogeneous m-ary substitutions in the
6rJP[j) w ], p > 2, which leave invariant a quadratic form f belonging to
the field and of determinant not zero, can be transformed by a linear
homogeneous m-ary substitution belonging to the field into the group of
all linear homogeneous m-ary substitutions in the 6rJF[j) n ] which leave
invariant
ml
where ft = 1 for m odd, but ft = 1 or a particular not-square v for m
even.
LINEAR GROUP WITH QUADRATIC INVARIANT.
159
170. The conditions that the substitution
m
S: -
(* !,..., m)
shall leave F^ invariant are the following:
h ?-
143)
144)
It follows readily that the inverse of S is
771-1
S- 1 :
II (j-1, ... ,-!)
If 0'-)
(j, A = 1, . . ., w; j 4= K).
m (* = 1, ..,' 1)
7W 1
The determinant of S L is seen to equal the determinant D of 8.
Hence D 2 = l ? being the determinant of S~ 1 S = I.
Writing the relations 143) and 144) for the substitution $~~ 1 ,
we obtain the following relations, which are evidently together
equivalent to the set 143) and 144):
145) = -
146)
>m-l*m-l+ -
P
=
(j, fc = 1, . . ., m; j + I).
171. The substitutions leaving F^ invariant were proven to have
determinant 1. Among them occur substitutions of determinant 1, as
The group 0^(m, ^ n ) of all linear substitutions leaving J^ t invariant
has therefore a subgroup of index two 0^ (m, p n ) composed of all
linear m-ary substitutions in the GF\j) n ~\ of determinant unity which
leave F^ invariant. The latter substitutions will be called orthogonal. 1 )
For ft = 1, we have the first orthogonal group 0-^(m y p n ); for m even
and p = Vj we have the second orthogonal group O v (m, p n ).
1) This unusual restriction of the term orthogonal to substitutions of
determinant -f- 1 is done in the interest of the later terminology and notation.
We will be concerned with such substitutions alone. If it became necessary to
consider substitutions of determinant 1 which leave Fp invariant, they might
be designated extended (erweiterte) orthogonal substitutions and the group O
designated the extended orthogonal group.
160 CHAPTER VII.
172. Theorem. - - The order ^(m, p n ) of O^m, p n ) is, for m odd,
fpn(m \) _ ;[) yn(m 2) fpn(m 3) \^\ p n ( m ~ 4 ) . . . (p 2n
and, for m even,
L ^L\
n(m 2) w ( m 3 )
where the sign + ^'s or + according as ^ = 1 or v,
= + 1 according to the form 4Z 1 o/" ^? n .
We notice first that the number of substitutions $, 5", ... of
Op( m > P n } which leave ^ fixed is ^^(^--1,^). In fact ? they have
n = 1, 12 = cc 1B = - - = cci m = ; and therefore by 146) for j = 1,
a*i = (A; = 2, . . ., w).
Hence they belong to the group 0^ (m -~ 1 7 p n ) leaving invariant
u + j+-"+a-i+a-
Let T be a general substitution of 0/u. (m, p n ) and let it replace ^ by
where, by 145) for j = 1,
147) aj t + a\, + . + a\
The ^(m 1, y) substitutions T>S, TS', . . . and no others of the
group will replace ^ by o^. If, therefore, P^^m, p n } denote the
number of distinct linear functions o l by which the substitutions of
Op (m, p n ) can replace | t , we have for the order of the latter group,
Q^u(m, p n ) = Pn(m, p n ) Qfi(m 1, p n ).
This recursion formula gives
^(m, p n ) = P^(m,p n ) P^(m - 1, p n ) . . . P^P"),
since the identity is the only substitution of determinant unity on
one index which leaves jijjj, invariant, so that ^(l,jt) n ) = 1.
It will be shown in 174 180 that P^tyjp") equals the number
of sets of solutions in the GF\j) n ~\ of the equation
^ + 2 2 + --- + !-i + yi = i,
and hence, by 65 66, Ppfap*) =
1 n (^l\
pn (k- 1) q: 2 V 2 ^ even )
k1
p n(k-V -2- p n(k-l)/2 (ft O dd)
the upper signs holding if fi = l, the lower signs if ft = v, and f
denoting +1 or -- 1 according as 1 is a square or a not -square
LINEAR GROUP WITH QUADRATIC INVARIANT.
161
in the GF[p n ]. Whether the integer t be even or odd, we find
that the product
p nt (p nt F (t ~ l} (P nt T
* 1)
We derive at once the expression for ^(rn^p 91 ) given in the theorem.
The orthogonal group 0^0,1^) is generated ~by
173. Theorem.
the substitutions 1 }
Of;/:
4-
~7"*rt
following exceptions:
1. For p n =o, m > 3, f*=l, we may take as the necessary
additional generator
s ' = 4- -4-2L
51 Si i 62 i ^ 3 y
I 7?2 _ r
) | "<9 -*'< oo ~~~ *-
^s-
3?
2. JP9r ^) w = 3, m > 4, ^ = 1, we
additional generator
choose as the necessary
1234
= J
a '
1, we mai/ fo&e as %e necessary
3. JPor jp w = 3, m > 3, fc u = v
additional generator
Vi2m''
For m = 2, the theorem is readily proven. If any orthogonal
substitution replaces 1 by y^-\- d^ y then $=0ij2$i> where S
leaves ^ fixed and is therefore the identity.
For m = 3, the theorem follows from 174 179. For m > 3,
it foUows from 180.
1) For simplicity we write only the indices altered by the substitution.
DlCKSON, Linear Groups. H
162 CHAPTER VII.
174. Theorem. If c^, cc z , cc s be any set of solutions in the
6rF[j> w ], p > 2, of the equation
9 I 9 I -^ 9 -i
2 . + *l + -I = l,
there exists a substitution S, derived from the generators of 173 which
leave JjJ + Jf + ft If invariant, such that S replaces ^ % a^H- a a 62+ a 3^s-
The proposition follows if 1 a\ or 1 a\ be a square =|= in
the GF[p n }. For example, if 1 | = r 2 , then
"1 "3
\t' t
- - it 3 i, 2
The proposition will be true for a 19 a%, cr 3 if true for the quantities
where { EEE c^p - a 2 ff, a' 2 = a^ + cc 2 $, a' B = 3 ,
? 2 +<J 2 =l,
so that we have
148) { 2 +< 2 + -i-< 2 =af + l + |! = 1.
In fact, if the group contains a substitution S' which replaces |j_ by
{?!+ 2^2+ a s?3? ^ W ^ contain the product S=Oi]2~ a S f which
replaces | t by a^ + or 2 | 2 + a s | 3 .
175. Consider first the case in which --1 is a not -square in the
6r.F[j> n ]. By 64, there are p n + 1 sets of solutions p, <s in the field
of the equation 2 +G 2 =l. Not more than two of these sets of
solutions give the same value to
Indeed, upon eliminating tf, we obtain a quadratic for Q. Hence a 2
takes at least -^-(p n + 1) distinct values. But, by 67, there are
exactly -^-(p n 3) distinct marks ^ =f= for which rf 1 is a square 1 ),
so that 1 rf is a not -square. Hence there exist at least two values
of cc' 2 for which 1 cc 2 2 is a square or zero. If it be a square, our
theorem follows from the previous section. There remains the case
ffj a -=l, for which, by 148),
: 2 =-i; 2 .
|U/ rf
If fi = 1, we have a[ = K^ = 0, since 1 is a not -square, and the
required substitution is $=0i',2 2 - If ^ be a not -square, we may
take ft = 1 , so that
1) Zero is not reckoned as a square.
LINEAR GROUP WITH QUADRATIC INVARIANT. 163
But the theorem is true for a[, cc' 2 , cc' 3 if true for the quantities
where /3 2 y 2 =l. In fact, if S n replaces ^ by a"^ -f- aJSg -f- s 3 ,
then 0'JS" will replace & by <&+&+{!,. Thep-l sets
of solutions in the G -Fjj/ 1 ] of the equation /3 2 y 2 = 1 are given by
(* + I/*),
y =
where r runs through the series of marks =)= of the field. Hence
ft ^f Y may be given an arbitrary value r =f= in the field. The
theorem being evident if a[ = 0, we exclude this case. Then " =
i (/3 + y) may be made to assume an arbitrary value except zero, and
hence, if p n > 3, a value for which 1 a'/ 2 is a square in the field
( 64). For p n = 3, aj, e^, tfg are each +1, so that we may
evidently take
S=
where C and K are products formed from C 19 C 2 , (7 3 . But, if C be
the product of an odd number of the d, we note that
^1 ^123 = ^2 ^3 ^123 Vl ^2 ^3
We may therefore assume that C and K are each products of an
even number of the d and therefore derived from the given generators.
176. Suppose next that 1 is the square of a mark i of the
GF[p n ~\, while ^ is a not -square. There exist # n + 1 sets of solutions
in the field of the equation
149) 0t+.iy_i.
But the theorem is true for a lf # 2 , a s if true for
Indeed, if >S" replaces j^ by a^^ + a' 2 % 2 + a^Jg, then Os^S" will
replace g t by a^^ a a |g+ a6 8 -
There are at least -^-(i> n + 1) sets of solutions of 149) for which
the values of a' 2 are distinct; for, upon eliminating /?, we obtain a
quadratic for y. But, by 67, there exist only -^-(p n 1) marks ^|,
and hence as many distinct marks |, for which
(^i) a + 1 = 1 | 2 = not -square.
11*
154 CHAPTER VII.
Hence at least one set of solutions of 149) will make 1 a' 2 * a
square or zero. If it be a square, the theorem follows from 174.
If it be zero, we have by 148),
,v
W j
'2
Hence a{ = as = 0, c^ 2 = 1, so that we may take S = 0^2 2 .
177. For the case 1 ) --la square in the 6rJF[p n ] and fi = l, it
follows from 178 179 that 1 (3,p") contains a subgroup of order
at least p n (p 2n T) generated by the substitutions O"/, together with
-^123 H p n =5, all of determinant + 1. But, by 172, the order of
0i(3?JP ? i g p i( 3 >.P n ) P i( 2 ># n )- H ere P x (2, ^) w ) =^) n 1, being the
number of functions
by which the substitutions of 0^2,^") can replace |j. Also
In fact, if a substitution of (^(3, p") replace | t by
,,
150) J + | 4- J = 1.
By 66, this equation has ^ 2 " + l? n sets of solutions in the G-F[p*],
-1 being a square. The order of Oj(3, _p n ) is thus at most
^pinj r pn^pn_ \^ f From the two results it follows that this number
equals the order of 1 (3, ^) w ) and that for every set of solutions
of 150) there exists a substitution of L (3, p*) t derived from 0'/
and R, which replaces ^ by o^.
178. Theorem. The first orthogonal group O x (3, p n ) contains a
subgroup 0{(3, p n ) holoedrically isomorphic with the group LF(2,p n )
of linear fractional substitutions of determinant unity.
Let 1 = i 2 , so that i belongs to the G-F[p n '] if and only if
1 be a square in that field. Introduce in place of 1? 2 , | 8 the
new indices
%=-*ll, %=is *(J8> %=S2+*Ss>
so that
_ -jS l ^ ^ =^ fc2 i 2 _l_ fc2
T /l r /2 r /3 i T^ $2 3 '
1) For a more direct treatment of this case , but one involving considerable
calculation, see Amer. Journal, vol. 21, pp. 202 204, in which the proof of
Jordan, Traite", pp. 164 166, for n = 1, is corrected and generalized.
LINEAR GROUP WITH QUADRATIC INVARIANT.
The following substitution of determinant unity,
165
Y:
ad + fiy ay fid
2 2
a
2yd y 2 d 2
leaves %% y 2 absolutely invariant. Written in terms of the indices
1? la; s> ft takes the form
< ad 4- By
ap yd
It follows that X has determinant unity ( 101) and leaves |J H- || + 3
absolutely invariant. Giving to the substitution Y the notation
151)
we readily verify the formula of composition
The group of the substitutions X, being isomorphic with the group
of the substitutions Y, is isomorphic with the group of the linear
fractional substitutions 151). But Y and therefore X is the identity
if and only ifa = d = l, P = y = 0. Hence the isomorphism is
holoedric.
If 1 be a square in the GF[p*\ 9 so that the coefficients of
X belong to that field, the substitutions X form a group 0[(S,p n \
a subgroup of A (3, p n ), which is holoedrically isomorphic with
LF(2, p n ).
If 1 be a not -square in the GF[p*\ 9 the coefficients of X
will belong to the CrF\_p n ] if we choose a, /3, y, d in the GF\_jP*~\
such that K is conjugate ( 73) with d, p with y, with respect to
the GF[p n ~\. By 144, the resulting substitutions 151) of determinant
unity form a group holoedrically isomorphic with LF(2,p n '). The
corresponding substitutions X form a subgroup 0{(3,^ n ) of 0, (3>J>*).
In each case, the subgroup #i(3,j) w ) has the order -zrp n (p 2n l),
since it is holoedrically isomorphic with LF(2,p n '). We proceed to
prove that this subgroup does not coincide with 0^(8^jP). In order
that OJ 3 shall be of the form X. it is necessary and sufficient that
/ V
166
CHAPTER VII.
According to the definition of a, /?, y, d in the ahove cases, the
expressions
belong to the 6r-F[_p n ]. The above conditions then give
a = 2 A 2 -1, b = 2AB, A 2 + B*=1,
so that O^s must be the special substitution 2,'s S defined in 181
Any orthogonal substitution 02,3 not of the form Q 2 ^, and therefore
not of the form X, will extend Oi(3,# w ) to a larger subgroup 6r of
O l (3,p n '). The order of G is therefore a least p n (p* n 1). From
the remarks at the end of 177, it follows that G has -exactly this
order and hence coincides with 1 (3, p n ).
179. We proceed to the proof that, if - 1 be a square i 2 in
the G-F[p*], the group O t (3, p") is generated by the substitutions
0"/ together with E i23 if p n = 5. If p n > 5, there exist ( 64) marks
/3 and r in the CrF[p n '} such that
Then the product
08 + 0, r + 0).
l, 2 2, 3
-
' 2
, 2
ft
ft
o.-.i
[ o ri 6~] . ri &a/
_ 1 , which is an 02,3, transforms n .j into ~
Furthermore,
[-| Q 2T P "1 #2~1 P1 X?/^2l -,.'
i /3n ri /3n i p(ai + a
01 Lo i Lo i
Since /? =J= 0, we can ( 64) find marks c^ and a 2 in the field such
that /3(2 + a|) === x, where K is an arbitrary mark =^= 0. Also
r o 11-' ri --ir on ri 01 r o ii_ ,,.
Ui oJ Lo iJL-i oJ = L iJ' L-i oJ = 0l ' s ;
Hence, if p n > 5, we have reached from the O"'/ the substitutions
[J l]' [I l] ( arbitrary).
100 and 108, these substitutions generate the group LF(2,p n \
Hence the 0%'f from which they were derived generate the isomorphic
LINEAR GROUP WITH QUADRATIC INVARIANT. 167
group 0{ (3, p n ). Then, by the last section, all the OfJ generate
Oj (3, p n ). For p n =5, i = 2, we have
f i i 9 '
[2 11 [1 31 [2 ]
= LO 3j == Lo IJLO 2-iJ*
Li L \-
Hence from JR 123 and L 2 -i =^M' we reach Q j It follows
as above that .R 123 and 0^1 ' = C 2 C s and C^Q generate 0{(3, 5).
The latter is extended to 0!(3, 5) by any Of/* 1 .
180. Theorem. - - If a J9 tx z , . . ., a m be any set of solutions in the
GF\p\ of tt2 + tt2+ ... + a 2_ i+ JL a 2 i:=1
^ere e^sfe substitution S derived from the generators 1 } of 173 which
replaces ^ by a?! = ^ | t + or 2 | a H [- a m | m .
The proposition having been proven for m = 2 and Wi = 3, we
will give a proof by induction from m 1 to m, supposing m > 3.
Consider first the case in which every sum of three of the terms
af, |, . . ., m_i ? m is zero. These terms must all be equal and
* (t
therefore rt
= square.
Hence p = 3, while m is of the form 3& -f 2 or 3& + 1.
If m = 3fc -f- 2, we have 1 a\ = a\ =f= 7 so that the theorem
is reduced by 174 to the case of m 1 indices.
If w = 3fc-fl, we must have a\ = 1. But the product Oi'JS
will replace ^ by cc[^ -{ ----- \- i,m> where
a{ = aa^ |3cf 2 , | = fta + acf 2 ? J = K i (J = 3, . . ., w).
Of the 3 n l sets of values in the 6r_F[3 n ] satisfying
(K 2 +/3 2 =l,
at most two give the same value to a[ and hence at most four make
2
ct[ = 1. Hence, if w > 1, we can avoid the case a\ = 1. For p n = 3,
we may take
1) For the case pn= 5, m ^ 4, ft = not- square, it would appear that the
generator JS 123 were necessary in addition to the O&f. We can, however,
express J?^ in terms of the generators
leaving invariant |f + || -] ----- [- i^ _ 1 -f 3 i^. Indeed,
7? - f) /I /I 1 O /I ^ /I
-"123 ^I^m^Bm^im^^m^S
168 CHAPTER VII.
where and .BT denote products of an even number of the d [com-
pare the end of 175].
Suppose next that the above sums are not all zero, for example 1 )
. aJ + aj + iol+0.
We have proven that, for every set of solutions of
152) a +0+ly = i,
there exists a substitution X of the group,
which therefore satisfies the relation 152) and the following:
4. 0' + -iy' = 1 2 *f
If there be a substitution $' in our group which replaces f x by
ffl i
where
/ f | rtf
p
then the group will contain Z$ f which replaces | x by & v The prop-
osition is therefore true for the quantities ay if true for a[, cc^ a' m ,
a zi a i - -> a mi> We may thus make our proof by induction from
m 1 to m by showing that it is possible to choose a, /3, y among
the sets of solutions of 152) in such a way that i = 0. We may
suppose that a^ =J= 0, since otherwise the proposition is already proven.
If a\ + J = 0, then ff 2 4= 0. From i = 1, it follows that k a
is a square, say f* = 1. Then the values
~ a m - a m 1
a = > p == -5 ; y = l
satisfy 152) and make i = 0.
If a\ -f | 4 s ^ *^ e condition 152) combines with ai == to give
a single condition for /3 and y:
1) The treatment for a case like } -f- f -f a l = i = 1S ^ u ^ e s i m il ar i taking (i = l.
LINEAR GROUP WITH QUADRATIC INVARIANT.
Multiplying this by a\ -j- a\, it may be given the form
169
= a(a + I) -
Since the coefficient of y 2 is not zero, this equation has in the
GF[p n ] (by 64) p n 1 sets of solutions for y and
and hence as many sets of solutions /?, y.
The structure of the first and second orthogonal groups, 181 198.
181. The group 0/*(flw,|) n ) contains the substitutions
(o * 1 o
P 2 + T o'=
leaving ^f -f ZJ| invariant, where A = 1 if *, j < m, but A = f* if
i<j = m. For ^ and,; fixed, while p, (? take all possible values in
the field such that ^ 2 + y (y2=== . 1> the substitutions O z \'/ form a sub-
group denoted by Oij. Its substitutions are commutative since the
following product is unaltered if we interchange Q with Q' and 6 with o' :
By 64, the order of 0^- is p w s iJ7 where f o - = -f- 1 or 1 accord-
ing as I/A is a square or a not- square in the GF[p n '].
The squares of the substitutions of 0/j form a commutative
group Q it j, composed of the substitutions,
The order of ft^ is -^-(p n ij)- Indeed, the identity
tolds if and only if ()'= + (), a' = a.
170 CHAPTER VII.
Let OfJ be a particular substitution which extends Q^ to 0^,
the values p, 6 depending on A. Consider the subgroup 0^(w,# n )
of Oftfltojf*) generated by the substitutions
?/, Of/ Of,' ft }, *, I - 1, . . ., M; t + j,Tt 4= I)
where a, /3 take all the values in the GrFfp 71 ] satisfying a 2 4--r/3 2 =l,
A
the generator J2, TFor V being added in the exceptional cases of 173.
182. Theorem. The order of Oi(w,# n ) is at least half of the
order of 0^(m,p n ).
By the theorem of 173, every substitution of 0^(m,f)*) has
the form a
S = \0^ \0i\ihz.. .
where the hi (and the ft', h", h, . . . below) are derived from the
generators of O'^m, p). For m > 2, 0[; J and 0%T* = 0l\{ are
reciprocal. Hence
/)?<* __ f)Qi a r\Qi<* rfi, _ -L r\Qi a
U iJ ' U *,j ' ^2, 1 ^1, 2 == A&4 V 1, 2 .
Hence
= ^0^0?;^...
Furthermore, 0?;? is commutative with every Q% and every C?/>
i and j > 2. Since the square of 0?;J is 1;^, whose reciprocal is
6?> .
i,k y wn a >P- n^ (n& a }* n^~ n a >P
1% 2 Vl,A: = = ^1, 2 Wi, Jc) (^1, j Vl, k
- w n> a nft>~ n a 'P n^ __ T' n^ -- w nP* a
- ^i, 2 ut,i vi,i Vi,* v% "- ^i, A = ft ^i, 2 .
Aside from the above exceptional cases, we may conclude that S is
of the form h or else h - Of;J. We treat next the exceptional cases.
1. For^) w 5, m > 3, ft = l, the additional generator is J2 123 ,
and the only <#/ are Q^f^dCj and Cf/ |0 =I. Since
where T,-^ = (;)? is not in fty, it may be taken for OfJ. To complete
the proof that S = h or /&0?jJ, we note that
7 r T? - r r H T r . T r r r T r
^12^1 - V 123 ~~ " ^1 ^3 - ti !23 -^23^2 *'18* y l ^1 ^3 -*-lf^l
2. For p n = 3, m ^ 4, ft = l, the additional generator is W m .
The remarks of 1 apply here, if we replace the last formula by
-Z 12 Cj W ! 23 4 = Gj C 2 rKjj^^jj C7| .
3. For j9 w =3, m>3 ; p = v = 1, the additional generator is
F 12m and the only Oj (a 2 - j8 2 =l) are O.-Cm and J, the only <#
being I. We may take O^/ = T^d (%, j < m) and 0$[ = Cift,. To
complete our proof, we use the formulae
^12 Ci FI 2 m = Ci ft Fi 2 2 m Tig Ci, Ci C m V\ 2 m = J 7 !^ m T^ C 2 .
LINEAR GROUP WITH QUADRATIC INVARIANT.
171
183. Theorem. The group 0{(3 ; p n ) just defined is identical
with the subgroup of 0i(3,# TC ) of index two defined in 178.
It is only necessary to show that every $?/ an d every 0*,'/0*'*
are of the form X or, if we prefer, 151). We have
l, 2 ^2, 3 -
K
0'
~K -\-\-\-iG K 1 -|- * (
H -\-l-i a
-L 10 -L is
In particular, we reach T^T U and ^Tgg. These suffice to transform
Oils 0?" into O^sOSi and Ol\lOl\l. Transforming these products
by C C i9 C^Cz and C 2 (7 3 , we obtain every 0/0j*f, since (7; trans-
forms 0*// into 0/;f. Transforming Ql\l (which is of the form X
by 178) by T 12 T 13 and T 12 T 23 , we obtain every <#/.
184. Theorem. The group 0i(3,p n ) is of index two under the
second orthogonal group O r (3, p n ).
Consider the substitutions, in which a? -f- ft 2 = v,
0:
O" 1 :
2
Since transforms ^ + %\ + v|| into v (|f -f " 4- JJ), it transforms
the group 0!(3,p n ) of the latter into the group O v (3,p n ) of the
former. But is commutative with Ofjg. Hence if Oi,'| serves to
extend the subgroup 0{(3, p n ) to 0j(3, p w ), there exists a subgroup 6r
of O v (3,^) n ) which 0J| J extends to the latter. We readily prove that
6r is identical with the 01(3, p n ) defined in 181. For example,
transform Ol\lO\\lC^ into 0?;ICiC 8 , where
Here 0?; 2 is not a 1,2 since (1 + p)/2 = 2 /v is a not -square. But
QCj is a 1>3 in O^S,^ 71 ), hut not in O r (3,p n ). It follows that G
contains the product OillC l C S) neither factor being a Q.
185. It will be shown in the following sections that 0(w,# n )
is not identical with O^m^p 7 *) in the cases m = 4, 5, 6 and there-
fore, by 182, that its index under O^m^p^ is exactly two. By
181 184, the same result is true for m = 2 and m = 3. For
172 CHAPTER VII.
various reasons it would seem that the same result holds true when
m > 6, but no explicit investigation has yet been made. The devel-
opments in 191 193 are made on the assumption that this index
is 2. Moreover, if this conjecture prove false, very simple alterations
would be necessary in the treatment.
186. We continue the investigations begun in 163 on the
senary group 6ri,2, whose substitutions leave absolutely invariant the
Pfaffian [1234], viz.,
-^4 = * tt ^34 -*13 ^24 ~J~ *14 ^23
Denote by 6r 6 the group of all substitutions of determinant unity in
the 6rF[# n ], p > 2, which leave F absolutely invariant. We will
prove that 6r 6 is holoedrically isomorphic with 0^(6, p n ), where ft= 1
or v according as _p n =4Z-fl or 4Z + 3. Hence 6r 6 has. the order
( 172)
153)
It will therefore follow from the theorem of 163 that #4,2 is a
subgroup of index two under the group 6r 6 .
187. Let p n = 4:1 + 1, so that 1 is the square of a mark i
belonging to the 6rF[p n ]. We make the following transformation
of indices:
-M2 == 1 H~ *fe2J " -*ll '3 ~^~ ^64? 1-^= | 5 + ^5e?
Then .F 4 takes the form
Hence ^ 6 is holoedrically isomorphic with 1 (6,^ n ). By 164, the
following substitution of 6r 6 (leaving four of the indices fixed):
Zl -TT V 1 * l~V
13~ TJ -137 J '24~ -^24
belongs to the subgroup 6rl,2 if and only if r be a square in the
field. Expressed in the new indices, it has the form
m~-T(*-*- 1 )! 3 +i-(*+*- i )!4.
For T an arbitrary mark =J= of the field, 155) may be written
156) 01,1 e^-kr + T- 1 ), tf =-L( T _-i), , +
LINEAR GROUP WITH QUADRATIC INVARIANT.
173
For r = 2 , 155) may be expressed in the form
For r a not -square, 155) is not of the form 157), since that would
require y 2 = (t -f l) 2 /4r. It follows that to the subgroup 4,2 of 6r 6
of index two there corresponds a subgroup of 1 (6,jp n ) of index
two, where is extended to 1 (6,^) w ) by any substitution Of; 4 not
of the form Q 3i . We proceed to prove that is identical with the
group 0((6,# ra ) defined in 181. We first show that contains all
even substitutions on the six letters | t , . . ., | 6 . Expressing the sub-
stitution
(t V '
^Si^Ss;- ?i b2>
in terms of the indices I
= I 3 , SJ-*f !}-& (<-4, 5, 6)
it takes the form
12
13
14
23
24
34
r
12 ~
i/2
-i/2
i/2
i/2
f
-1/2
1/2
1/2
-1/2
f
1
;
23 ~
1
/
1/2
1/2
1/2
1/2
f
-i/2
i/2
~~ V I <U
i/2
By inspection this substitution is the second compound of
(1 .- j\ 1 H - *\
o V 1 l ) 9 V 1 l )
having determinant unity. Hence contains the substitution (^^s)-
In the transformation of indices 154), the pairs | t and 2 , 3 and | 4 ,
S 5 and | 6 enter symmetrically. Hence contains the substitution
(jjtii/i*), two of the distinct integers *, j, A;, each < 6, being chosen
from one of the above pairs. But the linear substitution denoted
by (liMs) transforms (^^5) ^ nto (^i^^s)- Hence contains every
cyclic substitution (%r% s %t) on the six indices and therefore every even
permutation of the six indices. 1 )
1) Netto-Cole, The theory of substitutions, p. 35.
174
CHAPTER VII.
A *T
Having every Ql] 4, contains their transformed Qlj (i=^=fy by
the even substitution (isfe)((U&)- By 164, Note, contains the
product Of; 2 #3,' 4 and therefore also every Of/Cjjf, where i,j,k,l
are distinct. Hence contains
09,
1,
2 3,4 ' A, 3 i, 5 = i f 2 l/i, 5
and therefore every Of/Ofjf in which two of the subscripts are alike.
For the case p n = 5, = 2, there is an additional generator, viz., -R 123 .
Expressing .R 123 in the indices T^-, we obtain the substitution
T4 3 2 41
010003
001000
000100
000012
000004
By inspection, this is the second compound of the following sub-
stitution of determinant unity with coefficients mod 5:
2
4
3
41
3
The group therefore contains all the generators of 0{(6, p").
Since is of index 2 under } (Q,p n } and 0{(6,^? n ) of index at
most 2 under 1 (6,jp) ( 182), it follows that = 0i(6,p"). We
have therefore, by 163, the theorem:
Iw p*= 4J, -\-\, the group 0[(6,p n ) has a maximal invariant
subgroup { I, T= C^C^ . . . 6 ) of order two, the quotient- group being
holoedrically isomorphic with the simple group LF(4,p n ). 0[(Q,p n ) is
of index two under the first orthogonal group 1 (6,p n ) and is extended
to it by any OfJ not a Q
188. Let p n = 41 4- 3, so that 1 is a not -square in the GF[p n ~\.
We make the following transformation of indices:
- fe
V
- 1 14 fe5 6
where a and /3 is a suitable set of solutions in the field of
159) a 2 +/3 2 =-l.
LINEAR GROUP WITH QUADRATIC INVARIANT.
Under this transformation, F takes the form
175
Hence 6r 6 is holoedrically isomorphic with the second orthogonal
group 0_i (6,^)*). Reversing equations 158), we find
The following substitution leaving F invariant,
14?
becomes, when expressed in the new indices 160),
I X 24 -MS,
= Y Y
i - L 2S J -14'
o
It is always an 5j6? but is of the form Ql\ if and only if r be a
square. It follows that 0_i(6, p") contains a subgroup 0' of index 2,
which is the form taken by 6r 4) 2 when expressed in terms of the |,-.
The subgroup 0' may be extended to 0_i(6,p n ) by the substitution
C 5 C 6 , the new form of T_I.
We proceed to prove that O r is identical with the subgroup
O.!_i (6, p n ) defined in 181. Expressing the orthogonal substitution
Of] 4 in the indices Y f j, we obtain the substitution, denoted for the
moment Of,' 4:
13
24
34
y
-4(1-
1
1
1
1 ,1
- (1 ()
)
(1 -J- Q
) -4
2
' 2
1 (
X 1
3) -0
1
-an
= $ we see that Of;* is
Y' =
X
-34 ~
For p = 2y s -l, tf-=2yd, whence
the second compound of the substitution of determinant
7 (
y -d
d y
-tf
176
CHAPTER VII.
As shown above, T_i corresponds to (7 5 C 6 . The product
seen to be the second compound of
s
X
G
x
G
2x
G
Zx
X
G
x
But x belongs to the 6r_F[j) w ], in which 1 is a not -square, if and
1 a
only if y(l-f p) is a not -square, which occurs if and only if 0^4
is not of the form Ql\l. Hence 0' contains Ol]lC^C 6 if OJJJ is not
a 2,4, but not in the contrary case. As shown above, O 1 contains
every 2*4- To P r ve that 0' contains all the generators of 0-!_i(6,^) n ),
it evidently remains only to prove that 0' contains all even substi-
tutions on |i, | 2; | 8 , | 4 , g , and, if jP n =3, also F M) 6.
Expressing the linear substitution (Si^Ss) ^ n ^ ne indices Yy, we get
1
1
K/3) y(a-
This substitution is the second compound of
x y '
* w
-W Z
-Y Z
having determinant unity, where
161)
cc 8 ccB
'
-A. =
Zx
In order that 161) shall belong to the 6r-F[_p n ], it is necessary and
sufficient that x be a mark =)= of the field. We proceed to prove
that, for every set of solutions in the field of a 2 + /3 2 = 1 , the
expression
LINEAR GROUP WITH QUADRATIC INVARIANT.
177
is a square in the field or else zero. 1 ) Eliminating /? between the
two equations, we find
or
or
(1 +.+ a 2 ) 2 - 4as + 4s 2 =
(1 + a + a 2 + 2s) 2 = 4s(l + a)
Hence will s be a square. 2 ) Solving, we find
The linear substitution (i 2 i 4 i 5 ) expressed in the indices Y f j takes
the following form, say F:
1 J
T"'
-ft
2 1*
y
1
i
1
i
Y'
" Y
i
i
Y
Y
i
i
The product FJ5J will be simpler than F, if we take as E:
24
which is recognized as the second compound of
7?'-
JLJ =
1
a
P 0'
1
A
a
1
1
1) The case s = requires 1 -{- -f 2 = 0, and may thus be avoided
2) For pn= 3, 7 or 11, there exist solutions of 2 -f (3 2 = 1 for
is an arbitrary square in the field. Is this always true?
DlCKSON, Linear Groups. 12
178 CHAPTER VII.
The product VE has indeed the simple form U =
f J^
i
1
i '
8
q> \i
"Y
Y
1
1
i
i
Y
Y
"Y
" Y
2
a/2
i
2
1/2
-0/2
-a/2
1/2
-1/2 a
1/2
-1/2
1 1/2
1/2 -ft
1/2
1/2,
which is seen by inspection to be the second compound of
U' =
-1
1/2
-1/2
1/2
Hence F
1 is the second compound of V
Having the linear substitutions (^Sglg) and (| 2 | 4 | 5 ) 7 0' contains
every even substitution on | 1; . . ., J 5 . It will suffice to prove this
for literal substitutions (123), etc. Transforming (245) by (123) and
by (123) 2 , we reach (345) and (145). We then get '
(124) = (154)(245), (314) = (132)~ 1 (124)(132),
(12)(34) =(124) (134), (12)(45) - (12) (34) - (354).
But (123), (12) (34) and (12) (45) generate the alternating group on
five letters (Of. 265 266).
For p n = 3, O.!_i(6,p n ) requires an additional generator F 126 .
Expressing the latter in terms of the indices Yy defined by 158),
where we may now take a = /5 = + 1, it becomes V:
1) The reciprocal of E' is given by changing the signs of a and (3.
LINEAR GEOUP WITH QUADRATIC INVARIANT.
179
12
13
14
23
24
84
Y 12 =
V -
JL, -t o - *
Y r =
Y' =
-JL Q A
34
1 -1
1
1
111-1
1-1 1-1
1 1-1-1
-1 -1 -1 -1 -1
1 1 1-1-1
-100000
having determinant unity. F is seen to be the second compound of
-1 -1 11
-1 -1 -1
-1-100
1 -1
of determinant +1. Hence 0' contains F 126 when p n 3.
Since 0' contains the group 01_i(6, p n } but is of index 2 under
0_i(6,p w ), it follows from 182 that 0'= 0.^(6, #). Applying
163, we have the theorem:
For p n = 41 + 3, the group OLi(6,p n ) is holoedrically isomorphic
with the simple group Z.F(4 ; p w ) and is of index two under the second
orthogonal group 0_i(6,p w ), being extended to it by C 6 C 6 .
189. Theorem. The subgroup 0{(5,_p n ) is of index two under
0^(5, p n ) and is holoedrically isomorphic with the simple Abelian
group
By 161, A(4;,p n ), p > 2 is holoedrically isomorphic with the
second compound A^ of the quaternary special Abelian group.
A } 2 leaves absolutely invariant the Pfaffian [1234] and Y 12 +Y M .
By the introduction of the new indices ( 162)
*= ~2\ 12 ~~~ 34 ' ; ^i ~2~( 12 ~^~ 34 '' -M2 = ^i+-^> -^34 ^L~~-^?
^U,2 takes a form not involving Z and so becomes a quinary group Q
leaving absolutely invariant the quadratic function
The group Gr of all quinary linear substitutions of determinant unity
which leave O absolutely invariant will be proven holoedrically iso-
lorphic with 0^(5, p n ) and therefore ( 172) of order
12*
180
CHAPTER VII.
Since Q is holoedrically isomorphic with J.(4,p n ), its order is, by
115, half of that of O^b,^). To complete the proof of the
theorem, we then show that Q is holoedrically isomorphic with a
subgroup of 1 (5,p n ) containing all the generators of the group
0{(5,i>) defined in 181.
Let first 1 be the square of a mark i of the GF\jp n ~\. Set
162)
whence
54?
= 55 +
= fe5 ~~
Y = I & Y
_ (h 2 i 2 _|_ 2 i 2 j .2
^ 2 i *3 ' 4 T^ 5 T^ '
Hence 6r is holoedrically isomorphic with 1 (5,jp ra ). Proceeding 1 ) as
in 187, we find that Q 1 (Q expressed in the indices ,) contains
the substitution Of' J if and only if it be a 3 , 4 , a l so that Q f contains
(i 8 S 5 te). The latter with (^islU) w ^ generate all even substitutions
on | 2 , .
indices
preceding section. But
expressed in the
s
12
13
14
23
24
34
12 =
1/2
-i/2
i/2
1/2
13 =
-1/2
-i/2
-i/2
1/2
r f
1
23 =
1
rf
24 ~
-1/2
i/2
i/2
1/2
7 4 =
1/2
i/2
-i/2
1/2
This is seen to be the second compound of the following special
Abelian substitution:
o
o
4(1
R
It follows that Q' contains every Qfy (i, j = 2, . . . 6; i =)= j)- Also
<)' contains 0J0e and hence every OfjO^l For ^=5, we
take * = 2. Expressing the additional generator E Z45 in the indices
Yij, we reach the substitution (mod 5)
1) Comparing the transformations of indices 154) and 162), we note that
they are identical as far as || 8 , | 4 , | 5 and | 6 are concerned.
LINEAR GROUP WITH QUADRATIC INVARIANT.
181
'1
0'
1
3
3
1
1
2
1
2
^
.0
1,
which is the second compound of the special Abelian substitution
3^
J_
2
01
3 1
0002
Hence Q' coincides with 0{
Consider next the case
la not -square in the 6rjF[p n ]. Set
163)
*i = ii~~' a
54;
where a 2 -f- /3 2 = 1. Then O becomes | 2 -f 2 + | 2 + || -f J 2 . Hence
6r 6 is holoedrically isomorphic with O i (5, p n ). Reversing equations 163),
we get
m*} vt - - v v ofc .
As in 188, we find that Q l (Q expressed in the indices | f ) contains
every Ql\ 4 and the linear substitution (li^is) and consequently also
4, the transformed of the former by the latter.
Expressing in the indices | f the following substitution of 6r 6 ,
jr. -yf yr -y' . V
we get (7 1 C 4 Of 4 ""^ 2a/S . This 2}4 is not a <g;J since 2a; 2 -l==a 2 -/3 2
requires o; 2 = /3 2 . But C' 1 (7 4 =i ) ' 4 belongs to Q v Since ^" does
not belong to Q ( 164), it foUows that Of; 4
extends & to O^S,^). If Of;? denotes_0f;? when expressed in the
indices 7 ij} we find that the product K0i\ I has the form
182
CHAPTER VII.
y y y y
-*- -10 -*- 1X -* 03 -Let
13
14
23 J -24
V
*!&
y
-'
/3 2
aft
aft
-a/S aft
-ft 2 -a 2
-a*
-aft
a
-a 2 -
and is the second compound of the special Abelian substitution
ft a 0*
a' -ft
ft -a
-a -ft
Hence Q contains OllJOiJ?. We next show that Q l contains the
linear substitution (tgl^s), so that with (iifkis) Q will contain all
even substitutions on | 1; . . ., | 5 . Expressing (Ig^ls) in the indices Y/,- ;
we get
f i
a 13
2 2
2
a
1 1
i
2
2
2
P H
-a|3 a 8
a 8
1 a
-p
2
a
2 2
2
2
2
cu
2
2 2
2
2
P 2
2
a
2
2 2
2
2
2
-P 1-
- a P a 8
-a 8
1 + afJ
P
2
2 2
2
2
2
1
f?
g
a
1
L 2
2 2
2
2
2 J
which is the second
compound of
the special
Abelian
substitution
1
Y
-i
2
~" 1 (n ft
^ A r/y4
2 t a P
>> 2 ^ H
i
1
j
1 .
,
2
iifa
2
i
/A / /y , 1 /A
l
l
1
2 ^
PJ 2 { K ^P)
2
2
Lr4-
a\ ! / /3\
1
1
2 V a T-
W fCtfp)
2
2
For _p w = 3, 0^(5,^) requires an additional generator TF 1234 . For
a = /3 = 1, the following substitution
LINEAR GROUP WITH QUADRATIC INVARIANT. 183
y y _i_ y y - y t y
-'IS ^13 i~' 23; - 14 U ' "*24
when expressed in the indices J 1? . . ., | 5 becomes (mod 3)
fl 2 2 2
11210
11120
12110
00001
In every case it follows that Q l coincides with 0{(5,p n ).
190. Theorem. 1 ) If p n 41 + 1; Ov(6,# w ) is holoedrically iso-
morphic with the simple group HA(4:,p 2n ). If p n = 41 -f 3, 0{(6,p n )
has the maximal invariant subgroup {I, C^C^C^C^C^C^} of order 2, the
quotient- group being holoedrically isomorphic with HA(4, p 2n ). 7w
either case, 0*^(6, p n } is of index 2 under 0^(6,p n ) owe? is extended to
the latter by any substitution 0/j- not a Qij.
Consider the group H 1 of quaternary hyperabelian substitutions
in the GrF[p 2n ~\ of determinant unity. It has the order
h' = (# 4 n l)p 3 n (p* n -f l)^? 2 n (p* n l)p n .
The special Abelian group SA(4:,p n ) is a subgroup of H'. Denote
their second compound groups by A^ and JB^a respectively. By
161, ^2 leaves absolutely invariant the functions
For an arbitrary mark a? =J= in the (rjP[^ 2n ], the substitution
(0
'
CQ~~ pn
CD" 1
G>P n
is hyperabelian and of determinant unity. Its second compound is
Qt. I *lf
I y _ . nP n i
13?
14;
24
Taking p > 2, we introduce in place of F 12 , r 34 the new indices
165) fe = i(r u - rj, & = =(r 12 + rj, t . - ,
where J is a mark of the GF[p* n ~\ satisfying the equation
166) J^-^-l.
1) Bulletin Amer. Math. Soc., May, 1900; Transactions, July, 1900.
184 CHAPTER VII.
Reversing relations 165), we find
167) FuSfe-c^. JuzE-fe
Written in the new indices, the substitution Q f becomes
where .
= "
The coefficients Q, JQ, g/J belong to the 6rF[_p n ] since
Hence Q" belongs to the GF[p*\ and has determinant unity.
If , in 162, we set T=^ lf Z = 7J 6 , we obtain the present
transformation of indices 165). Hence, if we express any substitu-
tion [a] 2 of ^4,2 in terms of the new indices, we obtain a substitu-
tion, not involving 6 , the matrix of whose coefficients is given in
162. Hence A,% is transformed into a group A!' of substitutions
belonging to the GF[p*\ which do not involve | 6 and which leave
absolutely invariant ^
fel ~~ -M3-^24~~ -M4^23'
In order that A" shall contain the substitution
A Y' = a>P n + l T Y ' tnP* 1 - 1 Y
- fL - rJATV* *U ^23" J1 23?
it is necessary and sufficient (by 164) that o)* w + 1 be a square in
the GF[p n ~\ and hence that o be a square in the GF[p*"\ 9
Hence the group G", given by the extension of A 11 by Q", will contain
if and only if o be a square in the 6r.F[j) 271 ]. Now j& leaves
~ || + 7 2 || invariant and is therefore an 16 . We proceed to prove
that, if a be a square in the 6rF[p 2w ], every K is a Q%$ and every
Qife is a K, where a, /3 belong to the G-F[p n ~\. Let, in fact,
168) a =
Since o^^- 1 )/ 2 ^ 1, we see that a and /3 belong to the GF[p n ]. Also
169) a*-plj*=l
170) 2a/3 = y J(c)^- 1 - (o-^+ 1 ), 2 2 -l = i (w^- 1 +a)
Hence '^ has the form 1,'^, where a, /3 are defined by 168).
LINEAR GROUP WITH QUADRATIC INVARIANT. 185
Given, inversely, a Qf&, where a, ft are marks of the GF[p n ~]
satisfying 169), we can determine a square a in the GF[p 2n ~] which
satisfies 170). In fact, 170) may be written in the solved form
nP*- 1 - 2 2 - 1 2
(a, - ty\
of which the second follows from the first in virtue of 169). That
the first can be satisfied by a square a in the GF[p 2n ] follows
from the relation
For 03 a not-square in the CrF[p* n ~], Q" is the product of
an 16 , not a 16 in the GF[p n ~\, by the substitution A, neither
factor belonging separately to G".
Under the transformation of indices 165), JP 4 becomes
Y U Y U ,
where, by 166), J" 2 belongs to the GF[p n ~\ but is a not -square in it.
We introduce in place of the 3Ty new indices such that
Y Y -Y Y = 2 4- 2 4- t* 4- 2
*- 13^24 *- 14^23 2 "^ '8 "^ 4 ^ 5'
Then Y becomes J 2 | 2 - | 2 . Therefore, by 189, JL" wiU be
transformed into 0{(5,p w ).
For 1 the square of a mark i in the 6rF[p n ], we may take
* r _ < I < ^7" --_- ^ * ^ T^ _. ^ I * ^ "Ty^ _ - ^ * J*
-'IS == b2 ~T *fe3 ? -^24 :::::r fe2~*fe3? ~"-^14 = 4+ *5? -^23 :=r ?4 *?5'
As in 187, ^1 becomes an 4) 5, which is a 4,5 if and only if o
be a square in the 6rJP[p 2ra ]. Hence 6r" is isomorphic with a sub-
group of O v (6,p n ). The subgroup contains every i,s and every
16 45 , neither factor a $, but does not contain the separate factors.
For 1 a not -square, so that p n = 4? + 3, we may take co so that
Then A multiplies Y u and Y 2S by 1. The required transformation
of indices, transforming 6r" into a subgroup of 1 (6,j w ), is the
following:
/ 2 i /32 _
TT" _ . i^ ,q <j i^ fr V> ~i P ! T
J -23 == ?3 ~T P4"T a b5'
As in 189, A becomes in the new indices C^C^O^ a the last
factor being not of the form $ 4)5 , while (f^C^ = Qll^ belongs to
0{(5,^) w ). Hence G-" is isomorphic with a subgroup of 1 (6,jp w ).
The subgroup contains every Q 1Q and every OjgO^, neither factor
being a , but does not contain the factors separately.
186 CHAPTEE VII.
It follows that G" is holoedrically isomorphic with Oi(6, p n ) or
Oi(6,p w ) according as #"=4:1 + 1 or 42 + 3. But, for p > 2, the
order of the second compound H^ 2 of If' is h' and therefore
z
equals that of 0^(6,^). Hence 6r", #4,2 and 0^(6,p w ) are holoedric-
ally isomorphic.
By 132, we pass from H* to the quotient -group HA(4c, p* n )
by making the substitutions T x (106) correspond to the identity. The
corresponding substitutions of H^* are the identity I if p n = 4:1 -\- 1,
but are I and the substitution T changing the signs of the six indices
ify=4Z + 3. Hence O' v ($,p n ) is holoedrically isomorphic with
HA(,p* n ) if i>*=4Z + l; while, for p=4Z + 3, Of (6,1)*) has the
maximal invariant subgroup { J, C^ (7 2 (7 3 CC$ C & } of order 2, the quotient-
group being isomorphic with HA(4:, ^ 2ra ).
191. We proceed to determine the structure of the orthogonal
subgroups 0^(m, ^) w ), m > 7. Every m-ary linear homogeneous sub-
stitution is commutative with
C =&... C.: --& ( -!,...,)
(7 belongs to the group Oj^m,J^) only when m is even and ji = 1
(see 185). Suppose that 0^(m,^ n ) has a self - conjugate subgroup G
containing a substitution S neither the identity I nor C:
m
Suppose first that 5 reduces to the form
171) 65-^fc (^ = l,.. v m)
where J t - = 1. Then /S is merely a product of an even number of
the C 1 /, in which certain ones as Ct are lacking since S =)= C. If ^ v
and therefore m even by hypothesis, we may suppose that both C m
and C k (k <wi) are lacking, since dC m does not belong to Ol(m,p n ).
But if S= GiCjC r C s . . ., its transformed by TijT i1t (always in the
main group) gives S' = CkCjC r C s . . ., so that 6r contains the product
S S EE C/kCi.
From it we obtain in 6r the substitution OjC^ and are thus led
to the case treated in 193.
Suppose, on the contrary, that S is not of the form 171). We
may assume that 12 , 1S , . . ., lw are not all zero. In fact, either S
or its reciprocal will have at least one cc^ =j= in which i <j.
Transforming the one or the other by T^Tn 9 if j < m, we obtain a
substitution in Gr which replaces ^ by
LINEAR GROUP WITH QUADRATIC INVARIANT. 187
If j = m, we transform S by Ti k Tu (k not 1, i or m) and obtain a
substitution in G which replaces | t by
From the resulting substitution S in which a 12 , 18 , . . ., I OT are
not all zero, we derive a substitution S^ belonging to 6r and having
ii -*- ft ?2 =H L We S et #1 immediately if a^ + ^ ^ x f or j = ^ ^ ?
or m 1. In the contrary case, we have
172) ?i +!,-!, !.! = = !-i.
If 12 = 0, then a^=l and therefore i TO =0 by 147), contrary to
the assumption that a 12 , cr 13 , . . ., i m are not all zero. Hence a 12 =f=0.
Transforming $ by a suitable product of the ft, we can take
Transforming 1 ) the resulting substitution by Oj?, we obtain a sub-
stitution which replaces ^ by
If ^) w > 5, we can determine a and /3 in the 6r_F|j? n ] such that
2 +/3 2 =l, ^ +(,, + /!)+ 1.
Indeed, since ai 2 =a i3=H^; an( ^ a ii+ a i2 = ^? ^ e secon d condition
becomes 2a/3 =)= 0. But, of the jp re s sets of solutions in the GF[p n ]
of the first condition, where s == 1 according as 1 is a square
or a not -square in the field, only four sets of solutions have either
a or /3 equal zero. Hence, if^) n >5, there exist other solutions.
For p n = 3, we transform S, in which
a i\ *"* 0? ^12 "* ^13 ^ a i4 a is *** i lj
by TFg| 45 and obtain a substitution in 6r which replaces ^ by
for which therefore af t + J 2 = 0.
For _p w =5,'5 has a n = 0, J 2 == J 8 = | 4 = 1 in virtue of 172).
Transforming S by a product of the ft, we may take
The resulting substitution is transformed by E 2U into a substitution
of 6r which replaces | x by 2| 2 + 2| 3 -f a 15 5 H ---- , for which
4
!2
1) If the transformer does not belong to O'(m, #), we afterwards trans-
form by #4'^- Since the product Og'^OJ'f? belongs to the main group, the
transformed substitution will belong to <r. A like remark is to be understood
throughout this section.
188 CHAPTER VH.
Taking the reciprocal of the substitution of G which has
we obtain in G a substitution S in which af, -f- aL 4= 1. Then G
J.1 * -1 I
contains the product
o - c i/") ri o t fi r\ - o /~i ri
where S a = S~ C l C 2 8 is of period two and has the form
(m 1 \ / rn 1 \
2ji +^ 1 J ^*{2f> + ** m ***)
= 1,2, . ., m).
S* is not the identity since S would then be commutative with C, C 9
J. 4/ L &
and would therefore break up into the product of
by a substitution on 3 , . . ., | m .
We readily obtain the transformed S a > of S a by an orthogonal
substitution 0, in which i,j<k:
where by 145)
173) 2 + /3 2 + 4-y 2 = 1 (* - 1 if ^ < w; A - ft if A; = m).
A*
We have S a > = (SO)~ l 0^(80). But S' = SO has the coefficients
= 1, 2, . . ., m)
+ /J'
;, EE K S I (s = 1, . . ., m; s 4= i,j, fy.
If afi+ |i+ A>ttki=^= 0, we can find solutions in the GF[p n ~\
of 173), which make 'i = 0. We suppose a/i =f= 0, the trans-
formation of S a being unnecessary if an be already zero. Eliminat-
ing a from 173) and
174) ccccn + fittji -f- y^ii = 0,
we find the single condition on /3 and y,
~i T C.\ /)2^2 i 2\ i
1 io) p \otii-r Mji) H
If a4-|i=0, so that ^i=|=0 and a^i =)= 0, this equation deter-
mines /3, when y is assigned any value =|= in the field. Then 174)
determines a in the field. But, if afi + Ji be =|= 0, we multiply it
into 175), which then takes the form
4-
LINEAR GROUP WITH QUADRATIC INVARIANT. 189
The coefficient of y 2 being not zero, this equation has solutions for
in the field and hence solutions /3, y.
Transforming S a by the orthogonal substitution
|,+ y&+ *6*, 6} = *&+ '!,+ J/
176) + 02+ y*+ -.*!,
we obtain as above a substitution^ S% in which
We proceed to show that solutions of 176) exist in the GF[p"\
which make an = 0. We may suppose that a,-i, a/i, an are not zero,
since otherwise the result follows by inspection. If either of the sums
be not zero, the problem is solved as above. If both sums be zero,
men 9*9* -< 9 9 9 /-\
afi = AAI, A = square = 1, an 4- /i + all = 0.
Then the following set of solutions of 176) will make an zero:
a = <xji/<xii, P = OH/ <*n, <y =
192. Transforming S = SaC^ by 53 4m, 68 47n, ., O m _i 3 4m in
succession, we obtain in 6r a substitution S' in which 5i, 6i? ?
aj,_n, are all zero. Then by 143),
22 2.2 2
11 + 21 + 31 + 41 4-
Also
22 2
Hence si + ii + ^mi 4= ^; so ^ na ^ we can transform /S' by a
suitable Os 4 m into /S"' = & Q C 2 in which
Transforming S" by 0/453 (j = 7, 8, . . ., m 1) in succession,
we can obtain a substitution $ 2 = S^C 1 C 2 which leaves | 7 , | 8 , . . ., m _ i
fixed and has /3 41 = /5 61 = /3 61 = 0. If /3 42 , /3 52 , /3 62 are all not zero,
we transform $ 2 by O^g and obtain a substitution $2 in which we
can make 062=0 except in the case 1 )
1) If ^ 31 = or ^mi= 0, we transform $ 2 by 06345 or Oem45 and make
Ws-0.
190 CHAPTER VII
In the latter case, we transform S 2 by O^J&^e and require that
& = A 4- &/3 w2 + c/J 42 + d ft, 4* e/? 62 = 0,
#, = <*&! 4- &0i=0.
I akiTisr
a = - bfmijfa* c = - rfA/&s>
the second condition becomes an identity and the first takes the form
'81
The further condition a 2 H -- & 2 4- c 2 + d 2 -}- e 2 = 1 then becomes
177)
Since /S?, + !3i,H=l, 0L + &+ &+ p/i-l, it follows that
178) 7 + if+-
The coefficient of & 2 is zero for at most two values of /3 62 . In view
of 178), these two values of /J 62 can be avoided, if # n > 3, by an
earlier transformation of $ 2 by 4567 an operation not affecting the
previous argument. Also the coefficient of d 2 is not zero. Hence 177)
has solutions d, ~b in the field. The conditions 0g 2 = p' Bl = can thus
be satisfied.
For p n = 3, 178) requires p 1. The coefficient of W in 177)
is then zero only when ft =(= 0. If =(= 0, we can determine a and 6
(each 4= 0) such that
a @32 + bPm2 = 1, ^fti 4- &0rol= 0.
Since a 2 =6 2 =l, ^t=l, the remaining conditions become
These are satisfied modulo 3 by taking c = /3 42 , ^ = /3 52 , e = 0.
193. We have thus reached in G a substitution X which leaves
fixed 6 , J 7 , . . ., | m _i and which is not the identity. If
- Vl ^2 ^3 ^4 ^5 ^"l
we obtain from it the substitution C C 2 as at the beginning of 191.
From the known structure of the subgroup 0^(6,^"), it follows that
Gr contains all the substitutions of this subgroup. Transforming these
by suitable even substitutions on the &, we obtain all the generators
of 0(w, p n ), with which Gr therefore coincides.
194. In stating our results concerning the structure of the
orthogonal groups on m =j= 4 indices, we introduce permanent
notations for the simple groups reached. For the first orthogonal
LINEAR GROUP WITH QUADRATIC INVARIANT. 191
group the case m 4 is shown in 195 196 to be quite except-
ional. We denote by F0(m,p n ) the first orthogonal subgroup
0[(mjp n ), when m is odd, and the quotient -group of OiOwjjp") by
its maximal self - conjugate subgroup { J, (7), when m is even and > 4.
By 72, F0(m,p n ) has the order
for m odd; while, for m even, m > 4,
( \
n
The second orthogonal group on an even number m > 4 of indices
has a simple subgroup 1 ) $0(m,^), previously denoted by O' v (m, p n )
of order
n i
n
It will be shown in 197 198 that this result holds true for w = 4 2 ).
In both places, s equals 1 according to the form 41 1 of p n .
Theorem. The first orthogonal group O x (w, p n ) has for m even
and > 4 the factors of composition 2, F0[m, p 71 ], 2 and for m odd the
factors of composition 2, -F0[w,^ w ], the case m = 3, p w =3 &em#
exceptional. The second orthogonal group O v (m,p n ) on an even number
m > 2 of indices has the factors of composition 2, S0[m,p n ]. The
orthogonal groups on 2 indices are commutative groups.
195. In virtue of the identity
e + 8 + + B - e+i - e+i ----- a. ="(fc - t+o(b + {.+),
it follows from 169 that the group 3 ) L 8tp n of 2s-ary linear homo-
geneous substitutions of determinant unity in the GrF[p n ~\, p > 2,
s
which leave ^ X^i invariant is holoedrically isomorphic with 1 (2s, j p > *)
if 1 be a square in the GF[p n ~\, p > 2, or if 1 be a not- square
while s is even, but is isomorphic with O v (2s,^ n ) if 1 be a not-
square while s is odd. In particular, L 2 , p n is, for p > 2, holoedrically
isomorphic with 1 (4,j w ). In determining the structure of L^, p we
do not exclude the case p = 2.
1) In view of the not -square factor in its invariant, it first appeared in
the literature with the notation NS(m, pn).
2) This result is readily verified for the case pn = 3 not treated in 197 198.
3) The structure of this group was first determined by the author without
making use of its isomorphism with orthogonal groups, Proc. Lond. Math. Soc.,
vol. 30, pp. 7098.
192 CHAPTER VII.
196. Theorem. The factors of composition of L^n are
(if li > 2) 2,o 2
(if p = 2) 2, (2 2 - 1) 2* (2 2 - 1) 2",
>"=2 or 3, wftew the composite numbers 6
respectively are to be replaced ~by their prime factors.
To determine the quaternary substitutions leaving
absolutely invariant, consider the two pairs of equations 1 ),
12
+
The most general quaternary linear homogeneous substitution, leaving
invariant the pair of equations 179), for every value of H in the field,
is readily seen to be
181)
having the determinant (a^ /3y) 2 . For it we have
The group of the substitutions 181) is therefore simply isornorphic
with the binary group on the variables j^ -f- aejjg and rj 2 Krj v Since
the transposition (| 2 ifa) transforms the pair of equations 179) into
the pair 180), we obtain the most general linear homogeneous sub-
stitution, leaving invariant the pair of equations 180), for every x,
if we transform the set of substitutions 181) by ( 2 %)> gi y i n g the set
182)
The product of any substitution 181) by any substitution 182) gives
183)
i{ =
a^L yC
aC
r4
ll-
_ /?7? AT)
L/ JLJ \J -If
-ftD
-dB
s-
ccB yD
ccD
yB
^ =
ft A -dC
PC
dA
1) They give the two sets of generators on the ruled surface
= 0.
LINEAR GROUP WITH QUADRATIC INVARIANT. 193
The same result holds if the substitutions be compounded in reverse
order, so that the substitutions are commutative. Further, the only
substitutions belonging to both of the sets 181) and 182) are seen to be
184) 11 = ii, *?{*=%> 6J = &, ^2 = "?2-
The substitution 181) leaves J^ -+- | 2 ^ 2 absolutely invariant if
and only if ad ($y = 1. Hence there are (# 2n T)p n such substitu-
tions. It follows that there are
, (if 2> = 2)
distinct substitutions 183) for which
185) tf-/3y = l, AD-BC = 1.
The substitution T 2)X , defined in 114, wiU be of the form 183)
only if
Therefore A= a 1 , D = xa- 1 , # = # *, so that
ad - = fc- 1 ^ 2 AD - BC = jc- 2 .
It will thus satisfy the relations 185) only when x is a square
in the Gf [p n ]. Hence there are at least {(p 2n l)p n } 2 substitu-
tions 183) which satisfy the single relation
186) (ad - fty)(AD-BC)~l.
For p > 2 , Zf 2 ,p is holoedrically isomorphic with (4, p n ) and
therefore, by 172, has the order (p Sn p n )(p 2n T)p n . Hence
L^ p n is composed of the substitutions 183) alone. Those of these
substitutions which satisfy 185) form a subgroup L'^ p n of index two.
It is extended to the main group L^ p n by a substitution T 2jX -
For p = 2, the substitutions 183) which satisfy 185) form a sub-
group Z 2 ,2 of index two under L 2 ^ n - In fact, by 204, the order
of L^n is 2(2 2 -l) 2 2 2 , which is double the order of Z 2 >. The
transposition (Si%) serves to extend -L 2j2 to L^^ n ] for, if 183) reduces
to the form (ii%), then a A = aC = JB = 0, D = 1, whence
J.= 0=5 = 0.
For either p > 2 or j? = 2, the group -L 2)P of the substitutions
183) satisfying 185) has an invariant subgroup formed of the sub-
stitutions 181) which satisfy the relation ad fty = 1. The quotient-
group is holoedrically isomorphic with the simple group LF(2,p n ).
Indeed, it is clearly the quotient -group of the group of substitu-
tions 182) satisfying A D BC = 1 by the group of the substitu-
tions 184), a 2 =l, common to the two sets 181) and 182) under
the conditions 185).
DlCKSON, Linear Groups. 13
194 CHAPTER VH.
197. Theorem. For p n > 3, the second orthogonal group O v (^p n )
is holoedrically isomorphic with the group E^ p n of quaternary linear
substitutions in the GF[p n ] of determinant unity which leave absolutely
invariant the function
in which q = | x % + If + ^rfc is irreducible in the field.
For# n =3, the theorem necessarily fails, since q then becomes
(1 ~~ %) 2 - ^ or P n > 3, there exists a quaternary substitution in the
which transforms the invariant of the orthogonal group ,
= g + g + g + V Q (v = not-square)
into the function /i = t^ -f- Sa% + ^? + ^i* But, ^ or anv ^ /I i g
transformed into h~ l f by the substitution SJ ""* * *tu ^2 = ^"^s-
If 1 be a not -square in the G-F\_p n ], we may take i/ = 1.
Then the substitution of determinant a3
converts into the function
Of the p n - 1 sets of solutions in the &FO*], p > 2, of 2/3 2 - 2a 2 = 1,
two sets make a/3 = O. 1 ) Hence there are p n 3 substitutions which
reduce O to /J. The irreducibility of q follows from that of f -f g 2 ,.
If 1 = J 2 , where J belongs to the GrF[p n ], the substitution
of determinant Ja/3 transforms O into the function
Of the p n -f- 1 sets of solutions in the GF[p n ], p > 2, of 2i//3 2 - 2 2 = 1,
two sets make a/3 = 0. Hence there are p 71 1 substitutions which
transform <t> into f v The irreducibility of q now follows from that
of g + v g 2 .
198. Theorem. Whether p = 2 or p> 2, the group E^ p n contains
a subgroup El, p n of index two which is holoedrically isomorphic with
LF(2,p 2n ). .According as p = 2 or p > 2, -E^w & extended to
% (61%) or T 2)V .
1) According as 2 is a square or a not -square, the solutions are given by
a = or (3 = respectively.
LINEAR GROUP WITH QUADRATIC INVARIANT. 195
Let o be a root of the equation
which is irreducible in the GF[p n ] in virtue of the irreducibility
of q. The second root is therefore 0P n = a, so that (5 <F = A 2 . The
substitution
transforms the function F=XY-{- | 2 ^ 2 into f. Let a, /3, y, $ be
any set of marks in the 6rJ^[# 2M ] subject to the condition ad (ly = 1.
Then jP is absolutely invariant under the substitution [see 181)]
U: X'=aX+ r r] 2 , Y'=97-
If we regard 1 ) 1? ^, | 2 , % to be arbitrary marks of the GF[p n ~\,
Y will be conjugate to X with respect to the GJP[p"], while F will
be absolutely invariant under the following substitution conjugate
to U [see 182)]
U: X'=dX-M 2 , r'=
If therefore the product UUlae expressed in terms of the indices
%i> %> f2> %? *^ e resu lting substitution W will leave f absolutely
invariant and have its coefficients in the GrF[p n ~\. To give the
explicit form of W, let U and U become E7J and C^ when written
in the indices &, ^ Since the reciprocal of Z is
(tf - <j) % == z- r,
we find for J7 the substitution
OCC-6d
;2, _. 12^ ffft
A, U A CC O|J
oy
a-S
08 tia /3
7
-r
6y K
$
^/3
d
The coefficients of U are conjugate to the corresponding coefficients
of U r The product TF= U^U V is readily found to be the substitution
1) This interpretation is not a necessary one in view of the later explicit
calculations. .0:
13*
196
CHAPTER VH.
S
P
E
Q
G^d-^d
where x~ 1 = G G and
6yd)
yy
p'p
= xa~d da
Since every coefficient of TF equals its own conjugate with respect
to the 0-F[p], TF belongs to that field.
As in 196, the substitutions U form a group {U} holoedrically
isomorphic with the group of binary linear substitutions of determinant
unity in the fiiFJjr*}. The substitutions W form an isomorphic
group {W} leaving f absolutely invariant and therefore a subgroup
of E p n. Indeed, if we take U~W and U'~W, then to UU' will
correspond
UJJ1 - U^ = UJJlTJiU{ = U& - UJ U f ~WW,
since the set of substitutions U is commutative with the set U by
196. Moreover, an identity UU=U'U' or U'~ 1 U==U'U~
requires U 1 = U or CU 9 where C merely changes the signs of the
four indices. In fact, the groups {7} and {U} have in common only
the identity and (7. Hence CU is the only substitution in addition
to U which corresponds to the product W^U^U^^CU^ CU^ It
follows that the quotient - group of {7} by { J, C} is holoedrically
isomorphic both with the simple linear fractional group LF(2,p* n )
and with the group {W\ In particular, the order of {W} is
Y(p 4n I)p 2n or (2 4n l)2 2w according as p > 2 or p = 2. For
p > 2, # n > 3, .E 4j yi has the order (p* n + p w ) (p 2n l)_p n , being holo-
edrically isomorphic with r (4,^ n ), whose order is given in 172.
For p = 2, JE^ 2 is holoedricall/ isomorphic with the group leaving
li%+ 2% ~^~ ^^i H~ ^i absolutely invariant, whose order is shown in
204 to be 2(2 4 *-l)2 2 ". Hence {W} is of index 2 under E^ p n.
According as p > 2 or p = 2, {W} is extended to E^ p n by T 2 , K
or (ij%), where x is any not -square in the 6rF[p w ]. It is only
necessary to show that these substitutions are not of the form W.
If (g^) were of the form TT, then y^= 00 = 0, 5 = = 0. Hence
/3 == y = 0, (Fad = Wet, "dad = Gad. Hence would tidcc and Gad
CHAPTER VIII. LINEAR HOMOGENEOUS GROUP IN THE &F[2] etc. 197
and consequently also their product belong to the 6rF[p n ]. But
a^accdd belongs to that field only when a or d vanishes, so that
a d-p y = 0.
If W reduce to the form jP 2 , x , then ace = x, dd = Jt" 1 , /3 = y = 0,
S = Q = 1, JR =P = 0. By the latter, ad ad. Then 5 = 7 gives
d~a = 1. But ad = ad /3y = 1. Hence a = a 7 , so that H = a 2 ,
a belonging to the GF[p n ].
CHAPTER
LINEAR HOMOGENEOUS GROUP IN THE &F[2] DEFINED
BY A QUADRATIC INVARIANT.
199. Theorem. If a quadratic form with coefficients in the 6rF[2"]
can not ~be expressed in the field as a quadratic form in fewer than m
linear homogeneous functions of | 1? . . ., m , it can be reduced by a linear
homogeneous substitution belonging to the field to one of the canonical
forms
JF^gjIg + g^H ----- him-stm-i+Bi (m odd)
where 'k is zero or is a particular one of the values A f for which
r\ fe fc I 2/fc2 j_ iffc2
V ^^ bwi 1 w i * fern 1 ~r ^ 5m
^'s irreducible in the GF\2 n ~\.
We first prove that, if m > 3, / can be transformed into a
quadratic form having a n =0. If every a^- (^,j = l, . . ., m; <j)
were zero, /* would reduce modulo 2 to the form
This being contrary to our hypothesis, we may assume that cr 23 =f= 0,
for example. We may also suppose that ff 22 =|= 0, since otherwise the
transformed of f by (Jj^) would have a n = 0. The terms of f which
involve | 2 may be written thus,
a 2ai| + 2 0*18 & + "23^3 + ^SA H H 8m6j)-
198 CHAPTER VIE.
Hence the inverse of the following substitution,
bs === #12 il i #23 3 i #24 b4 ~r ' ' ' ~T~
' =
bt - bz
will transform /" into fc g $- v
^ * 1 ' *
V y ' ' '} )
summed for i, j = 1, 3, 4, . . ., m; * <j. Applying the substitution
we obtain as the new coefficient of |J the function a 22 A 2 + j8 11; which
may be made to vanish by determining A.
We may therefore suppose that o^ = in our original function /.
Since the KIJ are not all zero, we may assume that a 12 =4=0. Apply-
ing to / the inverse of the substitution
H
= i (* = 1, 3, 4, . . ., m)
we obtain the function
t,>*=
^ i
2
Replacing ^ + y 22 2 + y 23
by | 1? we get
Similarly, if m ^ 5, we can transform /"' into
& _1_
l $2 i^
If m be odd, we reach ultimately the form
c fc I c c I i c c I -. c2
fel b2 ~T 3 4 ~i ' ' i STO 2 fern IT * bm
Applying to it the substitution which replaces | m by x
obtain F.
If m be even, we reach ultimately the form
= | 1
we
If 6 w _i + /J6 8 ,-i8 J +y6^ be reducible in the GF[2*], i.e., be the
product of two linear homogeneous functions of % m i and {,, an
evident substitution will reduce to F Q . In the contrary case, a, /3, y
are certainly distinct from zero, so that the substitution
LINEAR HOMOGENEOUS GROUP IN THE F[2n] etc. 199
will belong to the 6r-F[2*]. It transforms into
187) y 2 + I 3 i4 + + 8m-sSm-a + 65.-1 + 8-l6m +
$ being such a mark that the equation
188) | 2 +i + d =
is irreducible in the GF[2 n ]. It follows from 188) that
Hence 188) has a root | in the GF\Z n ~] if and only if
The left member being its own square in the GF[2*] and hence
either or 1, it follows that 188) is irreducible in that field if and
only if
189) d f 6 8 +a 4 +..-+d* | - 1 -i.
Applying to the quadratic form 187) the transformation
Sii-i Im-i+lSm, 15 lo ( !,.. .,w; ^4=m-l)
the constant d is replaced by
d' = d + A + A 2 ,
which is therefore a root of 189). Giving to K all possible values
in the F[2], we obtain the 2"- 1 roots of 189). Indeed, if in
the
we must have A x = A or A -f- 1. Hence all irreducible quadratic forms
in two variables of the GF[2 n ] can be transformed linearly into each
other. Applying, finally, the transformation
187) becomes JPi T .
200. Changing the notation used in exhibiting F, the canonical
quadratic form for an odd number 2m + 1 of indices may be written
The conditions upon the coefficients of the substitution S:
200
CHAPTER VIII.
>=1
m
= 0, 1, . . ., m)
( !,..., m)
in order that it leave V absolutely invariant are seen to be the
special Abelian relations 1 ) 76) for ft = 1 together with the following:
771
190)
191)
It follows from 114 that every set of solutions a {j , /3 l7; y^-, d o -
in the (rJP[2 w ] of the relations 76)^=! leads to a special Abelian
substitution
whose determinant A is unity. 2 )
The determinant of the coefficients of the 2m quantities *, tf< in
the 2w equations 190) is seen to equal A. Hence, since A =j= 0,
tt; = 0* (* = !,..., m).
It follows that S takes the form
S':
the coefficients of S 1 being subject to the Abelian conditions 76)
only. The group of the substitutions S is therefore holoedrically
isomorphic with the special Abelian group SA(2m, 2") of the sub-
stitutions Z. The structure of the latter group is given in 1 IX
201. Changing the notation employed in exhibiting the function F*,
the canonical quadratic form for 2m indices may be written
1) Since p = 2, we have 1 =& -f- 1 i n the field.
2) For a direct proof that A =|= 0, see American Journal, vol. 21, p. 244.
LINEAR HOMOGENEOUS GROUP IN THE GF[2] etc. 201
We study the group 6r;. of 2m-ary linear substitutions in the 6rF[2 n ],
192) 8: ft.
which leave fi absolutely invariant. The conditions upon the coeffi-
cients of S are the Abelian relations *) 76), for ft = 1, together with
( j < m)
(j-m),
193)
/ /" ^ / . ., \
(jf < m)
(j - m).
Since S must be an Abelian substitution in the 6r.F[2 n ], its reciprocal
is obtained by replacing c^, /3/y, y^-, d/^ by respectively d^- t -, ft-/, ?>,;,
<tyf. Writing for 8~ [ the conditions 76) and 193), we obtain the
equivalent set of conditions 78), for ft = 1, and
( j < m)
1 (j = m),
194)
(/<
Among the simplest substitutions leaving fa invariant occur
< m if
(if A
which reduce, when A = 0, to the N m j ti( , Rm,j,x, etc., defined in 114.
According as A = or A = A', 6a is called #&e /2rs or ^Ae second
hypodbelian group 2 ). The name arises from the fact that
subgroup of the special Abelian group SA(2ni, 2 71 ).
s a
1) This also follows from the fact that the invariance of f^ implies that
m
of its polar. Hence, if jp = 2, Cr^ leaves invariant ^ (l^^-f- i^a)' w ^ ere i t -i
i? fl and ^. 2 , r) i2 are sets of cogredient variables.
2) For the case n = 1 , these groups were studied at length by Jordan,
Traite" des substitutions, pp. 195 213 and p. 440. For general n, they were
set up and investigated by the author in the papers, Quarterly Journal, 1898,
pp. 116; Bulletin of the Amer. Math. Soc., 1898, pp. 495 510; Proceed. Lond.
Math. Soc., vol. 30, pp. 70 98; American Journal, 1899, pp. 222 243.
202 CHAPTER VHI.
202. Theorem. If m > 1, 6a may be generated ~by the sub-
stitutions 1 )
195) M^ Ni t j iX (i,j = 1, . . ., w; H arbitrary in the field).
We note that Jfi transforms J^,,-, x into >, ? -, x and
y.. Further, for i,j < w if A = A f , we have
Ry EE 6;, t, 1 (?,/, i ft, /, i, .
2 2 =M MP
T^ = T^ T^ Jf - 1 T^ T
But every mark of the Gf [2 n ] may be expressed as a square p*.
Except in the case m 2, A = A', we thus reach every T i}X . In the
latter case, we derive every Ti fX from the formula
196) ^,l,xem,l,x-^-l Nm^^LMiMmT^,^.
Taking first K A" 1 / 2 , we find that i may be derived from the sub-
stitutions 195). Applying 196) again, we reach every TI^X*.
To prove that every substitution S satisfying the relations 78)^=1
and 194) can be derived from the substitutions 195), we first set up
a substitution T derived from them which, like S, replaces ^ by
where by 194),
m
197) ^ Kl . yij+ ifi m + i y \ m = o.
.7=1
a) If a u =f= 0, we may take as T the product
*i an $1,2, a 12 ^2, 1, y 12 Ql, m,
since it replaces | x by
/-
b) If <x n = 0, y n =j= 0, we may take for T the product
/w, 1, i m
which replaces | x by
1) The structure of 6rj being evident from 203 if m = 1, we exclude this
case henceforth.
LINEAR HOMOGENEOUS GROUP IN THE F[2] etc. 203
c) If !>= YIJ= (j = 1, . . ., "k - 1), but ait and 7/1* not both
zero, we may, for Jc < w, proceed as in case a) or b) and obtain a
substitution T' which replaces * by /i and is derived from the sub-
stitutions 195). We then take T = T'P k .
d) If iy=yi^=0 (j = l, . . ., m 1), the proof given in c)
applies if A = 0, since then Pi m is generated by the substitutions 195)
of ft. For A = A', this case cannot exist, since the equation
requires K lm = yi m = on account of the irreducibility of Q. Then
would /j ^ 0.
It follows that S = TS 1 ^ where S leaves ^ fixed but is a sub-
stitution belonging to ft. Let ^ replace i^ by
nt
where, by 78), ft = 1, and 194),
m
198) d u = l,^/Miy + A^f m + A^f m = 0.
j=i
The product
^" = ft, 1, /9 ia 2, 1, (T lt - -R/n, 1, /9 l7n Cm, 1, d l
replaces |j by | A and % by
which equals /"' since the coefficient of | t equals j8 lt by 198).
We may therefore set S i =S l S Z) where $ 2 is a substitution of
ft which leaves | x and % fixed. Then by 78),
an = fti = y,-i = d<i =0 ( = 2, . . ., m).
The relations holding between a^-, /3 t ^, y^-, d;y (*, j = 2, . . ., m) are
seen to be the relations 78) and 194) when m 1 is written for m.
Proceeding with $ 2 as we did with S, etc., we find ultimately that
$ = jT'Z, where T' is derived from the substitutions 195), while Z
is a substitution of ft which affects only | m and ^ OT ,
I: %m= a%m+ rVm, tyn' ^ & + ^
The conditions 78), 193) and 194) become, for m = 1,
199) ad + j3y = l, /3 + A 2 + A/3 2 = A, yd + V+ Atf 2 = A,
200) d/3 + Ad 2 +A/3 2 =A, ya + Ay 2
Combining 199) with 200), we may replace 200) by
201) 0(a + d) = y( + d) = A( -f d) 2 .
204 CHAPTER Ym.
Suppose first that a -\- d =%= 0. By 201), Z becomes
a 4-
202) OS : ~\ (ad + tfv? + >M 2 - 1).
Suppose next that /J + y =f= 0. Applying the above procedure to
it follows that Z x = y . Hence I = 0' y JLf m .
Suppose finally that a-\-d P + y 0. Conditions 199) and
201) become rf+^-l, / = 0,
so that Z=I or-if m .
In every case, I = 0% 6 or Z = 0% Y M m . If * = 0, 06'=!^.
and the theorem is proven. If A = A f , A' being suitably chosen, we
prove in the next section that every 0% is a power of L = Om
and may therefore be derived from the substitutions 195).
203. Let Q be a primitive root of 2W+1 = 1. It will satisfy an
equation belonging to and irreducible in the GF[2 n ],
If we set = A" 1 , Q = % m /r] m , we find that ^ 2 n -|- ^m~\- Im^m is
irreducible in and belongs to the GF[2 n ]. Changing the variable
from Q to = Ap, we obtain for the irreducible equation
203^ g 2 I h >l 2
Since the roots of 203) are and 2 ", we have + a 2 " = 1.
We make the transformation of indices:
204) | m = A 8 / 2 *- 1 F 12 H
Solving, we find, for ^) = 2,
Then ^ + ^ + A a = y y
The substitution 202) takes the form
906") Y" f = +Y Y r *- r^Y
4\juj - L i2 TJ -12) *I4 . -^iiJ
where , __ / , ^\ _, _ ! A \ ?
T = CC -f- ( CC -f- ) 0j X =; O -j~ ( CC
, 1 ^ I ^ - / I c\\ O t _ O / t 5^\ 9 ^
XT == CCO -f- 0(CC + O) -j- ( -f~ O) rrzad
We have T 2W + 1 = 1 since (mod 2),
*1
In particular, L = 0m takes the form
207) F/ 2 =9r,
, 2 ,
LINEAR HOMOGENEOUS GROUP IN THE 6?F[2] etc. 205
The substitutions 206) are evidently powers of 207), Q being a
primitive root of a? 27l + 1 = 1. Hence the substitutions 202) are powers of L.
Inversely, every substitution 206) for which r 2 "+ 1 = l may be
transformed by 205) into a substitution 202) of the GF [2*]. In fact,
a + d = r + T- 1 , a = x -K + ^K <? = r- 1 -f (T + r" 1 ) ff,
so that a + d belongs to the GrF[2 n ~\ and likewise a since
The number of substitutions 206) is 2 n +l. The number of sub-
stitutions 202) is therefore 2 n +l. Furthermore M m ~ (im^m) takes
the form .
We have therefore a new proof of the results at the end of 202.
It is worth while to verify independently that the number of
substitutions 202) is 2* + 1 according as A = A' or A = 0. We have
only to determine the number of sets of solutions in the GF[2 n ] of
208) a<? + A 2 2 -M 2 <J 2 =l.
The result for the case A = being evident, we suppose that A = A'.
The left member of 208) vanishes only when a = d = 0; for, otherwise,
(1 + _|_ A 2 ^- 1 ) 2 =0, a 2 = tfa/d
would be reducible in the field, contrary to the irreducibility of 203).
Hence each of the 2 2n 1 sets of marks a 17 d\, not both zero, in
the GF[2 n ] will make
X 2 = 0.
Then will ajn, ^/Jc be a set of solutions of 208), and inversely
every set of solutions of 208) may be so obtained. Hence, if A = A',
the number of distinct sets of solutions is (2 2 " l)/(2 n 1).
204. We can now readily determine the order Q m ] n of ft. The
number of distinct linear functions /j by which the substitutions
of ft can replace | A is P$ n 1? if P? denotes the number of sets
of solutions in the 6rJF[2 w ] of 197). For m > 1, the pair of equations
has (2 n + x l)P^-i jn sets of solutions when r = and has
sets of solutions when r runs through the series of marks =J= of
the 6rF[2 n ]. We have therefore the recursion formula (m > 1)
2/n 2)
'
206 CHAPTER VIII.
According as . >L = or k = 1J, the number of sets of solutions of
is Pi=2"+ 1 1 or Pjf2 = l- We find by simple induction,
p W _ 1 = (2* _ 1) (2<- - 1> + 1) , P$ - 1 = (2" + 1) (2< - 1).
The number of distinct linear functions f is 2"( 2m ~ 2) . In fact,
198) determines /3 U in terms of fa, dj (j = 2, . . ., w), so that the
latter may be chosen arbitrarily in the GrF[2 n ].
It follows therefore, from 202, that
Q W = ( P W - - ;n 92(m-l) QW . fw > 1"!
^m, n \J- m,n *-) & '*m 1, n \7t* ^ *-)
By 203, we have the initial values
We now readily obtain the formulae
2 )... (2 2w l)2 2w ,
2 )... (2 2w l)2 2w .
205. Theorem. Those substitutions of ft which satisfy the
further relation
1, . . .,m
209) JO, ft y, d) =
a subgroup of index 2 wte/i awi/ Jtf,- extends to ft. If m> 2,
s subgroup is identical with the group generated as follows:
Ni,j, y .} (i, j = 1, . . ., m; ^ arbitrary in field).
If m = 2, ^ *s identical with the group
We first prove that every substitution of Ji satisfies 209). To
do this, it suffices to show that, if Z be any substitution of ft
which satisfies 209), the products MiMjT., N f j iX I. will also satisfy 209),
the case m = 2, being treated later. Let Z have the form 192).
a) If the product MjL be expressed in the form
210) 81*
- ____ 4 _ -*
we have
(/ - 1 A01
ft'r-l|!'.'.,';
LINEAR HOMOGENEOUS GROUP IN THE JF[2] etc. 207
Hence
upon applying 209) and 76). Hence MjL does not satisfy 209),
while Mi Mjl. does.
b) If the product N m j tX T. be expressed in the form 210), we have
r., firs ft-. (V, S = 1, . . ., m)
= $r f S = 1 . . . m S m
(r = 1, . . ., m).
Hence I(t( f ff f yf f f) equals
r, s = 1, . . . , m
s==m,j
^mji^Pmj I
But the last two sums are zero by 76) and 193).
c) An analogous proof holds for .$/,/, x .(i,j < m), the above
terms involving A# 2 not being present.
d) Since the substitutions Qi,j,*j -B/,/,x (*> j 1> > w) and P t -y,
jF* y (^ 7 < m, if A = A') may be expressed as a product of the N t - ,- x
*> *" \ / t/ / J v M. - " M
and an even number of the MI ( 202), the products T;, X Z, (?,/, X Z,
etc., will satisfy 209) if Z does.
Inversely, every substitution S of Gri which satisfies 209) belongs
to Ji. In fact, by the proof given in 202, S is of one of the two
forms Kj KM m , where K is derived from 1 ) M { Mj, &/,> ^,y, x ,
jg.^^ (^ j = 1, . . ., m); P, v , r f , x (*, j < m if A = A'). Since 5 shall
satisfy 209), it is not of the form KM m . It remains to show that
these substitutions MiM j} #/,/ >x , . . ., 2^, x belong to Ji.
For m > 2, J* contains Qi,j, y ., the transformed of JVJ,/, X by
MjM k (k=^=ij j); also U,-,/, x and Q&t,*, the transformed of J/i,/ fX and
C-,^ x respectively by MiMj. Applying the formulae at the beginning
of 202, we reach P fj and T^T^ (i,j<m, if A = A'). Then Ji
contains Tf^I^p 1 , the transformed of the latter by Jf^JM}. The
product of the two gives T^*.
p
1) By 196), L and therefore every 0* is derived from MfM^, Q% ly and
208
CHAPTER VIII.
x,
For m = 2, Ji contains M M 2 , T^, -2V 2 ,i,x> JR 2> i,
If A == 0, Ji contains P 12 = Q^i, i $1, 2 , i Q%, i, i.
The fact that M M 2 and JVi, 2)X do not generate 7 , for m = 2,
follows readily from 196. Since Jf t M 2 transforms .2Vi, 2 , x into -Ri, 2 , x ,
every substitution derived from the two former may be given the
form V or VM M 2 , where V is derived from JVi. 2)X and JRi, 2>x . The
latter two are of the form 181). Hence the group of the substitu-
tions V is a subgroup of the group of the substitutions 181) having
the order v = (2 2 w 1) 2 n .
Hence M 1 M 2 and the -2V"i, 2 , x generate a group whose order is at
most 2v. But 2v < (2 2 l) 2 2 2n , the order of J" for m = 2.
It follows similarly from 197198 that M M 2 and JV 2 ,i, x do
not generate Jr for m = 2. This result may be shown directly for
the case n = 1, when Ji' has the order 60 ( 204). In fact, setting
M = M M 2 , N = Nt t i t i, E=E 2> i,i= M~*NM, the group generated
by M and N contains only ten distinct substitutions:
J, M, N, E, NM, EM, EN, NE, NEM, ENM.
For m = 2, the structure of J Q was determined in 196 and
that of Jx in 197198.
206. Theorem. Ihe senary first hypoabelian group J" in the
GF[2 n ] is a simple group holoedrically isomorphic with LF(, 2 n ).
We obtained in 163 a senary group G-[,z, leaving absolutely
invariant Y Y Y Y -\-Y Y
which is holoedrically isomorphic with the simple group LF(4, 2 n ).
To identify Gl, 2 with J Q (m = 3), we set
Y - Y=Y=Y='n Y = n Y n
X 1S 91) - L 13~ 27 -*-14 '3? J -23 '/3J -*-24 '/2? -*-34 '11'
The general substitution [] 2 of Gl, 2 , given in 164, may be written
1 ? 2 ?S ^?3 ^?2 ^1
6i
Ai
Ai
s
22
723
33
23
fll
732
A
721
731
32
^
22
In this form the notation agrees with that employed in 201 for
the substitutions of J" . In view of 165, the above general sub-
stitution of 6r4, 2 must satisfy the relation (mod 2)
a l Al+ a ! 2 ^12+ C 13 ^13+^21^21+ 22 ^22+ ^23 ^23+ a 31^31 + a 32 ^32+ ^33 ^33 =*
But this is relation 209) for A = 0, m = 3, which defines the sub-
group JQ of the first hypoabelian group. Hence #4, a = T .
LINEAR HOMOGENEOUS GROUP IN THE
etc.
209
207. Theorem. T}te senary second hyperabelian group J^ in
the GF\2 n ~\ is a simple group holoedrically isomorphic with HA(4,2 2n ).
We begin as in 190, but make the following transformation
of indices, including the transformation 204) for m = 3:
* = * = ^ = -*
The invariant of the second compound group is transformed thus:
V V V V J_ V V - fc ~ ' fc -*- i fc ~. i i 2 i 3 ^2
* 12-^34 - t !3- L 24 "I" - t !4- z 23 Si r /i i ?2 T /2 ' '3 r /3 ~T~ * 3 i A7 /3'
If we take
the substitution 206) becomes in the new indices a substitution 202)
with coefficients in the GF[2 n ]. In particular, if o be a suitable
primitive root of the GF[2 2n ], x will be the primitive root g of
X 2 n +i === i^ w e thus reach, by 207), the substitution L.
We next express in the new indices the general substitution [] 2 ,
given in 164, of the second compound A t 2 of the group of qua-
ternary Abelian substitutions of determinant unity in the GF\2 n ~\.
For example, it will replace | 2 EE Y IB by
13
12
snce
/t
V
13
1
3
.
13 13
-r <*
A
^3
'
13
fe2+ i
4
61 +
23 ^ ~ 24 1
h
13
,
18/2 -1
+
34
(A ? 3
+
A ^ s ).
dent
of
| 3 is = (mod 2) and that of t/ 3 is A" 1 /
2 13
12
13
13
#11 #10
Cv
#1/1
12
+
34
81 ^32
.
4
3 #34
=
by one of the Abelian conditions, while A 2 ^ -1 +(?-fl = by 203).
Proceeding in this manner, we find that [] 2 takes the form
,_
14
14
A 2
14
14
14
1
14
13
12
24
23
-j
fef _
13
13
A 2
13
13
13
'2
14
13
12
'
24
23
211) ,,
?3 =
tf
12
14
A"" Y
12
13
1 A- 1
12
12
+A- jft
12
24
A"^
12
23
V 3 =
0001 00
1
f _
24
24
A 2
24
24
24
/2
14
13
1 2
24
23
i
i? =
23
23
A~ T
23
23
23
'1
14
1 3
12
24
23
DlCKSON Linear Groups. 14
210
CHAPTER VIE.
To prove that this substitution satisfies relation 209) for m = 3 7
consider it to be expressed in the notation used for the general
substitution [a] 2 of 206. The condition 209) built for the sub-
stitution 211) therefore becomes
14
14
23
23
14
13
23
24
13
14
24
23
13
13
24
24
12
12
= 3 (mod 2).
The left member may be written (mod 2):
a.
n
a a
BU a u
13 14
#32 #33 #34
+ #31
#22 #23 #24
+ #41
#22 #23 #24
#42 #43 #44
#42 #43 #44
#32 #33 #34
4- 2 a 2 4- a 2 a 2
~ WU * UW
1122
1221'
Upon expanding according to the elements of the first column the
determinant on the left of the following identity
"11 "12 "13
#
14
#21 #22
#
23
#
*31
K
32
a
a
41
a
42
33
"43
24
*34
*44
1
*>
we obtain the first three terms in the above expression together with
r $l
#12 #13 #14
#32 #33 #34
+ #11 #22
#33 #34
+ #81 #12
"23 "24
+ 41 #12
"23 "24
#43 #44
"43 "44
"S3 "34
"42 #43 #44
It remains to show that the sum of these terms together with
#11 a la + "12 "21 * s zero - Upon applying the Abelian relations (mod 2),
"33 "34
"11 "12
"32 "84
"12 "14
"32 "33
"12 "13
"43 "44
"21 "22
"42 "44
"22 "24
"42 "43
"22 "23
"23 "24
"21 "22
"23 "24
"21 "22
"13 "14
"31 "32
y
7
"43 "44
"41 "42
"33 "34
"31 "32
"23 "24
"41 "42
the sum is seen to be congruent to zero (mod 2). The substitutions
211) therefore belong to J^. Their number equals the order
(2 4?l 1) 2 3n (2 2n 1) 2 W
of the quaternary Abelian group SA(i, 2 ra ) ( 115), which was shown
above to be holoedrically isomorphic with the group of the sub-
stitutions 211) leaving ^ 3 fixed. We prove in the next section that
this number equals the total number of substitutions belonging to
Ji' (m = 3) and leaving ^ 3 fixed. It follows that the substitutions
211) include the following substitutions of Jv not altering r] m :
LINEAR HOMOGENEOUS GROUP IN THE GF[2n] etc. 211
These substitutions must therefore belong to the group C, the second
compound of HA(4:, 2 2n ) when expressed in the indices ,-, 77,-. Also
C contains L and therefore also MiM m by formula 196). Hence C
contains all the generators of Jj' (m = 3). But the order of C, being
equal to that of HA (4, 2 2 "), is
(2 4w - 1) 2 3 * (2 3 + 1) 2 2 (2 2 * - 1) 2",
which equals the order of 7i- (m = 3). Hence JV = C.
208. Theorem. If m = 3, tffte number of substitutions of Jx
which leave % m fixed is
(2 2n + 1) 2 2 w (2 2 w I) 2 2 2n .
If a substitution S of Ji' does not alter \ m and replaces ??, by
fm =
>
we must have, in virtue of the relations 78) and 194) ,
m
212) d mm = l,^T/3 mj ,d m ,+ Aft w = 0.
We proceed to prove, inversely, that if /3 m _,-, d m j be any set of solu-
tions in the 6r.F[2 w ] of 212) there exists a substitution X in Ji'
which leaves | m fixed and replaces rj m by /* m .
If fimj = d w; - =0 ( j = 1, . . ., m 1), then /3 OT7n = or A" 1 . Hence
we may take as Z the identity or M^M m L respectively.
In the contrary case, let /3 m2 =f= 0, for example. Then Jv contains
a first hypoabelian substitution T leaving | OT and t] m fixed and re-
placing % by
since /S ro id TOl + /5m2<J = in virtue of 212). Then we may take
For m = 3, the number of sets of solutions in the GF\2 n ~] of 212):
Pmldml + /?m2#m2 + ^m7w+ ^/^mm =
is (2 2 *-|- 1)2 2 ". Indeed, there are 2"- 1 distinct values in the GF\2 n ] of
^ == Pmm\ A>p mm .
By 204, /3 m id ml +/5 m2 d m2 =T has 2 3w +2 2w -2 sets of solutions
if x = 0; while, if T have any one of the 2 n ~ 1 1 possible values
=j= , it has
14*
212 CHAPTER VIII.
9 4n f 9 3 i 9 2 nn\
2 -12 +2 -2 j =
2 W 1
sets of solutions, and therefore in all
sets of solutions. But each value of r furnishes two values of /3 mwz .
209. Theorem. The liypodbelian groups J% on 2m > 6 indices
are simple.
Let K be a self- conjugate subgroup of Ji containing a substitution
6! = (u & + y<^), *?' = (fofe + ft/ty) 0' =
not the identity I. We first prove that K contains a substitution =4=
which multiplies ^ by a constant. Let S replace | t by
where by 194),
in
213) Wiy+ A! m + A r f m = 0.
If /^ =(= a iiSi? we nave ne f the following three sub -cases.
a) ^j =f= 0. Then Ji contains the product
T = TI } n 1 R 2 , 1, a la V2, 1, y 12 -*C, 1, ! TO fe,
which replaces ^ by y^ 1 ^ and ^ by the function
This equals f ly since the coefficient of ^ is congruent to or n modulo 2,
in virtue of 213). Hence K contains ^^T" 1 ^! 7 , which replaces
If Ji contains a substitution j^ which leaves ^ and ^ fixed
and is not commutative with 8 lt K will contain the product
which leaves |j_ fixed. Suppose on the contrary that 8 1 is commu-
tative with every substitution of Jx. which leaves | x and ^ fixed.
Among the latter are jR 2 , s,x and 3, 2, * If we equate the two
expressions by which SiE 2j s,y. and ft, 3, x$i replace ?; 3 , we find
v.+*i-v.+( )&+( )t-
Similarly, if ^ be commutative with 3, 2, x; we have
LINEAR HOMOGENEOUS GROUP IN THE GF[?n] etc. 213
Hence ^ = ( ) | 2 . Transforming ^ by P 12 we obtain a substitution
=J= J which multiplies | t by a constant and belongs to K.
b) Let 7ii = 0, 12 = a 13 = = i m _i= and, if A = 0, also
= 0. If A = A', we must have i m = j>i m = 0, since 213) reduces to
whereas Q ( 199) is irreducible in the field. Since /i=|=ffiii> we
cannot have y 12 = y 13 = = y\ m \ together with yi TO = 0, if A = 0.
Transforming $ by a suitable P 2%; ( j < m, if A = A') , we reach a
substitution $' having y 12 =(= an d belonging to K. Transforming /S"
by M 2 M S , we reach a substitution of K in which y u = 0, 12 =4=
[case c)].
c) Let 7> n = ; 0^2, . . ., ai m _i, i m be not all zero if A = 0;
let y n = 0, 12; . . ., cfi OT _i be not all zero if A = A'. Transforming S
by a suitable Pay, we reach a substitution $' of K having 12 ==(= 0.
Then J^ contains
T ^T^a^Q^a^ ' VI, ft, 0* -4^8, 4, ft, ' fe m, aj w ^2, m, y t m
which does not alter | x but replaces 2 by
Since y n = ? this reduces to /i in virtue of 213). Hence K con-
tains $ 17 the transformed of S' by T, which replaces | t by | 2 .
If jSj be commutative with both Rs^i t y. and J^s, 2, x 7 it merely
multiplies | 3 by a constant, so that its transform by P 13 gives the
required substitution. In fact, SiRj^^ x and -Ra,/,x$i replace ^- by
respectively . . x N} . , Nf .
^j + Aii, i?} + ( )6 8 +( )&.
In the contrary case, " contains the two products
sr l - R^X Si j? 3) /, x ( j = i, 2)
which leave ^ fixed and do not both reduce to the identity.
Next, K contains a substitution =j= 1 leaving | x and ^ /?^ec?. We
have previously reached in K a substitution S =)= J which replaces |j_
by alj. Let it replace ^ by (j3i,-|,--f- ^%). By an Abelian
.7=1
relation 78), d n = or 1 . By 194), we have
214
CHAPTER
a) Let U = 0, ft,- (?i,- = (j = 2, . . ., m - 1), and, if A = 0,
also ft m =di m =0. If A = A', then must ft m =d lwi =0 by 214).
Evidently S = T^ a 8, where 8 leaves | x and % fixed. By the
Abelian relations 78), 8 involves only the indices |/, ^ (i 2, .;.,m)i
If $ x be not commutative with every substitution Z t of Jj, which
does not involve | , then .ZT will contain a product
1?
which leaves ^ and ^ fixed. In the contrary case, S l is commu-
tative with H^Z,K and Q$,<z,y., so that, as shown above, ^ will replace
| 2 by p| 2 . Since S L is to be commutative with M%M 5 also, it will
replace % ^J 0^2- Hence, by an Abelian relation, s =1; whence
Q = 1. Transforming S by P 12 , we obtain a substitution =f= -^ which
leaves |j_ and -ft fixed and belongs to 5".
b) Let /3 n == 0, ft,-, d 1>? - ( j = 2, . . ., m) be not all zero if A = 0,
but let /3 n = 0, /3i^, di^- ( j = 2, . . ., m 1) be not all zero if A = ti.
Then by 202, Jj contains a substitution T, affecting only
vi. i
which replaces
^' = 2, .. ., m),
Hence K contains 8 19 the transformed of S by T.
ali and % by - 1 ^ 1 +| 2 .
If J;i contains a substitution F, leaving | 1?
is not commutative with S f K will contain
replaces ^ by
J 2 fixed, which
which leaves 81 and ^ fixed.
In the contrary case, /S^ will be commutative with E 2 , 3, i and
R m a and M*M m , Equating the two functions by which $i-R 2
* //fcj Wj ** t* "* i ^J t/
and -R 2 , 8 ,x#i replace %, we find ig = ( )6 8 +( )5 2 . Equating the
two functions by which SiR mt 3, x and R m ,z,xSi replace ^ m , we find
that 6g = ( )I 8 + ( )8 TO . Hence |g = ^| 3 . Since ^ is to be commu-
tative with Jf 3 Jf m , ^' 3 =P%- Then 0=1. Transforming S by P 13 ,
we have a substitution =j= J in ^" which leaves ^ and ^ fixed.
c) Let /3 n =f= 0. We can determine a substitution 5' of JiT
form similar to that of S but having also d J2 =J= 0. In fact, if A = 0,
the products ftytfi^ (j == 2, . . ., m) are not all zero by 214). Trans-
forming by a suitable P 2t -, we have ftg^is^^- If A = ^/? the sam(
result follows unless ft y = ^= ( j = 2, . . ., m 1), in which cas(
either ft m =}=0 or di m =%=0l>j 214). In the latter case, we can tab
LINEAR HOMOGENEOUS GROUP IN THE 0F[2n] etc. 215
tf lm =|= 0, transforming by M 2 M m if necessary. Transforming the
resulting substitution of the form
(fti 4= > <?l
by the substitution !2 OT , 2 , x, we obtain a similar substitution having
in ?/j the additional term xdi m % 2 -
Recurring to S', in which # 12 =(= 0, we transform it by J^d" 1
and obtain a substitution $ t of K having the form
Consider the following product, leaving | t , T^, | 2 fixed,
TFEE 3, 2,^,^3,2,^, C,Ml
It replaces ^ 2 by the function
in which the coefficient of 2 equals /3 11 a~ 1 +ft 2 V 214), since
^ 12 = 1 and d n = a" 1 . Hence W transforms S into the substitution $ 2 :
Let p = ftitt"" 1 =(= 0. If among the substitutions 3 , 2 , ^ N*, 3, i>
2j, ^ Jfg JKf, etc., of Ji, leaving | 1? ^ and ^| 2 + ^ 2 invariant, there
exists one, say V, which is not commutative with $ 2 , then K contains
which leaves | x and T^J fixed. In the contrary case, we find, on
equating the functions by which $ 2 8 , 2 , n N^ 3) t and 3,2,^-^2,3,1^
replace | 2 , that
By one of the relations 194), we find a 23 = 0. Then, if $ 2 be also
commutative with T^^M^M Zj we must have 3 = | 3 , Vs^'fo-
In proving that ^T contains a substitution $ =^= I which leaves j
and ?h fixed, we assumed the existence of the indices
only. But, by the relations 78) and 194) S is a hypoabelian sub-
stitution on the indices | f , 17,- (i = 2, . . ., m). Hence, if m > 4, a
repetition of the previous argument shows that K contains a sub-
stitution =f=7 involving only the indices |/, f?j (* = 3, . . ., m). After
m 3 such steps, we reach in 5" a substitution =j= J and affecting
only six indices | f , ^- (i = m 2, m 1, m). In view of the sim-
plicity of the senary hypoabelian groups, K will contain all the sub-
216
CHAPTER VIE.
stitutions of Ji will affect only the last six indices, and, in particular,
MiMj, Nij^y, (i, j = m 2, m 1, m). Transforming the latter by
suitable substitutions P rs (r, s < m, if A = A'), we reach all the
generators of J; v . Hence K=Ji, so that J;. is a simple group.
In view of the importance of the subgroups J" and 3% of the
first and second hypoabelian groups respectively, they will be
designated by the more explicit notation FH(2m, 2 n ) and SH(2m, 2 W ).
They are both simple when m > 3. The second is simple and the
first is composite for m = 2 ( 196 198).
210. MISCELLANEOUS EXERCISES UPON CHAPTERS I VIH.
1. Every m-ary linear homogeneous substitution in the 6rF[2] leaves
invariant the function S 1 + S 2 + \- s m , where s r denotes the sum of
the products of the m indices taken r at a time.
2. An m-ary linear homogeneous substitution in the G-F\_p n '\ of
determinant D multiplies by D the function of the indices
Si
if
k
2
::.*
Si
2
w 1) v>n(m 1)
K
5m
Hence Y is a relative invariant of the group G-LH(m,p n ).
3. The structure of the m-ary linear homogeneous group in the
G-F[2"\ which leaves |? + l + + n absolutely invariant may be
derived from that of the special linear group SLH(m -- 1, 2 n ).
[Take as new indices X = ^ -j- J 2 H ----- f- ^ TO and | 2 , | 8 , . . ., ^J.
4. Those substitutions of the hyperorthogonal group G~ m ^ n ( 143)
whose coefficients all belong to the G-F[2 n ] form a group Gr, a subgroup
of the group of Ex. 3. Prove that G is generated by the binary sub-
stitutions . - - -
and that G is a solvable group of order 2 wm ^
5. Consider the group C of 2 m-ary substitutions in the G-F[p n ],p > 2,
.7=1
common to the special Abelian and orthogonal groups. Being Abelian
its reciprocal is obtained by replacing t -^, y,-y, jS,-^, ^,7 by d) t -, - - %/,
- |3yi, ay respectively. Being orthogonal, its reciprocal is obtained by
replacing the former by ay,-, ft-,-, yy,, ^,-. Hence must
c)
LINEAR HOMOGENEOUS GROUP IN THE F[2] etc. 217
The conditions that an arbitrary substitution $, for which c) hold, shall
be orthogonal are the same as the conditions that it shall be a special
Abelian substitution.
6. The most general 2w-ary substitution commutative with the
special Abelian substitution M = M i M 2 . . . M m has the form
The group in the G-F[p n ], p > 2, commutative with M is identical
with C of Ex. 5.
7. Setting -1=I 2 , X t - & + 1^, A i$ = ct^ Iy ih S of Ex. 6
becomes
If 1 be a not -square in the 6rF[j> w ], we may pass, inversely, from
an arbitrary substitution Z in the G-F[p 2n ] to a substitution S in the
G-F[p n ] by equating the coefficients of J and I. Z leaves invariant
the function
m m
- l
- /,,) (a + ^) .
Hence , if p w be of the form 4 Z -j- 1 , the group C is simply isomorphic
with the hyperorthogonal group 6r n2) ^ . If 1 be a square in the
6rF[j7*]j we introduce the further indices Yf = & J^,-, J5,-y ^ ,-y + Jy,-y,
when >S becomes
leaving invariant^ ^-J 7 !. Inversely, from every substitution Z 1 we
z=i
derive a substitution of the form S. The group of "dualistie" substitu-
tions is simply isomorphic with GrLH(m^p n \ since the B,-j are determined
in terms of the A's.
8. The simple group A (4, p n ), p > 2, of order
contains just two sets of conjugate substitutions of period 2. The one
set contains ^-(p 2n + l}p* n substitutions conjugate with Tj _i. Those
21
of the other set are conjugate with .MjJfg and are in number
1) according to the form 4Z + 1 of # n .
218
CHAPTER IX.
9. The group of all quaternary linear homogeneous substitutions in
the G-F[p n ~\ which leave absolutely invariant the functions Ji^i+ig'fe
and Ij -j- i/j has a subgroup of index 4 holoedrically isomorphic with
10. The squares of the substitutions of the first orthogonal group
^i( m ? P n ) generate the subgroup 0{(m, p n ) of 181.
11. To the subgroup E^ p n of E^ p n corresponds, for p > 2, the
subgroup 0^(4, p n ) of O r (4,p w ) defined in 181.
12. In order that AI + fa shall be capable of transforma-
tion into |ii(i + 2 ~ ) by a binary linear substitution with coefficients
in the 6rJF[p 2 *], it is necessary and sufficient that the ratio /^/Ag shall
belong to the GF[p s ~\.
CHAPTER IX,
LINEAR GROUPS WITH CERTAIN INVARIANTS
OF DEGREE q > 2.
211. Consider the group 6r 3 of substitutions in an arbitrary field
x\ =
S
which leave absolutely invariant the function of degree g' = 3
It will be convenient to employ a symbol, analogous to a determinant,
ABC
a /3 y ~A(5c + Ayl) -\-Bccc -\-Bya + Cab -j- Cfia.
a & c
The conditions that S shall leave O 3 absolutely invariant are then
r r r
215) ^ -Z^-V'j = 0, ^ -Zlftf fty w y = 0, y 1 Ni^ijHij = 0,
LINEAR GROUPS WITH CERTAIN INVARIANTS OF DEGREE q > 2. 219
r
01 K\ ^^ (TiiA-Tii _L27^^-^
^IDJ ^ {JLjijAijbik -f- JUij(,jjA.jk-f- hijlijLiii:) =- U,
217)
ik ) = 0,
218)
0,
T
219)
i 1
Li j LI % LU
A'ij ">ik AH
v v I
"ij "ik fat
vf j lilt Wilt
220)
[*>ik
n it
(if j-fc =
(unless j = Jc
where , throughout, i, j, It 1, . . ., r, while k =}= j in 216) and 219),
and ={=,;> ^ in the first of the relations 219); together with relations
derived from 216), 217), 218) and 219) upon interchanging L, I, I
with M, ^, m or with $t f 'v,n. But relation 216) must also hold for
fc = j f being then derived from the first one of set 215) upon multiply-
ing the latter by 3. Similarly 219) must hold for &=j, being then
derived from 216), 217), 218). Lastly, the first of relations 219)
must hold for t = k = j, being then derived from the first of the
set 215) upon multiplying by 6. Hence the above conditions must
hold for i, j, k = 1, . . ., r independently.
Let j be any fixed integer <^ r and consider the 3r equations
216), 217), 218) for k 1, . . ., r. Taking as unknowns the 3r products
221)
(-!,..., r),
the determinant of their coefficients is seen to equal the determinant
of S and is therefore not zero by hypothesis. Hence the products 221)
are all zero. From the analogous conditions,
222)
223)
= U n,j
=
n- t j =
(i, j 1 , . . ., r),
(i, j = 1, . . ., r).
Expanding the symbols in 219) according to the last columns
and applying a similar reasoning to the resulting equations, we find
224) LI fht ~h Lik^ij = Lijlik -f Liklij = (/;* + hwj = 0,
We obtain similar identities 225) and 226) between the M, ji,m and
the N,v,n. From 220) for j =)= k and the following of type 219),
220 CHAPTER IX. LINEAR GROUPS WITH CERTAIN INVARIANTS etc.
T 1\/T T
-L'ij-LM-ik-L'it
r
2
= o, y
*J
each set holding for t = 1, . . . , r, we derive as above
227) Lij(i ik + Mi^j = iV Wf 4 + M ik l^ = A^-* + f*a-^ = .
By a similar process, we get, for 7c =j=^',
228) -LtfV,-* + N ik lij = Z^Wffc + N ik lij = A^w/jb + v t - t Z^ = 0,
229) Nij!i ik + M i1c Vij = Nijmik + M ik n if = v^m^ + p ik nij = 0.
212. Theorem. The group 6r 3 is generated by the substitutions
230) (flj/y/), (flj/0,-) , P,v = (a;,-^) (y/^) (^-^).
together with the substitutions of the type
231) xl = L { Xi, y'i = iity h z\ == /,, i/ftin/ = 1, (^ 1, . . ., r).
Let ^S denote any given substitution of 6r 3 . We can determine
a suitable product Z of the substitutions 230) such that Z$ = ^ will
have the coefficient n =)= 0. Then by 221), 224), 227), 228), we find
Hence ^ replaces ^ and ^ by the respective functions
The product Z-L/S^^, where Z t is the identity if jt u 4= ^
Z t = (^ ^) if ft n = 0, will be of the form S with the new coefficient
0. Then by 222), 225), 227) and 229), we find
Hence must n n =(= and therefore N n = v n = by 223). Hence
replaces x l9 y l7 % by L n x 1} ^ n 2/i, w n ^j. respectively. Also
Since the determinant of >S 2 is not zero, the coefficients i 2 ;>
iHfg;, j^2; ( j = 2, . . ., r) are not all zero. We may therefore determine
a suitable product Z' of the substitutions 230), in which i, j > 1,
such that Z'$ 2 = $ 3 will have L 22 =f= 0. Proceeding as above, we
find that S = Z"$ 4 , where Z" is derived from the substitutions 230),
while $ 4 merely multiplies x 19 y 1} lf x%, y 2 , z% by constants. After r
such steps, we reach a substitution of the form 231).
Corollary. Any substitution leaving 0s invariant may be
expressed as a product AB, where A is of the form 231) and S is
derived from the substitutions 230).
CHAPTER X. CANONICAL FORM AND CLASSIFICATION etc. 221
213. The preceding methods may be employed 1 ) to investigate
the group G q of linear substitutions S on rq indices with coefficients
in an arbitrary field which leave absolutely invariant the function
For g> 2, it is seen that S = AB, where A merely multiplies each
index |,-y by a constant, while S is a permutation on the indices , ;
having the imprimitive systems 2 )
-
The substitutions ^1 form a commutative group which is transformed
into itself by every substitution S and is therefore self - conjugate
under G q . The quotient -group is the group of the substitutions IB.
The latter has a self-conjugate subgroup E formed by the direct product
of r symmetric groups, the general one being on the q letters |/i,
&>?? t 9? the quotient -group {L}/E is a symmetric group on
r letters, viz., the r sets 232). The structure of the group G q , #>2,
is therefore completely determined. The result is essentially different
from that for the case q = 2 (see 195).
CHAPTER X.
CANONICAL FORM AND CLASSIFICATION OF LINEAR
SUBSTITUTIONS.
Canonical form of linear homogeneous substitutions 3 ), 214 216.
214. Consider a substitution with coefficients in the GF[p n ]j
m
S: ti= tt <& (.' - I, . . ., m).
In order that S shall multiply by a constant K the linear function
we must have
or
1) Proceed. Lond. Math. Soc. } vol. 30, pp. 200 208. On pp. 203 204 the
numerical factors are incorrect; C should equal ^!f 2 ! . . . tt\ The proof
however is valid.
2) J? replaces the indices of any set |a, if 2, . ., If? by indices all in one set.
3) For = 1, the results are due to Jordan, Traite, pp. 114 126. The
simple proof by induction of the fundamental theorem is due to the author,
American Journal, vol. 22, pp. 121137.
222 CHAPTER X.
Hence K must be a root of the characteristic equation
"
21
K . . .
-**
= 0.
Corresponding to each root K y we may determine at least one set
of solutions fa of the above linear equations and hence one invariant
function 77.
If A (K) = has m distinct roots K, K 2 , . . ., K m (not necessarily
in the initial G-F[p n ], we reach m linear functions ft, %,..., ^w,
which S multiplies by) K lf K%, . . ., K m respectively. These functions
are linearly independent with respect to the variables | f . For, if
constants exist such that
Ml + M2
= 0,
we have on applying the substitutions S, $ 2 , . . ., /
identities
>mnm= 0,
1 the further
1
1
1
= 0,
But the determinant
1 1 .1
Hence
= = / m = .
Introducing the linear functions ^ as new indices in place of
the {;, the substitution 5 takes the canonical form
S':
(i = 1, . . .,
If we take in place of ft a suitable multiple of ft, we may suppose
the reduction of S to S f to be accomplished by a transformation of
indices of determinant unity.
Suppose, however, that the roots of A(2T) = are not all
distinct. Let
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 223
where F k (K), Fi(IC)j . . . are the distinct factors of &(K) which
belong to and are irreducible in the GF[p n ]. Designate the roots
of F k (K) = 0, and of F t (L) 0, etc., by the notations
Theorem. Ify a suitable transformation of indices, S can be
reduced to a canonical form of the following type:
y - 2, . . .,
i) ( j 2, . . .,
2 +a l + > ;--l) (j = 2, . . .,
|(fi ^ +> -f & Ji+^-
where a l + a 8 '+ %+ = a, 6 X -f 6 2 H ---- = 0, . . .; aw^ where the
indices have the properties:
1) The indices ^ * (s =* 1, . . ., a) are linear homogeneous functions
of the initial indices | f having as coefficients polynomials in K with
coefficients in the G-F[p n ]i
2) The indices ??,, are conjugate to the r)o s , being obtained by re-
placing KQ by KI in the coefficients of ^ ;
3) The indices Jo* (s = 1, . . ., /3) are linear homogeneous functions
of the indices f whose coefficients are polynomials in L with coefficients
in the 6WF[jf]5
4) The indices g,-, are obtained from the g * &2/ replacing L by L^ etc.
5) T/^e ^a indices ^ (* == 0, 1, . . ., 7c 1; 5 1, . . ., a) may be
replaced by ka linear homogeneous functions yi of the initial indices ,-
with coefficients in the 6rF[j>*], such that S replaces each y it by a
linear homogeneous function of the y is with coefficients in the GF[ff f \\
6) The I /3 indices g,-, may be replaced by an equal number of linear
homogeneous functions Zi of the with coefficients in the 6r-F[j) n ], such
that S replaces each by a linear homogeneous function of the z ig with
coefficients in the field; etc.
224
CHAPTER X.
For the case a = /3 = = 1, we obtained above the canonical form
77-1 = KiTfin (i = 0, 1, . . ., fc 1)
where ^ 01 = /*(ii, . . ., ,; -BT ) and 17*1 = /"((;!, . . ., (U; JT/), and, similarly,
&i are conjugate with g 01 . The new indices therefore have the
properties 1) 4).
We will prove the general theorem by induction, supposing it
true for every substitution belonging to the GF[p n ] whose char-
acteristic determinant has no irreducible factors other than F^ (JE),
Fi(K\ . . ., and has these to a degree at most a -- 1, /3, . . . respec-
tively. We will prove that the theorem is true for any substitution S
for which these factors occur to the degree , /?, . . . respectively,
where # > 1.
Corresponding to the distinct roots K Q , K v . . ., Kki ofFk(XC) = 0,
we obtain as above a set of linearly independent conjugate functions
A , AJ, . . ., Afci which S multiplies by K Q , K Q , . . ., Kk i respectively.
We may introduce these in place of an equal number of the original
indices, e. g., | m _ A+1 , . . ., | TO . The substitution S then takes the form
(i = 0, 1, . . ., k - 1)
k 1
t; =
coefficients /3^- belong to the GF[p n ]. Indeed, we may set
A,- =
-f -f
where the X,- are linear functions of the | t - with coefficients in the
6r.F[jp w ]. Since the A/ are linearly independent, the X,- must be
linearly independent functions of the J t . Since
the X, can be expressed as linear functions of the A,-. Taking the Xj
as new indices in place of the A/, S' takes the form /S", a substitu-
tion on the indices X,- and | t - with coefficients in the 6r.F[_p w ]. But
S n replaces | f by
/fc 1
for * = 1, . . ., m
arbitrary
,- and X,-, the coefficients
Since these functions belong to the field for
, d,-^ must belong to the field.
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 225
Since the determinant of a linear substitution is not altered by
VL linear transformation of indices ( 101), the determinant of S'
equals the determinant of S:
We may, therefore, consider the following substitution in the GF[p n ]:
m k
(i = 1, . . . ? m
of determinant =%=0. Also, the characteristic determinant A(JT) of S
equals that of the transformed substitution $', viz.:
k 1
fti
ftl
'12
Hence, the characteristic determinant of $ x is
Hence, by hypothesis, 8 can be reduced to a canonical form of the
above type. Applying the same transformation of indices to $', it
takes the form S:
= 2,
L/ 0' = 2,
the expression for ^[-, being derived from that for r]Q S by replacing
K Q by .KJ; the expression for ' 5 from fo 4 upon replacing Z by Z^-, etc.
To simplify the form of S, introduce as new indices
Y =
(5=1,..., ),
k 1
= ^o 4-
and their conjugate functions
^02 > ^03 by
DlCKSON, Linear Groups.
Z i8J . . . Then S replaces 3T 01 .
15
226
CHAPTER X.
^o^oi + "lo^o + x, [i + OK
i = l
k l
(K t -
k 1
i=o
respectively. By choice of the Aj i} we can make the terms in brackets
all zero; those of the first sum by choice of A llf . . ., -4.1 j_i, those
of the second by choice of -4. 10 , A 21f . . ., A 2 k i, those of the third
.,
by choice of J^ 0? J. 31 ,
ing To* (s = l, . . ., a).
/S replaces Z 01 , Z 02 , .
*I. A like result holds for the remain-
by respectively
k 1
*
+
Since Kt L =%= ; the coefficients of A,- may be made to vanish by
choice of the BU. Hence, S takes the form $ 2 :
.
-l)
ii 0,1; :..,*-!)
(i = 0, 1, . . ., Z - 1)
If the constants
are a ll zero no furthei
suppos<
. . ., Y it
reduction is necessary. If any two are not zero, as gj and
for definiteness that a x ^ a 2 , and introduce in place of I
the new indices
"^ -rr ^ TT / .* _
-I- * = J- * ~ -*- 1 a, -4- *' (. .1 ~
CANONICAL FORM AND CLASSIFICATION OF LINEAR, SUBSTITUTIONS. 227
The substitution S 2 replaces 3^ 1; Y {j (j = 2, . . ., a^) by respectively
' Ki F, *i(F,, + F,,_0 ( J - 2, . . ., O-
Hence, the introduction of the Ytj has the effect of setting g? =
in S 2 . Proceeding similarly, we can suppose that <p, ifr, %, . are
all zero but one, say #> =j= 0. In the latter case, we set
and find for S 2 the canonical form
Yv-d, (j-2,...,aj
> ** i+2+J ~ -^-iv-^* ai+aa+^"T-^*a 1 + a +> V^
In every case we reach a canonical form of the type given in
the theorem, for which the indices Y is have the properties 1) and 2).
But the indices Z^ are linear functions of the | t - with coefficients
which certainly involve Z,- and apparently 1 ) also K { . If the K t be
involved, we proceed as follows. From the canonical form actually
reached, S = YS 1 , where Y is the partial substitution on the indices
Yij, not altering the indices ^-, etc., while S does not involve the
indices Y^-, but affects the ^-, etc. Setting
r,. = y s + y'sZi + $K? + $-*Et-\
(s = 1, , . ., a; i = 0, . . ., ~k 1)
where the /'s are linear functions of the J z - with coefficients in the
G-F\_p n ], we can evidently introduce the y's as new indices in place
of the Y is , so that Y takes the form of a substitution belonging to
the 6rF[jp w ] and affecting only kcc indices. Likewise, by introducing
in place of the Z ii9 etc., an equal number of linear functions %, etc.,
belonging to the G-F[p n ~\, it is possible to give to S the form of a
substitution in the field and affecting only m lea indices. Its
characteristic determinant is \Fi(lOj\P . . . Hence, by the hypothesis
made for the induction, ^ can be reduced by a linear transformation T
to a canonical form
1) By the considerations in the text, we may dispense with the difficult
proof, analogous to that of Jordan, Traitd, pp. 121 122, that the Zi$ do not
involve Kt, but the single imaginary Li.
15*
228 CHAPTER X.
where the Jy are linear functions of the ( with coefficients involving
the imaginary LI only. As the transformation I does not alter the
indices which Y affects, we obtain the desired canonical form.
215. Consider as an example the substitution in the GF\_p*~\ 9 p n
of the form 41 1 ,
O. fcf _ _ Ofc fcf fc f _ fc' _ fc
" i - ~ ^2 ?4? ?2 1> '3 63; 4 ~ = *3>
having the characteristic determinant
where K 2 -\- 1 is irreducible in the field. A root of i 2 = 1 belongs
to the GF\_p* n '] but not to the GF[p n ]. The functions which S
multiplies by i and i are readily found to be respectively
Introducing A 1? A 2 in place of the indices # 2 , # 3 , S takes the form
f ^_ * 1 /y ' _ /yi | * /O 1 * / O 1 if * 1 If
The partial substitution of determinant unity,
multiplies y = x ix by i and multiplies ?/ 2 = ^ -|- # 4 by - i.
Introducing y l and y 2 as new indices in place of x and # 4 , S takes
the form
f 4-. 4 " 1 t _ ' 1 1
.. === 1/A-tj A 9 - - t'An
Introducing as new indices,
2/2 =
>S takes the canonical form
where Aj and A 2 are conjugate linear functions of | 1; | 2 , 3 , | 4 , and
likewise for y i9 y 2 .
216. Theorem. Two linear homogeneous substitutions S and
in the G-F[p n '] on the indices 1? | 2 ; > %m have the same canonical
form C if f and only if, T is the transformed of S by a linear home
geneous substitution W in the GF[p n ] on the same indices.
If T= W~ 1 SW, then 8 can be reduced to 1 by the introduc-
tion of new indices defined by the transformation W and therefore
and T have the same canonical form.
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 229
Suppose, inversely, that two substitutions S and T in the G-F[p n ~]
on the indices |,- can be reduced to the same canonical form by the
respective transformations S' and T'. Let I' denote the transforma-
tion from the indices | 1? . . ., % m to the indices 7?,- 5 , &,, . . ., where
w : V _i_ V"' fT _1_ V" TiT? [ [ T/"(* 1) T7"* 1
/ 1 9 ^- I * 9 * ^i I * 9 * j I i *^
(*-!,...,; -o;i,.:.,fc-i)
(s = 1, . . ., 0; i = 0, 1, . . ., Z 1)
3T S , Y/, . . ., Z SJ Zsj . . . being linearly independent linear functions of
the I,- with coefficients in the GrF[p n ~\. Denote by x the trans-
formation of indices from ^ a , & s , ... to Y s > YJ, . . ., Z s , . . . By
hypothesis, 1 ' transforms T into the canonical form C. Let x trans-
form C into C t . Then I'r is a substitution in the GrF[p n ~] which
transforms T into C*, likewise in the 6rjF[j n ]. Similarly, let S'
denote the transformation from the indices | 17 .., | OT to the indices
flfc*, &., , where
Denote by 6 the transformation of indices from ij it> is , ... to
Y, 9 . . ., Z s , . . . By hypothesis, S' transforms S into the canonical
form C, which in the same substitution on the indices iy f ,, &,, . . .
that C is on the indices ^,, fe,, . . . Let <J transform (7 into C7 a .
Then, if E be the substitution in the G-F[p n ~] which transforms
Y s , . . ., Z t , . . . into Y t , ..., Z,, . . . respectively, then
GO = H C/ili"
It follows that the product I'rE(S t G)~ l is a substitution on the
indices | f with coefficients in the 6r-F[# n ] which transforms T into S.
217220.
Substitutions commutative with a given linear substitution*).
217. Let the given linear homogeneous substitution S on m
indices | f with coefficients in the GF[p n ] be brought to its canonical
form /Sp For definiteness, suppose there are three sets of new indices,
nij (*-0,l,. ..,*-!; j-1,. ..,); &v (-0,...,Z-l; j = l,. ..,);
* (* = 0, . . .,^ - 1; j = 1, . . ., y);
where
1) Amer. Journ., vol. 22, pp. 121137; Proceed. Land. Math. Soc., vol. 32,
pp. 165170.
230 CHAPTER X.
In order to express more compactly the canonical form S 19 we let
a, &, c denote an arbitrary one of the respective sets of integers
a) 1, % 4- 1, ty + a* + 1, . . ., % + a 2 H ----- h a r + 1;
b) 1, ^ + 1, fci + ftg + l,..., 6 1 + 6 a +...+ 6. + l ;
c) 1, q +1, Cj + <* -hi, . . ., q + c 2 H ----- he, + 1.
Also let A denote any integer < not an a, B any integer < /3
not a &, any integer < y not a c. The canonical form ^ may
now be written as follows:
-1 (i = 0, 1, . . ., & 1)
f fl = A' t + Li^iBl (i = 0, 1, . . ., I 1)
il>ic= Qt^cc + Qi^ic-i (i =0, 1, . . ., q 1).
An arbitrary linear homogeneous substitution I 1 on these indices
replaces ^ by a linear function
233) IDx^xu + T.E& &. + 1^1 ^ w ,
where (as henceforth) the summation indices have the series of values
K = 0, 1, . . ., k 1; Z = 0, 1, ...,-- 1; /it = 0, 1, . . ., q -- 1;
tt1,..., a; w=l, ..., 0; w = 1, . . ., y.
In order that T be commutative with S it is necessary that 233)
involve only the indices r) iu (u = 1, . . ., a). Equating the functions by
which TtSi and S 1 I 1 replace ^- a; we get
6'
/*, 0'
Equating the coefficients of the ?/s and J's in this identity, we get
Ml -JCrZft (
iDxAl = Ky. D K A 1 + Xe -^x 4
7" 7? L 7
l = JU2.-&IB 1 T J
Since Jf f =|= Z^ ? the third equation gives Elbi = 0, where & is
any integer > 1 of the set b). If & 1 is a J5, the fourth equation
gives JEJ]_2 = 0. In the contrary case, & 2=J=J5 1, and the third
equation gives JSJ?__ 2 = 0. Similarly, according as & 2 is or is not
a JB, the fourth or third equation gives -E]j_3 = 0. Proceeding in
this manner, we find that every E\ a v = (A = 0, . . ., I 1 ; v = 1, . . ., /3).
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 231
By a similar argument, the first and second equations give
Equating the coefficients of the ^'s in the above identity, we
find analogously that every F^ , = 0. Hence T^ replaces rj ia by
Consider any a such that a + 1 is an A and equate the functions
by which T^S^ and 8 T replace rji a +i. Among the relations occur
"c \ = F 1 1 + Q F c + '.
From these three pairs of equations we find (as above) respectively
Hence ^.replaces ^- a +i by a function of the ^ t - M only.
Considering any a such that a -f 1 and a-\-2 are of the set A y
we find by the same method that T replaces ^- a +2 by a function
of the tj iu only. We readily verify that, if T x replaces t^a-M ^7 a
function of the ^- M only, the same will hold for ^ a ^.d^_!. Since the
series a, a + 1, a + 2, a + 3, ... yields every integer, we have proven
that T! must replace each t?^- by a function of the i^ M only, if 1^
shall be commutative with S lt
Similarly, T must replace each &$ by a function of the g ff , only
and each ^ by a function of the i^ iw only.
When we return from the indices %, &j, tyij to the initial
indices f^, . . ., J m , ^ becomes, by hypothesis, a substitution $ having
its coefficients in the GF[p n ~\. Under what conditions will T, T in
the indices |,-, have its coefficients in the GF[p n ]? We have shown
that Ji replaces ^^ by a function of the form Z)??^. Recurring
M=l
to the properties 1) and 2), 214, of the indices ?fo, we must have
as the D l / u certain polynomials in the quantity KI with coefficients
in the GF[p n ~], such that
' ! _ / j^j \ p n l
= (JLfQ u )
232 CHAPTER X.
Similar remarks hold for the indices J/, and faj. We may now state
our results in the following form:
Theorem. To determine the most general linear homogeneous
substitution T on m indices with coefficients in the GF[p n ~\ which shall
be commutative with a particular one S, we apply the transformation
of indices which reduces S to its canonical form S 1 and T to some
form T. Then S t may be expressed as a product
where each substitution ^,-, g f , fa is defined thus:
ty : rfia = Kiijiat rj'iA = KiijiA + Kity A-I (for every a, A)
&: & b =L ib , iB=Li&B + L&B-i (for every I , B)
& 1>ic = Qifac, ty'ic = Qityic+ QiVic-i (for every c, C).
The most general T { must be expressible as a product
^i = H H! . . . H A _i Z Z x . . . Z t _i YQ Y x . . . M / 3 _i,
individual substitutions have the forms:
coefficients dj u , ^ r , (?y w &em^ polynomials in !$, L Q , Q , respectively,
with coefficients in the G-F[p n ]. Furthermore, H must be commutative
with ?? , Z with 5 , Y w^/i ^ .
Inversely, if these conditions on H,-, Z t - ; Y f &e satisfied, then the
substitution T corresponding to the product T will be commutative
with S and will have its coefficients in the GF[p n ~\.
218. In order that the substitutions H and ^ be commutative ;
it is necessary and sufficient that, for every a, A and A',
234) (^ = 0, *.!_!_!- (a >1), ^_i = 0, (^-i^-i = (W-
Indeed, ?? H and H ^ replace ??oa by the same function only if
every tf a ^=0. In order that they shall replace ^ ^ by the same
function, we must have
&A A' ^0 A'l =
If u is not of the form A' 1, it must be of the form a 1 or else a.
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 233
To take an example, let r = 2 and a t = 3, 0% = 3, a 3 2. Then
81 * = StA=8 7A = (A 2, 3, 5, 6, 8); <J 3o = # 5a = ( = 3, 6, 8)
<^' = ^-i A'-I (A, A' = 2, 3, 5, 6, 8).
Setting iio u = i}u, we find that H has the following form 1 ):
f
01
I
02
t
03
*
fli-
fli-
Vi-
%
Its determinant is readily seen to equal
'77
'41 U 44
In the general case, H is seen to take the form
^01 ^02 ^03 ..-^Oa, ^
...0
...0
..0
fll _ 21
...0
..0
^ d
-1 1
d"
If a 1= = a 2 , d' = d ai+11 and d"= ^i ai +i- If %> ^2? we nave
and d\ ai+i = # 2 fll
0. Finally, if ^ < a 2 , we have
1)
are zero bein g 6( l ual to
respectively.
234 CHAPTER X.
The matrix of the coefficients of H is made up of (r -f- 1) 2 rectangles,
of which the general one RIJ is of height a/ and of base a/. Let t
be the smaller of the integers i, j or their common value if i = j.
Then JR^ includes at its left or bottom a square array S t of coeffi-
cients a t to a side. The coefficients in its diagonal are all equal;
likewise those in any parallel to the diagonal. All the coefficients
in Eij which lie above or to the right of the diagonal of the square
S t are zeros.
219. The results of 218 will be applied only in such simple
cases that the determinant D of H Q can be simplified by inspection.
It will therefore be sufficient to state without proof 1 ) the simplest
expression which can be given to D. Our notations may be fixed
so that a > a 2 >%>> &/+ 1 Let
where
Aj
The determinant D equals D^ D^ . . -Dp, where, if (i,j} =
X
(1,1) (1,^ + 1) (1, 2^+1) ...(1,^^-^-1-1)
A, -A, + 1, 1) & ^ -^ + 1, 1) ... (\A, - A, + 1, ^ A -^ + 1)
Since the coefficients d { j are functions of K Q , a root of an
equation of degree & belonging to and irreducible in the GF[p n ],
the number of sets of values for the A 2 , coefficients entering D* a for
which this determinant is not zero is ( 99)
Excluding the coefficients of H which are always zero, there
remains the following number of distinct coefficients tf^-:
1) A method of proof is given by the author in the American Journal,
vol. 22, pp. 133134.
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 235
o = (% -f a 2 -f cr 3 H h r+i) + (20 2 + a s -\ h r+i)
the 2 th parenthesis giving the number of such d,-j in the # th row of
rectangles. On account of the equalities among the a's, we find
AJ + A^ fa + 2 A! + 2A 2 ) +
f JU*(X*4- 2^ + + 2A r _!).
Excluding also the Af -f A| H ----- h A? coefficients in the determinants
Dji g , there remains the following number of wholly arbitrary 8^:
t
Q = V ft (Aa - 1) + 24, V x + 2 ^3^3 (^ + ^l) + -
0=1
Each one of these Q coefficients may take p n * values. The total
number of substitutions H Q is therefore
o . . . o * > = Q A "* Q * . . . a a, *
number of m-ary linear homogeneous substitutions T in the
G-F[p n ] commutative with a particular one S, whose canonical form is
expressed in the notations of 217, is given ty the product^-)
Recurring to the above example, % = 3, 2 = 3, & 3 = 2, we have
/'(>!, a 8 , 05, A;, p n ) = (j) 2wA 1) (p 2w * - ^ ni ) (p nk 1) - p"*,
as is directly evident from the form of ^ and its determinant.
220. As an important example, suppose that S has the canonical
The most general substitution Tj_ commutative with 5 replaces ^ ,
, ..., ^ by x(^)i? , A(Zf )6,, . .., (>( )^o respectively, in which
the coefficients of the functions x, A, . . ., Q belong to the GF[p n ~\.
If Kj L, . . ., be primitive roots of the Galois fields of orders p nk ,
p nl , . . ., p n v respectively, we may set
1) This result is in accord with that of Jordan, who treats the case n = l.
His method of proof is merely illustrated by the consideration of a particular
example, Traite', pp. 128 136. Moreover, it does not give the explicit form of
the commutative substitutions.
236
CHAPTER X.
If, upon returning to the initial indices ,- upon which S is a sub-
stitution with coefficients in the GF[p n ~], I shall become a sub-
stitution with coefficients in that field, T must have the form
\i (i = 0, 1, ..., 1-1}
Distribution of the substitutions of the general linear homogeneous
group into complete sets of conjugate substitutions, 221 223.
221. The substitutions of the group G m = GLH(m, p n ) are to
be classified into complete sets of conjugate substitutions and the
number of substitutions in each set determined. Although a complete
solution of this problem is furnished by the preceding general theorems,
their generality and complexity make it desirable to consider in detail
the special cases m = 3 and m = 4 .
The classification employed is based upon the canonical forms
of the substitutions of G m . These in turn depend upon the character-
istic determinants of the substitutions (a,
vz.
A (A)
a ll~
a 21
- A Cf 12 fflm
1*22 A ... 2m
-i
*
Furthermore, G m contains a substitution in whose characteristic
determinant the coefficients Oj, or 2 , . . ., a m are any preassigned marks
of the GF[p n ] such that a m =%=6. The required substitution is
cc
a a
3
. . . cc m
1 K m
1
...0
1
...0
1
...0
...1
o ,
222. Consider first the group 6r 3 of order
By 214 215, every linear homogeneous substitution in the GF[p n ~\
on m = 3 indices can be reduced by a linear ternary transformation
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 237
(not necessarily in the GF[p n ~\) to one of the following five types
of canonical forms:
T* I f iTC I
>: x = [ix, y = p p y> z = az
C: x' = ax, y'=py, s=yz
D: x'=ax, y' = (ly, / = /3 (# -j- ?/)
E: x'=ax, y'=a(y-\-%), ^^(s + y),
where 1 satisfies a cubic equation and ft a quadratic equation each
belonging to and irreducible in the GF[p n ], while a, /3, y denote
marks =)= of the GF[p n ].
Upon replacing 'k by W n or by A* 2w , we obtain from A a sub-
stitution conjugate with A. Any other replacement of Z leads to a
substitution not conjugate with A ( 102, Corollary), since its
characteristic determinant differs from that of A. Hence the type A
includes -^(p 3n p") distinct sets of conjugate substitutions, those in
different sets being not conjugate under 6r 3 .
Let $ be a substitution of 6r 3 having the canonical form A,
where A is a definite mark of the GF[p 3n ] not in the GF[p n ~\. If
a substitution T of 6r 3 be commutative with $ and if we apply to T
the same transformation of indices which reduces $ to the form A,
then ( 220) T will take the form
where 6 is a primitive root of the GF[p' 6n ] and r is some positive
integer < p 3 n 1 . Hence 5 is commutative with exactly p 3 n 1
substitutions of 6r 3 , so that $ is one of N~ (p 5n 1) conjugate
substitutions within 6r 3 . The total number of substitutions of 6r 3
reducible to the canonical forms A is therefore
Type B includes (p 2 n p n ) ( p n 1) distinct sets of conjugate
substitutions. In fact, the replacement of ft by ft 2 '" leads to a sub-
stitution conjugate with B, while any other replacement of ft or any
change in a leads to a substitution not conjugate with B. A sub-
stitution of 6r 3 commutative with a particular substitution reducible
to a type B has the canonical form
238 CHAPTER X.
where Q is a primitive root of the GF[p* n ] and d belongs to the
GrF[p n ], r being an integer <# 2n 1. The number of such sub-
stitutions is (p 2 n 1) (p n 1) . Hence the total number of substitu-
tions of 6r 3 reducible to the canonical forms S is
b) (p* n p n ')(p n l)(p Sn l)p* n .
Type (7 includes p n 1 canonical forms with a = /3 = y;
(p* 1) (p n 2) canonical forms with a = ft =|= y ; a like number with
a = y =)= /3; a like number with /3 = y =4= ; and (p n 1) (^) ra 2) ( p n 3)
with a, fi, y all distinct. By a suitable transformation of indices the
multipliers a, /3 ? y in are permuted in an arbitrary manner. We
have therefore the following numbers of distinct sets of conjugate
canonical substitutions C:
p n l of type Cj with cc = /3 = y;
(^) n 1) ( j? TC 2) of type C 2 with only two equal multipliers,
say a = /34=j>;
yO n l)(> n 2)(p w 3) of type <7 3 with all three mul-
tipliers distinct.
The most general substitution of 6r 3 commutative with C s is
y' = by, z' = cz (a, 5 ? c in the
Hence (7 S is one of N-^r (p n I) 3 conjugate substitutions within 6r 3
The most general substitution of 6r 3 commutative with C 2 is
-'by f y' cx-\-dy, d = es.
Hence C 2 is one of JV-f- (p* n l)(j) 2n ^ w )(^) w 1) conjugate sub-
stitutions. Finally, C^ is commutative with every substitution of 6r 3
and thus is conjugate only with itself. The total number of sub-
stitutions of 6r 3 reducible to the canonical forms C is thus
(p n 1) + O 3n l)O n
Of the substitutions of type Z) 7 there are p n 1 with a = /3 and
1) (j) w 2) with K =j= /3, no two being conjugate under 6r 3 . A
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 239
substitution D with a = /3 is commutative only with the p Sn (p n I) 2
substitutions of 6r 3
x 1 = dy -f ex, y 1 = ay, d = by + as -f- c# (a, &, c, ^ e in the 6rJF[jp n ]).
A substitution D with a =)= )3 is commutative only with the j? w (p M I) 2
substitutions of 6r 3
= ex =
The total number of substitutions of 6r 3 reducible to the types D
is thus
d) (p n l)O 3w l)Gp"+l) + (p n l)O n 2)O 3n
No two of the^ w 1 substitutions of type E are conjugate under 6r 3 .
Each is commutative only with the p 2n (p n 1) substitutions of 6r 3
x' = ax, y'=bx + ay, 0' = ex + by + az.
The number of substitutions reducible to the canonical forms E is
e) (p n 1) O 3 n 1) O 2% - l)p\
A check on the above enumeration of the substitutions of 6r 3
consists is verifying that the sum of the numbers a), b), c), d), e)
equals the order N of 6r 3 .
223. Consider next the group 1 ) 6r 4 of order
N = O 4w 1) O 4w p
By 221, 6r 4 contains a substitution in whose characteristic deter-
minant A (A) = X 4 e^A 3 a 2 A 2 3 A 4 the coefficients x , ec 2 , 3 ,
4 are arbitrary marks of the G-F[p n ~\, 4 =)=0. According to the
possible factorizations of A (A) in* the Q-F[p*] t we distinguish the
cases: I) irreducible; II) linear factor and irreducible cubic; III) two
distinct irreducible quadratic factors; IV) equal irreducible quadratic
factors; V) irreducible quadratic and two distinct linear factors;
VI) irreducible quadratic and two equal linear factors; VII) XI) four
linear factors , according to the number of equal factors. Denote by
l t , V>t marks of the G-F[p nt ] not in the 6rF[p n *], r < t. For simpli-
city, the subscript unity is omitted from the marks a, ft y, d of the
GF[p n ]. The types of canonical forms of the substitutions of 6r 4
may be exhibited in the following complete list:
1) Cf. T.M.Putnam, Amer. Jowrn. Math., vol. XXIII, pp. 4148. For the
author's treatment of the case n = 3, ibid, pp. 37 40.
240
CHAPTER X.
Type
Canonical substitutions 1 )
Number M of distinct
canonical forms
I
t
ify
A4 Z
Af M;
ie^-j^o
II
M
ify
AS Z
A,;
( p 3 n p n ) (p n 1)
III
^z
ify
ft'
flf W
(p 2 p n } ( j9 2 pn 2)
iVt
A-Q v
^(y+x)
ifz
Af (w+^)
!(**-!>)
IV 2
A 5?
*
tfe
Af w
Ttf-'-W
V
lift
M
a,
Af w
(p 2 n p n ) (p n 1) (p n 2)
vii
A< |A/
A I ni 1 sr \
1 \fj i^ *^/
2
Ag w
_L(_p a "-_p")(!> n -i)
VI,
A-i it'
^9
A.> ^r
Af w
j
(^) 2 n p n ^) (^p n 1)
vn
#
to
y^
d<;
__ (^pn_\^ ^_2) (p n 3) (p n 4)
vnii
a#
fry
y^
y (w+^)
IT (^ ~ 1) (#" "~ ^) (^ ~~ ^)
11
Vlllg
ax
Py
yz
y^
Y (^? n 1) (p w 2) (p n 3)
IX,
ax
Py
P(s+y)
fi(w+s)
(p n 1) ( p n 2)
1X2
ax
Py
p(z+y)
PW
(|) n l)(# w 2)
IX 3
ax
Py
P s
PW
(p n 1) (p n 2)
X l
ax
a(y+x)
a(e+y)
a(w+i)
p n 1
Xj
ax
a(y+x)
*{*fff)
aw
^> n 1
X 3
ax
a(y+x)
a(w^rZ)
_p" 1
X 4
ax
a(y+x)
a^
aw
# n 1
X 5
ax
ay
a5?
aw
^_ 1
XI,
ax
a(y+x)
yz
y(w-*rZ)
y (_p n 1) (p n 2)
XI,
ax
a(y+x)
yz
yw
(p n 1) (^> 2)
?
ax
ay
yz
yw
_ (r _!)(_ 2 )
1) The notation
stitution
, 70, y(w-f #), f r example, is used for the sub-
w' = y
CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 241
Table giving the form and number C of the substitutions of the
group 6r 4 commutative with the various types of canonical forms:
I
Arc
^y
^%
3 w
A Jj
^_i
n
px
^y
p p * n z
aw
( pB n 1) ( pn 1)
in
QX
9 pn y
GZ
G P W
f 7)2 n 1^2
IVi
QX
GX-\-Qy
. Q^Z
n n n n
G v Z-{- Q ^ W
( p2 n 1) p2 n
IV 2
GX+Qy
*x + ry
G P Z -\- Q P W
P n z i. P n ,
(pi n 1) ( p4: n t)2 n)
V
ax
by
QZ
n n
Q* W
( p% n 1) ( pn 1)2
KI
ax
bx-{-ay
QZ
Q P W
( p2 n 1) ( pZ n p n )
VL,
ax+by
cx-\-dy
QZ
Q P W
(p% n 1)2 ( p2 n W 7 *)
vn
ax
by
CZ
dw
(pn I) 4
VHI,
ax
by
cz
dz-\-cw
(.pn I) 8 pn
vm,
ax
by
cz-\-dw
ez-\-fw
(p2 n pn) (p2 n 1) (pn 1)2
IX,
ax
by
cy-}-bz
dy -\-cz-\-bw
(pn I) 2 jp2n
ix 2
ax
by
cy -\-bz-\-ew
fy + dw
(pn I) s p3n
1X3
ax
by+cz + dw
ey + fz+gw
hy-\-iz -{-jw
(_p3n_l) (^2n_l) (pn-l^pSn
x i
ax
bx-\-ay
cx + by + az
dx -\-cy-\-bz-}- aw
(pn 1) p3 n
x*
ax
bx-\-ay
ex -\-by-\-az-\-ew
fx-\-dw
(pn-iypLn
X 3
ax-\-ez
bx-\-ay-\-fz-{-ew
gx-\-cz
hx-\-gy-{-dz-\-cw
X 4
ax
bx-\-ay-{-fz-\-ew
gx-\- cz-{-Jcw
hx + dz + lw
(jp2 n_l) (p2 n-pn} (pn-l)p5 n
X 5
arbitrary
N
XI,
ax
bx-\-ay
cz
dz-\-cw
(pn I) 8 p2n
XI,
ax
bx + ay
cz-\-dw
ez-\-fw
(p2 _1) (p2 n-pn) (p_l) pn
n
ax-}-by
gx-\-hy
cz-\-dw
ez-\-fw
( ^ n _l)2 ( ^ W _^)2
Here A belongs to the GF[p* n ], ft to the G-F[p* n ], Q, 6, x, r to
the GF[p 2n ], and a, l,c,.. ., j belong to the GF[p n ~]. If M denote
the number of distinct canonical forms in a general type, and C the
number of substitutions of 6r 4 commutative with each, the number
of substitutions of 6r 4 reducible to that type is MN/G. The sum
of these numbers is found to equal N, the total number of the sub-
stitutions of 6r 4 .
DlCKSON, Linear Groups.
16
242 CHAPTER XL
OPERATORS AND CYCLIC SUBGROUPS OF THE SIMPLE
GROUP iJF(3,jp). 1 )
224. By 108 the group G- = LF(3,p n ) of all substitutions of
determinant 1,
is ,/ "is+gy+ g g u 1
y y = - j - j - * Kit = i.
in which the coefficients a^- belong to the GrF[p n ~\, is a simple group
of order
where d is the greatest common divisor of 3 and p n 1, so that
d - 1, if p n = 3" or 3Z - 1; d = 3, if p n = 31 + 1.
The equation T S = 1 has in the GF[p n ] a single root = 1, if d = 1;
but has three roots 0, 2 , 3 = 1, if d = 3. Hence, if d = 1, there is
a single homogeneous substitution of determinant unity
I: If = aii + i sis + a/sSs (* = 1, 2, 3)
which, when taken fractionally, leads to the non- homogeneous sub-
stitution S. If d = 3, let denote the homogeneous substitution of
determinant unity which multiplies each index by 0. Then there are
exactly the three homogeneous substitutions of determinant unity,
I, 01 = 10, 2 I = I0 2 :
0T: iS = 6 r (,-iii+ *8is+ Ofsis) (* = 1, 2, 3),
which, when taken fractionally, lead to the non -homogeneous sub-
stitution S. Combining the two cases, we may employ the group
of ternary linear homogeneous substitutions of determinant unity in
place of the group 6r provided we consider to be identical the d sub-
stitutions Z, 0Z and 2 Z. Under this convention concerning the
homogeneous substitutions, we employ henceforth the homogeneous
notation for the substitutions of the group 6r.
225. Any substitution of 6r can be reduced by a linear ternary
transformation of indices (not necessarily in the GF[p n ] and not
necessarily of determinant unity) to one of the canonical forms A,
B, C, D, E of 222. In the present case, the determinants of
A, . . ., E must be unity.
1) For n = 1, Burnside, Proceed. Lend. Math. Soc., vol. 26, pp. 58106;
for general n, Dickson, Amer. Journ., vol. 22, pp. 231252, where certain errors
in Burnside's paper are pointed out.
OPERATORS AND CYCLIC SUBGROUPS etc. 243
If two substitutions S and T of the group G have the same
canonical form, there exists ( 216) a ternary homogeneous substitu-
tion W belonging to the GF[p n ] such that T = W~~ 1 SW. It
remains to consider whether or not there exists a ternary homogeneous
substitution W l belonging to the GF[p n ] and having determinant
unity such that W transforms S into I. If the canonical form be
A 9 B, C or D, such a W will be shown to exist; while for the
canonical form E such a W t does not always exist.
It is first shown that any one of the types A, B, C, D can be
transformed into itself by a substitution V of determinant equal to
an arbitrary mark =)= of the GF[ p n ~] and obeying the same laws
in regard to the conjugacy of its indices as does the canonical form
in question. For type A we may take as V the substitution
where (5 is a primitive root of the GF[p Sn ~\ so that T = a 1 +-?"+.?" 2
is a primitive root of the GF[p n ]. The determinant of V is thus x r ,
which by suitable choice of r may be made equal to an arbitrary
mark =)= of the GF[p n ]. For types B and C we may take V to be
For type D we may take as V the substitution
Let W have the determinant w and choose V so that its deter-
minant is w~ ! . We may take as the required substitution W^ the
product V W, where V is the form taken by V when expressed in
the initial indices. In fact F x and W have their coefficients in the
GF[p n ], while the product V W transforms S into T and has the
determinant w~* w = 1. Hence, if two substitutions of G have the
same canonical form A, B, C, or D, they are conjugate within the
group G.
For type E there arise two cases. If d = 1, so that 3 is prime
to #*!, every mark of the GF[p n '\ is a cube ( 63, Corollary).
Hence an integer r may be determined so that T Sr shall be an
arbitrary mark =|= i n *n e field. Hence the above argument holds
if we choose as V the substitution
For d = 3, only yO n 1) of the marks =)= of the GF[p n ]
are cubes. Their products by ft and /3 2 will be not-cubes, if ft be
any particular not -cube. We can therefore determine F f , of deter-
minant a cube, such that I is the transformed of S by the sub-
16*
244 CHAPTER XL
stitution ViW=W belonging to the GF[p n ] and having as deter-
minant one of the three marks 1, /3, |3 2 . Consider the three sub-
stitutions of 6r
E r : x'-x, y'^y + p'x, s 1 = a + y (r = 0, 1, 2).
The following substitution of determinant /3:
E: x' = fix, y' = y, s ! = 2
transforms E into E Q and E 2 into E . If E has determinant unity,
it is identical with E in the group 6r. It follows from the proof
above that any substitution T of 6r, which can be transformed into
E Q by a linear substitution W belonging to the, G-F[p n ], can be.
transformed into E by a similar substitution W' of determinant
($ = 0,1 or 2). Also B~* transforms E into E t . Hence T is
transformed into E t by the product W'E~ which belongs to the
GrF[p n ] and has determinant unity. Hence every substitution of 6r
of canonical form E is conjugate within 6r to one of the types
^0? -^1? -^2-
We next prove that no two of the types E , E lf E 2 are con-
jugate within Gr, i. e., by means of a substitution of determinant unity.
The most general ternary homogeneous substitution which transforms
EQ into E! is seen to be
x' = fi l cx, y'=cy + bx, #' = cz -f- ly + ax,
of determinant (l~ l c 3 ; which can not be made unity. Transforming
the latter by R~ 1 , we obtain the most general substitution which
transforms E into E 2 , viz.,
x' = p l cxj y' = cy + (lbx, z^cz + by + flax,
of determinant /3~ 1 c 3 =(=l. Finally, by 102, E can not be trans-
formed into QE lf nor E into QE 2 , by a linear substitution. The
results now proven may be stated in the explicit form:
Every substitution of G can be reduced by a ternary linear homo-
geneous transformation to one of the canonical forms
A: x'
B: x'
C: x' = ax, y'
D: x f a~ 2 x, y 1 = ay,
E Q : x'=x, y' = y + x, 0' = Z + y
EI\ x' = x, y 1 = y + fix, z 1 = -f y (/3 not-cube in G-F[p n ])
E 2 : x'-x, y' = y + p*x,z' = z + y,
in which 1 satisfies a cubic and ^ a quadratic equation each belonging,
to and irreducible in the GrF\_p n '], while, .a, /3, y belong to the GF\_p n ],
OPERATORS AND CYCLIC SUBGROUPS etc. 245
Of the substitutions of G- reducible to the forms A and B, those and
only those are conjugate within G- which are reducible to the same
form A or to the same form B. Every other substitution of G- is
conjugate within G- to one of the types C, D, E Q) E lt E 2 and no two
of the latter types are conjugate within G-.
226. Type A. The substitution of determinant unity
has the characteristic determinant
A(T) = - A 3 + M 2 + M + 1.
Hence a 1 and or 2 may be chosen in the GF[p n ] so that a root h of
A (A) = is a primitive root of the equation
235) i*"+' F+1 -l.
The order of the corresponding substitution A is the least
integer m for which
i.e., for which m(p n 1) is a multiple of p 2n -{-p ri -}- 1. But the
greatest common divisor of p n 1 and p* n -\- p n -f 1 is also that of
p n 1 and 3 and therefore equals d. The order m is consequently
Moreover, the roots of any irreducible cubic of the form A (A) =
may be written A s , k spn , A*^ 2w , so that the corresponding substitution
is the 5 th power of the substitution just considered. Hence the orders
of all substitutions having irreducible characteristic determinants are
factors of ^(p* n +p n +l\
Consider a substitution 8 of G of canonical form A for which A
is a primitive root of equation 235). By 220, the only substitu-
tions of 6r which are commutative with S have, simultaneously with
the canonical form A of S, the canonical form
X > =
where a is a primitive root of the GF[p* n ~]. Hence r (1 -\-p n +p 2n )
must be divisible by p Zn 1 and therefore r divisible byj? TC 1.
Setting r = p (j) n 1),
(?r= ^_i^ ==;i ,^
since <?*>" x is a primitive root of 235) and hence equal to some power
t of A. The only substitutions of 6r which are commutative with S
are therefore the powers of S. It follows that S is one of a set of
246 CHAPTER XI.
N
s -
distinct conjugate substitutions, N being the order of G.
The only distinct powers of S which have the same character-
istic determinant as S are evidently S, S p and S p . To each set
of three substitutions such as S r , S rp , S rp contained in the cyclic
group generated by S and all belonging to the same characteristic
determinant, there corresponds a set of s distinct conjugate substitu-
tions. Hence there exist in G
such sets of s conjugate substitutions. It follows that G contains in all
236) - - rJny
\_d V
substitutions not the identity whose orders are factors of
Hence G contains 2 distinct conjugate cyclic subgroups of order
227. Type S. Since G contains substitutions in whose character-
istic determinant 1? + i ^ 2 -f 2 ^ + 1 both a x and 2 are arbitrary
in the GF[p*~\, we can choose
<*! = ? + 1/d, - ofg = d + y/d,
so that
A (i) = - (A - I/*) (A 2 - yi + d),
where y and d are arbitrary in the GF[p n ]. In particular, G contains
a substitution T whose characteristic determinant has an irreducible
quadratic factor which vanishes for a primitive root ^ of the GF[p* n ~\.
The canonical form of T is then B. The order of T is therefore
the least integer t for which
i. e. ? for which both t(p n 1) and tf(j) w -f 2) are divisible by _p an 1.
But 3 and t(p n 1) are both divisible by jp 2n -1, for a minimum,
if and only if
t=p* n l, when # w = 3 7i or 3Z 1; = CP 2 "" 1 )' wnen JP"=3Z+1.
Hence the order of T is -^-(p 2n 1).
OPERATORS AND CYCLIC SUBGROUPS etc. 247
By 220, the most general substitution of 6r commutative with T
has the canonical form
and hence is T r . Hence T is one of a set of dN '-i- (p 2n 1) distinct
conjugate substitutions. The only distinct powers of S which have
the same multipliers as S are S and S p . Hence G contains -= -=
2 p* n 1
distinct conjugate cyclic subgroups of order -r(p* n 1).
The number of substitutions of G whose orders are factors of
(p2n_]\ w ithout being factors of -r("- 1), and hence not of
Ci ^ Ct ^ '
p n 1, is
237) Np n /(p n -f 1).
In fact, such substitutions form in all
different sets, those in each set having the same characteristic deter-
minant. Each set contains dN^r (p 2n 1) distinct conjugate sub-
stitutions. The product of the two numbers gives formula 237).
228. We can exhibit 6r as a permutation- group on p* n +p n -{-l
letters. Every linear function A^ + J5 2 + C%s> *- which A, JB, C
are marks not all zero of the 6rjF[_p n ], can be put into one of the
forms, * v /ft j- \ fc
^(?3-t-?i 2 + ^ii); f* (la + pii), ^ii,
where ft, p, t? are marks of the G-F[p n ~] and ^ =[= 0. Combining into
one system {A^-}- B% 2 -}- 0| 3 } the p n 1 linear functions
[i denoting in succession the p n - 1 marks =)= of the field, we
obtain p 2 n + p n + 1 distinct systems,
{ I 3 + $^2 + a i }? { ^ + 0li }; { Si } fo arbitrary marks].
Any ternary homogeneous linear substitution replaces the functions
fi (A% -f 5| 2 + C%s)> comprising one system, by linear functions
all belonging to a single system. Hence it permutes the above
p z n -f- p n + 1 symbols amongst themselves. It follows that G- is
isomorphic with a permutation -group 6r' on these symbols. But a
homogeneous substitution altering none of the symbols must have
the form , _ , _ , _
?1 **Si; fe a ?2^ 5s a bS-
248
CHAPTER XL
If it have determinant unity, it corresponds in G to the identity.
Hence G is simply isomorphic with G'.
The permutation -group G' is doubly -transitive. We need only
prove that G 1 contains a permutation converting (IjJ, {S 2 +ii}
respectively
the latter being any two distinct symbols, viz.,
For the corresponding homogeneous substitution, we may take
where a, /3, y are chosen in any manner such that the determinant
of the substitution is unity, viz.,
B C
B' C'
C A
C' A'
= 1.
By hypothesis the determinants are not all zero, so that solutions
a, /3, y in the GF[p n ] certainly exist.
229. Type D for a 3 =|= 1. Let a be a primitive root in the
GF[p n ], the cases p n =2 and p n =2 2 being necessarily excluded.
For such an a, substitution D generates a cyclic group of order
Considered as an operation of the isomorphic permutation -group,
D belongs to a subgroup of G which leaves fixed the symbols { x }
and {y}. The general substitution of G possessing this property has
the form
E:
x =
= as
a"x
In order that E shall have the order -5-p(p n l), it is necessary
and sufficient that a be a primitive root in the GF[p n ] and that either
(j) a! 4= 0, cc = ft 4= r , or (H) a" 4= 0, - y 4= /?.
In fact, if both /3 and y differ from a, E may be given the form
whose (p n l) 8t power is unity, by introducing in place of 2 the index
Hence, if a 4= /3, we may take a == 7. Then "=)= 0; for, if a" = 0,
/y
multiplies z -\ -oV by > so that J? would have as order a factor
OPERATORS AND CYCLIC SUBGROUPS etc. 249
of p n 1. Similarly, if a ={= 7, then must a = /?, V =H 0- Finally,
if a = j3 = y ? each may be taken equal to unity. Then, by induction,
E r : x' = x, y 1 *= y, z'^z + ra'y + rd'x,
so that E would have the period p. Hence either (i) or (ii) must
be satisfied.
Suppose, inversely, that relations (i) are satisfied. Setting
TT- ' rr K " %
Y=y, Z=3 + j 9
and is thus of period -jp (p n 1) if, and only if, be a primitive
of
root of the GrF[p n ~], Interchanging x with y, the proof follows for
case (ii).
Using the theorem just proved, we proceed to determine the
number and conjugacy of the cyclic subgroups of order -j-p (p n 1)
which leave the symbols {x} and [y\ fixed. For case (f),
E: x'=a~ 2 x, y' ' = ay, z' = az + a'y + a" x (a' =)= 0, a 3 4= 1),
where a is a primitive root of the GrF[p n ~\. By induction we find
__ Q J
= a*g + tcfa'-^ + a" a*- 1 " x.
I
In order that Q r E t shall be identical with the substitution
x'=a 2 x, y 1 = ay, z 1 = az -f g'y + $"0,
it is necessary and sufficient that
Let Jfi denote any one of the (p n l)/(l> 1) distinct marks Jf 1?
Jf 2 , . . . such that no two have as their ratio an integral mark 1 ).
If a be a fixed mark 4=0 and Jfan arbitrary mark, ihep n (p n l)/(p 1)
substitutions
238) x' = a~ 2 x, y' = ay, ' = az + -Mi'2/ + -3tf^
have the property that no power of any one of them reduces to one
of the set. We therefore obtain that number of cyclic subgroups of
order -jp (p n 1).
Furthermore, every substitution V of the subgroup leaving {x}
and {y} fixed, and having = /3, and of order a divisor of -^p(p n 1)
1) The marks Jfj, Jf 2 , . . . are evidently the multipliers in a rectangular
array of the marks =]= of the 6r-F[.pw], the first row being formed by the
integral marks 1, 2, . . . , p 1.
250 CHAPTER XI.
without being a factor of p or p n 1 , is contained in one of the
above cyclic subgroups. In fact, by the earlier argument, we may set
F:. x' = a- 2s x, y f = ct s y, z' = a s z + a'y + a"x (a' =f=0, 3 *4= !)
Let Mi be a mark =j= such that its ratio to a'a 1 ~ s is an integral
mark. The power s -f & (p n 1) of 238) gives
* 1)] a 8
By choice of k and Jf, we can make the coefficient of y in z' equal '
and that of x equal ".
Hence there are p n (p n !)/(# 1) cyclic subgroups of 6r of
order -rjp (_p n 1) for which a = /3, and as many more for which
a = y> each leaving the symbols { x } and { y } fixed, and together
containing all substitutions having the last property and having an
order not p nor a factor of p n 1.
These cyclic subgroups are all conjugate within 6r and, indeed,
within the subgroup which leaves fixed [x] and {y} or merely
permutes them. First, the substitution
, , , M'-M
x' = x, y=y, z=
K a
transforms 238) into a like substitution with M f in place of M. Also
x' = I^Q*X, y' = gy, z' = kQZ
transforms 238) into the substitution
z' = az
Hence the cyclic subgroups given by a = /? are all conjugate within
the group leaving fixed { x } and { y }. These symbols are interchanged by
x' = y> y'=-x, s' = z,
which transforms 238) into the substitution
x r =ax, y ? = a~ 2 y, z f = az My + Mix.
Hence the set of cyclic subgroups given by a = /3 are conjugate to
the set given by a = y within the group leaving fixed the symbols
\x] and {y} or permuting them. The latter group consequently
contains 2p n (p n 1)/(^ 1) conjugate cyclic groups of order
-jj) (p n 1) and those substitutions of these groups whose orders are
not divisors of p or p n 1 are all distinct.' Since the permutation-
group isomorphic with 6r is doubly transitive, it contains
OPERATORS AND CYCLIC SUBGROUPS etc. 251
conjugate subgroups leaving fixed or permuting the two symbols.
Hence there are altogether
0n i i iW0Sn i 0.A =
2^ ~
conjugate cyclic subgroups of order -j-p(p n l)- Each contains
P + ~J (P n 1) 1 substitutions of period p or a divisor of -j (p n 1).
There remain in each cyclic group (p 1) (p n * 1) 1 sub-
stitutions. Hence G contains
239) N(f-l-d)+F (p n - 1)
substitutions whose orders divide -rp (p n 1) but not p or p n 1.
For the cases p n =2 and^) n =2 2 above excluded, formula 239)
reduces to zero. Hence the result is always true.
230. Type D when 3 =1. We are to consider substitutions of
period p having the canonical form:
x' = x ' =
From the investigation at the beginning of 229 it follows that the
only substitutions of period p which leave fixed the symbols {x}
and {y} have the form
240) x' = x, y' = y, z' = z + ax + fty ( and not both zero).
There are p* n \ distinct substitutions of this form. They are all
conjugate to D f within Gr. In fact, if /3 =(= 0, the substitution
x' = x ' = -}-% s' = 8
transforms 240) into
x'~x, y'=y, Z' = Z + (K PQ)X + fiy.
By choice of Q, we can make a /3p = 0. If /3 = 0, we trans-
form 240) by .
x' = y, y' = %, z'=-z,
and get
x { = x, y' = y, z 1 = z ay.
In either case we reach a substitution of the form 230) but having
a = 0, |3 =|= 0. It is transformed into D 1 by the substitution of Gr
252 CHAPTER XL
The p* n 1 substitutions 230) determine (p 2 n 1)1 (p 1) con-
jugate cyclic subgroups of order p contained in the subgroup of G
which leaves fixed the symbols {x} and [y] and hence also {x-^-^y},
Q being an arbitrary mark of the GF[p n ].
Each such group therefore leaves fixed p n + 1 (and no more)
symbols. But the p* n -f- p n -f 1 symbols furnish
such sets of symbols. Hence G contains
conjugate cyclic subgroups, all of whose substitutions are con-
jugate under 6r. Each such subgroup is therefore contained self-
conjugately within a subgroup of order -^-p Sn (p n l)(p 1). The
total number of distinct substitutions of G of order p of the type
considered has thus been shown to be
941 "i
231. Types E if By induction we find that
E*t x' = x, y' = y + tx, ^ f = s + ty + ^t(t- l)a?.
Hence J5J is of period p or 4 according as p > 2 or p = 2. The
most general substitution of G transforming E Q into itself is
x' = ax, y' = ay-{-1)x, s' = az -\-~by -{- ex (a 3 = l).
Exactly p 2n of these substitutions are distinct in the group G.
Suppose first that p > 2. For any positive integer t<.p, the
substitution
242) x'=^-x, y' = y- ~^tf, z'=tz
is of determinant unity and transforms E Q into E^. Taking
2 = 1, 2,...,^-!,
we see that G contains exactly p 2 n (p 1) distinct substitutions
which transform into itself the cyclic group generated by E . The
cyclic group {E } is, for p > 2, one of N/p 2n (p 1) distinct conjugate
subgroups of G. In particular, G contains N/p 2n distinct conjugate
substitutions of the type E Q .
Suppose next that p = 2. Then E is of period 4. Since
x' = x ' = z' = x
leaves fixed the 2 n -fl symbols {x}, {y + lx}, I any mark of the
GF[2 n ], while E Q leaves fixed but one symbol {x}, the two sub-
OPERATORS AND CYCLIC SUBGROUPS etc. 253
stitutions are not conjugate under G. But E Q is transformed into
E* by the substitution 242) for t = 3, viz.,
The cyclic group - generated by JE is therefore -transformed into itself
by exactly 2 2 2 n substitutions of G. For p = 2, { E } is one of a
complete set of JV/2 2n + 1 conjugate cyclic subgroups of G. Just two of
the four substitutions of every such cyclic group are of type E ,
while the remaining one not the identity is of type D with a 3 = 1.
Hence, for p = 2, G contains N/2 2n distinct substitutions conjugate
with E Q .
Since EI and E% are conjugate to E within the general ternary
linear homogeneous group in the GF[p n ], the number of substitu-
tions of G conjugate to E within G equals .the number conjugate
to EI or the number conjugate to E 2 . Hence G contains altogether
243) 3N/p* n
distinct substitutions of the canonical forms E^ they form three
distinct sets of conjugate substitutions under G. Also, E Q , E > E 2
each lead to the same number of conjugate cyclic subgroups of G.
232. Type C. The substitutions of canonical form "C are of
order a divisor of p n 1. Of the (p n I) 2 sets of solutions in the
GF[p n ] of /3y = l, d sets have a = /3 = y and hence each equal to
r (r = 0, 1, or 2). If a be any mark different from 0, 1, 0, 2 , and
if /3 = , then y = a~ 2 =j=a, Hence there are 3(p n d 1) sets of
solutions in which two and only two of the quantities a, /3, y are
equal. There remain
(p n - 1) 2 - 3O- d - 1) - d=p* n 5jp+ 4
sets of solutions in which a, /3, y are all distinct. Dividing this
number by 6 to allow for permutations, we obtain the number of
distinct sets of unequal multipliers of ternary homogeneous sub-
stitutions C. ,
If, for d = 3, a, /?, y do not form a permutation of 1, 0, 2 , the
three sets
, /, y; e, e/3, d r , ex VP, Vr,
are not equivalent sets of multipliers in the homogeneous group, but
are equivalent in the non- homogeneous group G. The number of sets
of unequal multipliers in G is therefore
254 CHAPTER XI.
We proceed to prove that the total number of substitutions of G-
of canonical form C with a, /3, y distinct is, for d = 1 or 3,
N >-.6
...
Q>-1) 2 6
By 220, the only ternary homogeneous substitutions commutative
with C with a, ft, y distinct are the (p n I) 2 substitutions
T: x'=ax 'b z 1 = cz
For d \, each set of unequal multipliers therefore leads to N/(p n T) 2
conjugate substitutions, so that we obtain the number 244). For
<2 = 3, the substitutions T give only -^-(p n I) 2 distinct substitutions
in 6r. Furthermore, by 102, C can be transformed into 0(7 if,
and only if, the multipliers a, /3, y form a permutation of 1, 0, 2 .
The special substitution (7,
x' = x, y'=dy, z'=Q*z
is transformed into (7, 0(7 or 2 (7 by exactly the 3(p n I) 2 products
T, (xyz)T, (xsy)T. The corresponding substitution is therefore one
of N/ (p n I) 2 distinct conjugate substitutions under 6r. Each of
the remaining substitutions (7 with unequal multipliers is one of a
set of N^--^(p n I) 2 conjugate substitutions under 6r.
Corresponding to the p n d 1 sets of multipliers a, /3, y of
which two are equal, there are -j- (p n d 1) substitutions C' of 6r,
no two of which are conjugate. Such a substitution
C': x r = ax, y' = ay, z 1 = y# (a?y = 1, y =j= a)
cannot be transformed into 0(7'. By 218, the most general ternary
linear homogeneous substitution which transforms C' into itself is
x' = ax + by, y' = a'x + b'y> 2 f =c"z.
The number of such substitutions in the CrF[p n ] of determinant
unityis "
Hence the total number of substitutions in G- of the canonical form C' is
245) i (^ - d - 1) - -j
7i _ 1) (_^2 npn)
Ui
233. As a check upon the accuracy of our enumeration of the
substitutions of 6r, we may verify that the numbers given by the
formulae 236), 237), 239), 241), 243), 244) and 245), together with
unity, to count the identical substitution, give as total sum the
order N of the group 6r.
OPERATORS AND CYCLIC SUBGROUPS etc. 255
234. To complete the enumeration of the cyclic subgroups of 6J>
it remains to determine those generated by substitutions of the
canonical forms C. The method will be sufficiently illustrated if we
confine the investigation to the case d = l. r ) If be a primitive
root of the G-F[p n ], we may set
C: x' = a r x, y' = a'y, z r =*a r *2,
where r and s are integers chosen from the series 0, 1, . . ., p n 2.
Let g denote the greatest common divisor of r and s. The period
of C is the least positive integer I for which Ir and Is, and therefore
also Ig, are multiples of p n 1. Hence C is of period p n 1 if, and
only if, g be relatively prime to p n 1. In general, C is the g ih power
of a similar substitution with the multipliers a r / ff , a.*l g y al~~ r ~ *>/ ff , the
latter of period p n 1. Hence, for d = 1, the substitutions of type C
are all included in the cyclic groups generated by those substitutions
of type C which have the period p n 1. We may therefore confine
our attention to these largest cyclic groups. The exponents r, s in
the expression of any substitution C of period p n 1 must occur
among the sets of two positive integers less than p n 1 and having
their greatest common divisor prime to p n 1. Denote by F(p n 1)
the number of such sets. A similar remark holds for the couples
5, r; r, r s; -r s, r; s, r s; r s, s; provided r s
be replaced by its least positive residue modulo p n 1. If r, s, r s
be distinct, the above couples form six of the F(p n 1) sets, but
lead to the same set of three multipliers in C. If two of the
exponents be equal and therefore diiferent from the third, we may
take them to be r, r, 2r. Then the couples r, r\ r, 2r; 2r, r
form three of the F(p n 1) sets, but lead to the same set of
multipliers in C. Here r may be any one of the <t> (p n 1) integers
less than and prime to p n 1. Hence there are 3 (p n 1) sets
leading to O (p n 1) distinct sets of multipliers two of which are
equal, while the remaining sets lead to [F(p n 1) 3<b(p n 1)]
distinct sets of three unequal multipliers, together yielding all the
substitutions C of period p n 1. The value of F(p n 1) is given
by the following theorem. 2 )
The number of sets of two integers, not both zero, chosen from the
series 0,l,...,klso that their greatest common divisor is prime to k is
where q lt cfa, . . ., q % are the distinct prime factors of 7c.
1) The case d = 3 is more intricate and the results quite complicated.
The results are given in the Amer. Jown., vol. XXH, p. 251; the proofs in vol. XXIY.
2) Jordan, Traite, p. 96.
256 CHAPTER XI.
Of the k 2 sets of two integers each < k, k 2 /qj have their integers
chosen from the Jc/qi multiples of q t and are to be excluded. We
thereby exclude , in particular , the sets of integers each of which is
one of the &/#,</ multiples of qi^j. Hence, in afterwards excluding
the sets of integers each of which is a multiple of q^ we subtract
the number k 2 /q* %Yfl?j3< After the required exclusions have all
been made, there evidently remains the number of sets indicated by FQc).
Among the latter sets, the couple 0, does not occur since
235. A cyclic group generated by a substitution C of period
p n 1 will be called special if two of its substitutions C a , C b of
period p n 1 ar-e conjugate within 6r, i. e., have the same set of
multipliers. Since a and & must be prime to p n 1 , the condition
requires that C and C bai shall have the same set of multipliers,
where a is determined from aa = 1 (mod p n 1). It thus suffices
to investigate when C and C m have the same multipliers, m being
prime to p n 1 and 1< m < p n 1. The three distinct ways in
which the two sets
r , *, a'; a mr , a ms , a mt r-j-s + tf = (mod p n 1)
may be identical in some order will be considered in turn.
i) If a mr = a r , a ms = a 8 , a mt = a*, then r(m 1), s(m 1), and
therefore also g(m 1), are divisible by p n 1. Since g is prime
to p n 1, m 1 must be divisible by p n 1, contrary to hypothesis.
ii) If a mr =a s , a ms =a r , a mt = K t , then must
r, m 2 r = r (mod p n 1).
Then r- must be prime to p n 1; for a common factor would divide s
in virtue of the first congruence, whereas the greatest common divisor
of r and s is prime to p n 1. Hence, by the last congruence,
246) w 2 EEl (mod^-1).
Inversely, if m be any solution of 246) and if r be any integer
less than and prime to p n 1 and if s be determined by
s = mr (mod p n 1),
then C and -G m have the same multipliers. Moreover, C is the r th
power of a substitution with the multipliers a, m , a" 771 " 1 , which
may therefore be taken in place of C as generator of the special
cyclic group.
If 2* be the highest power of 2 contained in p n 1 and if % =
when & = or 1, % = 1 when & = 2, ^ = 2 when A; ^ 3, and if fi be
OPERATORS AND CYCLIC SUBGROUPS etc. 257
the number of distinct odd prime factors of p n 1, then the con-
gruence 246) has exactly 2*+^ solutions w. 1 ) The solution m = 1 is to
be excluded. Consider the 2*+^ 1 substitutions with the multipliers
, a m , a~ m ~ 1 } m>l. They generate as many cyclic groups. In fact,
(a m )*= requires x = m (modp n 1); while (~ m ~ 1 ) y = a is im-
possible since m -f 1 has a factor > 1 in common with p n 1.
Moreover, the sets of multipliers of the substitutions of period p n 1
in each cyclic group are the same in pairs. Hence these special cyclic
groups contain altogether -^^(p n 1) (2*+^* -- 1) distinct sets of
unequal multipliers.
(iii) If a mr =a, a ms = a', m '=a r , we find that
r(w 2 -f w+l) = +s-|-r=0, s(w 2 -J-w+l)=r+-i-s^O (mod^) w 1).
Hence Jlf = m 2 + m -f- 1 must be divisible by _p n 1. Since m (m + 1)
is even, Jf is an odd number. Hence p n 1 must be odd and there-
fore p n = 2 n . Since d = 1, 3 is not a factor of p n 1. Hence each
prime factor # of j9 n 1 is of one of the forms 6&-f5, 6& + 1.
Now M and hence also m 3 1 must be divisible by q. If # = 6& + 5,
Fermat's theorem gives m 6 *+ 4 =l (mod q). Since w 3 ^!, we have
m = l (mod #) and therefore Jf=3 = (mod g), which is impossible.
Hence must q = 6k + 1. Inversely, if# = 6&-fl, m 6k 1 = (mod ^)
has 6& distinct integral solutions. But the left member is divisible
by m 3 1 and therefore by M . Hence M = (mod #) has two
distinct solutions. Each of these solutions leads to one, and but one,
solution of M = (mod #*). To give a proof by induction from
x = e to T = e + l let m 3 1 == . Then
(m + o^) 3 1 = <?g*+ 3m 2 xq e (mod # 2e )
and will therefore be divisible by q 6 ^ 1 if, and only if,
= Q (mod q).
Since 3 and m are prime to q, x is uniquely determined mod q.
Hence each m determines one solution y = m-\-x<f of
y s 1 = (mod # e + 1 ).
Hence, if m 2 -f m + 1 be divisible by # 9 , i/ 1 will be prime to q
and hence y 2 + y + 1 will be divisible by ^ c + 1 . Supposing that the
prime factors of 2 W 1 are all of the form 6& -f- 1 and that the
number of distinct ones is y, it follows that Jf=0 (mod 2 n 1)
has 2 y solutions m. But, if m be a solution, then m 1 will be
1) Dirichlet, Zahlentheorie, 37.
DlCKSON, Linear Groups. 17
258 CHAPTER XI.
a second solution. Hence C is the r ih power of one of the 2 y ~ 1 sub-
stitutions with the multipliers a, a m , ci~ m ~ 1 . These generate distinct
cyclic groups, since (a m Y=cc requires x = m 1. Hence there
are 2 y ~ 1 of these special cyclic groups and the substitutions of period
p n 1 in each give just (p n 1) distinct sets of multipliers.
Excluding the special sets of multipliers of types (ii) and (iii),
there remain
1 11 v
. \F ( v) n 11 3 ( t) n 111 ( t) n 11 (2*~^~' w 11 O ( f) n 11 2
sets of unequal multipliers, the last term occurring only for certain
values ofp n . The corresponding substitutions C lie in sets of 0(^1)
in cyclic subgroups not conjugate under 6r. Noting that F(p n 1)
is divisible by <$>(p n 1), giving the quotient
where q l9 q%, . . ., q Y are the distinct prime factors of p n 1, we may
combine our results in the theorem:
' If p n 1 be not divisible by 3, the substitutions C generate the
following types of cyclic groups of order p n l not conjugate under 6r:
a) one group generated by the substitution with multipliers
a, a, a- 2 ;
b) 2*+^ 1 generated by substitutions with multipliers a, a m ,
a ro i^ w here w 2 ^! (mod^ n 1), x and ^ defined in (ii);
c) 2 y ~ L generated by similar substitutions with
occurring only when p n 1 = 2" 1 has only prime factors (y distinct
ones) of the form 6j + 1 ;
L nu (* _ i) _ 3] _ l ( 2 x+^ _!)_!.. 2>'- 1 further groups.
6
236. As a first example, let p n = 8, so that ^==1, % = 0, y = 1.
There is just one cyclic group of each of the first three types. The
generators have the sets of multipliers a, a, a~ 2 ; a, a" 1 , 1; a, a 2 ,
cr~ 8 respectively.
As second example, let p n = 17, so that ft = 0, 3c = 2, while the
third type of group does not occur. There are three cyclic groups
of the second type determined by the sets of multipliers a, a"" 1 , 1;
a, a 7 , a 8 ; a, a 9 , a 6 . The two cyclic groups of the fourth type may
be determined by the sets of multipliers a, a 2 , a 13 ; a, a 3 , a 12 .
OPERATORS AND CYCLIC SUBGROUPS etc. 259
237. It remains to determine the number of cyclic subgroups
of G conjugate with each group of the types a), b), c), d). Type a)
is generated by the substitution
x' = ax, y' = ay, z' = a*0 (a~ 2 =J=)
and is commutative with exactly (p 2 n 1) (p 2 n p n } substitutions
of 6r, viz.. . . .
x = ax + oy, y = ex + ay, z = ez.
The cyclic group of order p n 1 generated by the substitution
x ' = ax, y = a m y,
is transformed into itself by 2(p n I) 2 substitutions, viz.,
S: x' = ax, y' = by, z 1 = cz
and the products TS, where T replaces x by y and y by x.
When cyclic groups of the third type exist, each is transformed into
itself by the 3(p n I) 2 substitutions S, (xyz)S, (xzy)S. Each cyclic
group of the fourth type is transformed into itself by exactly the
(p n I) 2 substitutions S.
238. For p n = 2 2 , the simple group G has the order N= 20160.
There is, by 244), a single canonical form C, not the identity, its
multipliers being 1, 0, 2 . The N/ (p n - I) 2 = 2240 substitutions
of G of period 3 are therefore all conjugate and generate a single
set of conjugate cyclic groups. Applying the results of 226 231
to the case p n = 2 2 , we see that G contains
960 conjugate cyclic groups of order 7 with 5760 substitutions of period 7
2016 5 8064 5
630 4 1260 . ,,4
630 4 1260 4
630 4 1260 4
1120 3 2240 3
315 2 315 2
^_ n it
20160
The substitutions of period 2 are all contained in the cyclic groups
of order 4.
The group G differs in structure from the alternating group on
8 letters, likewise of order 20,160. Indeed, the latter contains 5760
substitutions of type (1234567), 3360 of type (123456) (78), 1344 of
type (12345), 2688 of type (12345)(678), 2520 of type (1234)(56),
1260 of type (1234)(5678), 112 of type (123), 1120 of type (123)(456),
17*
260 CHAPTER XII.
1680 of type (123) (45) (67), 210 of type (12) (34), 105 of type
(12) (34) (56) (78), and the identity. The alternating group has sub-
stitutions of periods 6 and 15, while G does not. Both groups
contain the same number of substitutions of period 7, the same
number of period 4, the same number of period 2. But the distribu-
tion into sets of conjugates of the substitutions of period 2, or of
period 3, or of period 4, differs in the two groups. In particular,
G is not isomorphic with the alternating group on 8 letters, each group
being simple and of order 20160. 1 )
CHAPTER XII,
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LI (2, #). 2 )
239. In 108 was defined the group of linear fractional sub-
stitutions
8: ''- A
on an arbitrary variable z with coefficients in the GF[p n ], We
proceed to represent it as a permutation -group on p n -}- 1 letters.
Suppose z runs through the series of marks of the GF[p n ]. For
y = 0, e } will also run through the series of marks. For y =f= 0, the
_ A /v
value a = d/y gives #' '= - ^p*-i so that 0' can not be determined
as a mark of the field. We may, however, obtain a set of elements
which are merely permuted by S by adjoining to the series of marks
a new element 00 = -* necessarily the same for every mark p =j= 0,
since = -z = -fj and assumed to combine with the marks A =4=
|it
of the field according to the laws
oo A = A oo = oo
while the indeterminate fraction - 4-? is assumed to equal cc/y.
-
Setting henceforth s=p n , the group LF(2,s) of linear fractional
substitutions of determinant unity in the GF[s] may therefore be
1) Miss Schottenfels established this theorem by direct calculations, Annals
of Mathematics, (2) vol. 1, pp. 147 152.
2) Moore, Mathematical Papers Chicago Congress of 1893, pp. 208 242,
Math. Ann., vol. 55 (56?); Wiman, Sweedish Acad., vol. 25 (1899), pp. 147;
Burnside, Proc. Lond. Math. Soc., vol. 25 (1894), p. 132. The work of Galois,
Mathieu and Gierster is cited in the exposition for n = l in Klein -Fricke,
Modulfunctionen I, p. 411 and pp. 419 491.
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(*,p). 261
represented concretely as a permutation -group 6r^",) on s + 1 letters
and having the order
247) M(s) = g( ^~ 1} (2; 1 according as p > 2; p - 2).
The group of all substitutions S has the order (2; l)M(s). For p> 2,
it may be represented as a permutation -group (rstf,). For p = 2, it
is the former group.
The group G*M~(^ is doubly transitive. It is only necessary to
prove that a substitution T with coefficients in the field and of
determinant unity may be found which will replace two arbitrary
distinct elements Q, (5 by the elements 0, oo. If both Q and <s are
marks of the field, we may take as T
Z Q G
If Q is a mark and tf = oo, we may take T to be 2' z Q.
The inverse of S= ( ^-? ) of determinant unity is S~* = ( ! ),
\y, */ \-y, a/ ;
so that S is of period two if and only if a -f- d = 0.
240. A substitution S, not the identity, of the group Gy, leaves
fixed at most two elements. The fixed elements are given by the
equation
248) y* 8 + (*-)* -0 = 0.
By 15, it has at most two roots in the field GF[$~\ unless y = /3 = 0,
a = #, when S is the identity. Now S leaves oo fixed only when
oo = a/y, whence y = 0. The other fixed elements are given by
($ K)Z ft = 0, which, for S =%=!, is satisfied only by # = oo or
e = mark according as d a = or =f= 0.
If S leaves fixed two distinct elements and # 2 , it can be trans-
formed by a suitably chosen substitution T of the group into a sub-
stitution with the fixed elements and oo, having therefore the form
I: ,'- (&-!).
Its period is a divisor of y (p n 1) or p n 1 according as p > 2
or p = 2.
* i
If S leaves fixed a single element ^ = #, , it can be transformed
*' = * + /3 (0 in field)
leaving fixed the single element oo. Its period is therefore p. But
the condition for a double root of 248) is (a + d) 2 = 4
If S leaves no element fixed, the quadratic 248) is irreducible
-in the GrF[p*]. By the corollary of 31, its roots e t and # 2 are
262 CHAPTER XII.
marks of the GF[p* n ~\ conjugate with respect to the GF[p n ~\. Now S
multiplies the function (e z\)^r(z %) by the constant a/b, where
The product ab reduces to ad fty = l. Also a and ~b are con-
jugate ( 73). Hence
a-i-fr-a*", a^+i-l.
Hence 8 can be transformed into a substitution of the form Z, whose
period is a divisor of (p n +1) or p n + 1 according as p > 2 or > = 2.
In particular, the substitutions of period p are characterized by
the invariant (a + <J) 2 = 4.
241. Commutative subgroups of order p n . The substitutions
form a commutative subgroup 6ri of order s = p n , containing all the
substitutions of GM(S) leaving the single element oo fixed and con-
taining no other substitutions. Each of its substitutions except the
identity is of period p. Hence there are (p n l)/(jp 1) cyclic sub-
groups G p of order p in the flj"\ To determine the conjugacy of
these substitutions and subgroups under G- M (s), we transform 8^ (ft=)=0)
by F== (- L -^) an( l ( see formula of composition at end of 108)
obtain the substitution 1 )
-y
This substitution belongs to flfj*' if, and only if, y = 0, when it
becomes 80*^. In particular, 5^ is transformed into itself only by
the substitutions (;r-fK Within GM() any substitution 8^ (ft =|= 0) is
self -conjugate in exactly the G-* , ^Me ^e 6r, ^'s self -conjugate in
exactly fhe G\( s i) composed of all the substitutions leaving the element oo
"^TT"
invariant, viz., \ ' _ 1 Y As to the order of the latter group, ft may
Vo, a V
be any mark of the GF\_p n ] and a any mark =J= 0; but a, ft
gives the same substitution as -f a, + ft
\
1) This order of the factors of a product is employed by Wiman, the
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2, pn). 263
Within G M (s)j 8^ is conjugate only with the substitutions S&n.
Hence the s 1 substitutions, not the identity, of Cr, are all con-
jugate if p = 2, but separate into two sets of -^ (s 1) conjugate sub-
stitutions if p > 2. The p1 substitutions of a cyclic group G- p
generated by Sp belong half to one and half to the other set if p > 2
and n be odd, but all belong to the same set if n be even ( 62).
In place of cx> the fixed element may be any one of the p n marks
of the 6r-F[jp*]. Since 6rjf(,) permutes the p n +l elements K trans-
itively, it contains p n + 1 conjugate commutative groups G^\ This
result also follows from the numerical identity
Each 6r, is defined by any one of its substitutions not the identity
as the group in which that substitution is self -conjugate. These
p n -f 1 groups have therefore no substitution in common except the
identity and contain in all p 2 n 1 distinct substitutions of period p.
o _ J
242. Cyclic subgroups of order -^r- If Q be a primitive root
of the G-F[p n ~\, the substitution
generates a cyclic group of order (p n 1) if p > 2, but of order
p n 1 if p = 2. It contains all the substitutions
(a in tlie
Since it contains all the substitutions which leave fixed the elements
oo and and no other substitutions, it will be denoted by Gf-*?.
"271
Any new substitution transforming this cyclic group into itself must
interchange the elements oo and and hence have the form
Inversely, every S transforms X into its reciprocal Z 1 . These
o 1
- substitutions S of period twa together with the substitutions Z
*5 i
form a dihedron- group 1 ) 6r*!-i, which is the largest subgroup of
2 27i
within which the above cyclic group is self -conjugate.
1) See the definition given in 245.
264 CHAPTER XII.
Since oo, form only one of the -^p n (p n -\- 1) pairs of the
p n +l elements, G M (s) contains exactly -^p n (p n +l) conjugate cyclic.
groups G$Li, each self- conjugate in exactly a dihedron 6r (x ;^i. Each
"271 _ 2 ~27T
of these cyclic groups is defined by any one of its substitutions not
the identity as the largest cyclic group containing that substitution.
These -~p n (p n -f- 1) groups have therefore no substitution in common
i
except the identity and contain in all s (s + 1) (s 3) or
J 4:
s (s 4- 1) (s 2) substitutions (not the identity) according as p > 2
or p = 2.
s 4- 1
243. Cyclic subgroups of order -577^ ^ 144, LF(Z, p n ) is
holoedrically isomorphic with the group J3"E^JETO(2,p 2n ) of binary
hyperorthogonal substitutions of determinant unity in the GF[p 2n ~\
when taken fractionally, viz.,
where A~A P is the conjugate of A with respect to the GF[p n ].
The reciprocal of V is, by 142,
If J be a primitive root of J p ~ 1 = 1, so that J=J , the
following substitution of H,
,0,
generates a cyclic group G s +1 composed of the substitutions
o,
Any substitution V of H transforms ^^ into
This substitution belongs to the cyclic group generated by Q if and
only if AB = 0. Two cases arise.
If S = 0, tken ^L2 = 1 so that F= f ^^^ belongs to the cyclic
\0, At
group and evidently transforms every Q g into itself.
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2, p). 265
If A 0, then B B = 1, so that V= ( _ '- ) The latter trans-
(~j9 f\ \ '
-) = <2 9 , which is distinct from Q y unless the
0, J 9 /
latter be of period two.
The largest subgroup of H within which the cyclic group Gr,+i
s 4- 1
is self - conjugate is therefore a dihedron -group of order 2 -57-
Hence H, and consequently also GM(), contains
/ 9 -4 \ Id
C ( O * -- o I
6 v 6 v ^_ g & ~r 1 A / 1 \
I 2 ^ y
cyclic groups conjugate with 6r,+i. Each of these is defined by any
substitution lying in it (the identity excepted) as the largest cyclic
group containing that substitution. The -^ s (s 1) groups have there-
fore only the identity in common and contain in all s (s I) 2 or
i
Y s 2 (s 1) further substitutions according as p > 2 or p == 2.
244. To verify that we have now enumerated all the individual
operators of 6rj/(,) and consequently all the largest cyclic subgroups,
we note that
It was shown that if any substitution S of a cyclic 6r,q:i be of
^~i
period > 2 (viz., /S^/S^ 1 ), then $ is transformed into itself by no
substitutions of G M ( t } other than those of the cyclic G-,+I. Hence
"271
the latter is the largest commutative subgroup of GM(S) which con-
tains the substitution S. A commutative subgroup containing an
operator of period > 2 and different from p is therefore a
cyclic group. A commutative group containing an operator of
period p contains only operators of period p ( 241). Hence if a
commutative subgroup of GM(S)> p > 2, contains an operator of period
> 2, it contains at most one operator of period 2.
245. Cyclic and dihedron groups and their subgroups. The abstract
dihedron -group ft* may be generated by operators A, S subject
only to the generational relations
i
266 CHAPTER XH.
From the latter two follow the relations (holding for any integer r)
The cyclic subgroup G k generated by A is therefore self -conjugate
under 6r 2ifc . The latter is said to have the cyclic base G k . The k operators
BA (*-0,l,..., fc-1)
are of period two. For & odd, they are all conjugate under G ik
since B transforms BA into BA~ l = BA k 1 , which belongs to the
series Bj BA 2 , BA*, . . . For ~k even, they form two sets of con-
jugate operators
B, BA 2 , BA*,...,BA*~*;
,
BA, BA S ,
According as k is odd or even, they generate cyclic groups G 2 forming
one set or two sets of conjugate subgroups.
For every divisor d of &, Gk contains a single cyclic subgroup G d ,
which is formed by the operators
A*, A**, A**,'..,,A** = I (d = 1c/ct).
If ft be a given one of the integers 1, 2, . . ., d, the following d operators
extend the cyclic group G d to the same dihedron G 2d . There are
exactly d such dihedron -groups. If & be odd, these G% d are all
conjugate under G^. If d be odd, but k be even, the exponents p,
p. + $> p + 2 d, . . . are alternately even and odd, so that each G% d
contains operators of both of the sets 249); the groups G% d are
therefore all conjugate under 6r 2 *. If d be even and hence fc even,
the exponents are all even or all odd, so that the operators all belong
to a single one of the two sets 249); the groups G$ d thus belong
to two distinct systems of conjugate subgroups of G^.
If d > 2, G% d has a single cyclic G d and 6r 2 * a single cyclic Gk,
so that the above process furnishes every dihedron subgroup G% d
of 6r 2 jt. The theorem stated below therefore follows if d > 2.
We consider next the case d = 2, Jo even and > 2. The only
operators of period two in G^k are then A k ^ and
BA (i = 0, 1, . . ., & - 1).
Hence any dihedron 6r 4 must contain two operators BA r , BA S (r =f= s)
and therefore their product BA r BA s ^A 8 ~ r . Hence every 6r 4 must
contain A k/2 and may therefore be based on the subgroup G 2 of G k .
The theorem then follows as before. The Jc/2 possible groups 6r 4
in Gzie are given by the formula
[1, AW, BA r ,
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,pn). 267
Theorem. - - For every divisor d of k the dihedron G 2 k contains
exactly k/d dihedrons G* d forming one system or two systems of con-
jugate subgroups according as k/d is odd or even.
246. Cyclic and dihedron subgroups of GM(S) whose cyclic bases
are subgroups of the cyclic G S +I. By 242 243, GM& contains
i "^
- s(sl) conjugate cyclic subgroups 6r s qn each self -conjugate in a
i -
2;1
dihedron subgroup G s +i, but self -conjugate in no larger subgroup
2 77i >
of G M (s)- Hence these dihedrons are all conjugate under the main
s T" 1
group. 1 ) Let d+ be any divisor of ~L- and denote the quotient
1
by d+. GM( S ) contains s (s 1) conjugate cyclic groups 6r<?- , each
of which is ( 245) the cyclic base for d+ dihedron subgroups G% d -
Under G s +i they form one system or two systems of conjugate sub-
2 17i
groups according as dqp is odd or even.
For d+> 2, two subgroups 6r 2rf - of G +i are conjugate within
2 ~
the latter if conjugate within GM(S)'I indeed, the transforming sub-
stitution must be commutative with 6r d -, the only cyclic group of
order d+ in either 6r 2d -, and therefore commutative with the cyclic
6r 5 +i determined by it. Hence if d+ ~be any divisor > 2 of ;"
"sTI
and the quotient be d^, G M (s) contains in all M(s)/2d+ dihedron 6r 2 d-
forming one system or two systems of conjugate groups according as d+
is odd or even. In the former case, a G$ d - is self -conjugate only
under itself; in the latter case, self -conjugate under a dihedron Gz.2 d -
These G% d - are all conjugate within G^^) M (s)-
For d+ = 2, we have p > 2 since s 1 is not divisible by 2
for p = 2. Then s=p n is of the form 4/& 1 according as the
Jacobi-Legendre symbol ( J is 1; hence Is ( J is even,
say = 20. Then all the substitutions F 2 of period two of GM()
belong to the conjugate cyclic G^a- It remains to study the four-
groups 6r 4 , each a dihedron G 2 . 2 containing three cyclic 6r 2 . Now
G M ( S ) contains s Is + (- - ) conjugate cyclic G 2 . Each 6r 8 lies in
I s ( - - ) four -groups 6r 4 . Hence, ifp>2, G M (t) contains in
1) For every operator commutative with a group G is transformed into an
operator Commutative with G' by the operator which transforms G into Gf'.
268 CHAPTER XII.
\
all M(s)/\2 four -groups. Also the a four -groups contained in a
dihedron G a form (under the latter) one system or two systems of
conjugate subgroups according as <> is odd or even, viz., according
as s~p n has the form 8^ + 3 or Sh 1. Since the Gr a are all
conjugate within GM^, it follows, for 6 odd, that all the four -groups
of GM(S) are conjugate; while, for 6 even, they form at most two
systems of conjugate subgroups under GM(S)- For 6 even, each 6r 4
is one of 0/2 conjugate subgroups of a certain G a and is therefore
self -conjugate under a subgroup of order 8 of 6r 4ff . Suppose that,
for c? even, the subgroups 6r 4 of G M (s) form a single system of con-
jugate subgroups. Then each 6r 4 would be one of M (s)/12 conjugate
subgroups and consequently commutative with exactly the 12 operators
of a subgroup 6r 12 . By an earlier remark, the 6r 4 is commutative
with a subgroup 6r 8 . Since 8 is not a divisor of 12, our hypothesis
is untenable. Hence, for even, the 6r 4 form exactly two systems
of conjugate subgroups of G M (s)- For r )p>2, the M(s)/12 four-
groups 6r 4 contained in 6rj/(,) form one system or two systems of con-
jugate subgroups according as s=p n has the form 8/& 3 or 8fe 1.
In the former case, a 6r 4 is self -conjugate under a 6r 12 ; in the latter
case, under a 6r 24 . In the 6r 2J /( s) the 6r 4 form a single system of
conjugate subgroups and each is self -conjugate under a 6r 24 . Each 6r 12
is not a commutative group by 244 and so is of the tetrahedral
type ( 247). Likewise, each 6r 24 contains a tetrahedral subgroup 6r 12 .
The latter is of index 2 and consequently self -conjugate under (r 24 .
Since 6r 12 contains a set of 4 conjugate 6r s , the 6r 24 will contain a
complete system of 4 conjugate 6r 3 . Each is self -conjugate under
a 6r 6 , which is a dihedron since it is not commutative ( 244).
Finally, no operator of period 2 is self - conjugate under 6r 24 ; for it
is self- conjugate only under a dihedron 6r s q:i which contains no
tetrahedral subgroup and hence none of the present 6r 24 . Then by
248 each 6r 24 is an octahedral group.
247. A non- commutative group of order 12 having a self -conjugate
four -group is of the tetrahedral type.
Let the operators of the four -group be J, F 2 , F 2 ', F 2 ", so that
they are commutative and the product of any two F's gives the
third F. The 6r 12 contains at least one operator F 3 of period 3.
The products Tr . T _ . Tr Tr . , . ~
JV> V * Y 3> v v *> v " v * (t-o/ijS)
are all distinct and so give all the operators of 6r 12 . The 6r 12 would
be a commutative group if F 3 were commutative with F 2 , F 2 f , F 2 ".
1) For p = 2, the four-groups are determined in 249. There are
4- (2* - 1) (2n - 2) sets.
6
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP iF(2,p). 269
Since therefore V 3 does not transform each V into itself and since
it does not permute two of them, its period being =)= 2, it must
permute them in a cycle. Fixing the notation, we thus have
V~ 1 VV = V V~ 1 W = V" V~ 1 V"V=V
(F 3 F 2 ) 3 EE V S V 2 V~ 1 Fs^FgFs - F 2 = F 2 "F 2 'F 2 = I.
Hence F 3 , V% generate 6r 12 and satisfy the generational relations
F 3 = J F 9 2 = J (FF) 3 = J.
of the tetrahedral group, an abstract group of order 12 holoedrically
isomorphic with the alternating group on 4 letters ( 265).
248. A group of order 24 having no self -conjugate operator of
period 2 and having a set of 4 conjugate G s each self -conjugate in a
dihedron G 6 is of the octahedral type.
The 4 conjugate 6r 3 are transformed into each other by the
operators of 6r 24 . Hence 6r 24 is isomorphic with a substitution - group
on 4 letters. The isomorphism will be holoedric and consequently
the latter the symmetric group G$, if the identity be the only
operator of 6r 24 which transforms each 6r 3 into itself, i. e., if the
four G 6 have only the identity in common. But if a substitution of
period 3 were common to the dihedron 6r 6 , it would be common to
the 6r 3 , and these would be identical contrary to hypothesis. If
the G 6 contain in common two substitutions of period 2, they would
contain in common the product of the two which is a substitution
not the identity of the cyclic bases G 3 ( 245). Finally, if the con-
jugate G 6 contain in common a single substitution of period 2, it
would be self - conjugate under 6r 24 contrary to hypothesis. Now the
is of the octahedral type
249. Subgroups of the s + 1 commutative G-i*\ Since these groups
are all conjugate under 6rjf(,), it suffices to determine the subgroups
of G^ formed of the commutative substitutions Sp of period p. If
a subgroup contain S^, /S^ 2 , . . ., S^, it will contain ^, where
ft = q^ + C2fi 2 H ----- h Ctpt, the d running independently through the
series 0, 1, . . ., p 1. Hence to every subgroup G- p m of order p m <p n ,
there corresponds an additive -group in the GF[p n ] of rank m with
respect to the G-F[p~] and inversely. Hence, by 69, the number
of distinct subgroups G p m of G p n is
(p n l)(ffff p)(p n p 2 ) . . . (p n p m ~ *)
l~)(pm p) (pmp*) . . . (pm pm 1)
Let 6ry be one such group composed of the substitutions $/,
where A ranges over an additive -group [A 1; . . ., A m ] of rank m with
270 CHAPTER XH.
respect to the GF \_p\. By 241, G- p m is transformed into itself
only by substitutions of the form F= (^ L -^ rV Since F transforms
\0, a y
Si into S a *z, a further condition is that a 2 1 and A should run
simultaneously through the series of marks of the [A 1? . . ., AJ.
Suppose that there are in the GF[p n ] exactly e marks s lf . . ., s e
such that [A 1; . . ., A m ] = [?Ai, . . ., ?A m ]. Then, according as # > 2;
og
j) = 2, the ^ r substitutions
if\ = (ljj\ (
TV^Ao, i/\o, 5-1
o,
where /3 ranges over the GF[s] } constitute the largest subgroup H
of G M (s) under which 6ry is self -conjugate. But the multipliers H
of the additive -group [A x , . . ., A m J are ( 70) the marks x =)= of the
multiplier GF[p k ~\, k being a divisor of m and n. It remains to
distinguish which of them are squares of marks t - of the GF[p n ~\.
For the respective cases
p > 2 with n/k even, p > 2 with w/fc odd; ^ = 2,
there are ( 62) exactly e = (2, 1; l)(.p*--l) marks ,-, so that T is
of order ~- ^ Hence G p m is one of a system of
2; 1 1, <i; 1
conjugate subgroups of GM(}- Here the value of k depends on the
individual G p m chosen. Given ft, the number of the corresponding
sets of G p m follows from 71.
250. Non- commutative subgroups of the s + 1 conjugate 6r% p .
It suffices to study the group G given by x = oo. It is composed
of the substitutions
^
^_\ = /i^W, o
>, ~v \o, i/vo, -
For a given mark a =|= 0, ^ and /3 run simultaneously through the
series of marks of the GF[s]. A rectangular array of the substitu-
tions of G may be formed by taking as the first row the substitu-
tions Sft y which form the self -conjugate subgroup G s , and as right-
hand multipliers the substitutions P a = ( - = | of the cyclic Gt^i
\0, dTV -271
In any subgroup 6r' of G the totality of substitutions of period p
give rise to a commutative group G p m of substitutions &, where A
ranges over an additive -group [A 17 . . ., AJ. Hence 6yn is self-
conjugate under G r . A rectangular array of the substitutions of G 1
\ 271
with those of G p m in the first row has the property that the sub-
stitutions in each row are all found in a corresponding row of the
rectangular array for G. In fact, two operators A, B of G 1 lie in
the same or in different rows of the array for G' according as
AB~ l is or is not in G p m . But AB~ belongs to G' and hence
belongs to G p m if, and only if, it occurs among the substitutions in
the first row of the array for G. Hence each row for G' lies wholly
in a row for G. The quotient -group G' /G p m is therefore a subgroup
G d of the quotient -group G/G S , the latter being a cyclic 6r,_ i
i;i
Indeed, these quotient -groups may be obtained concretely as groups
of the permutations of the rows of G induced by applying as right-
hand multipliers the substitutions of G or G'. But all the substitu-
tions in the same row of G (and, a fortiori, all in the same row
of G r ) give rise to the same permutation. Hence G d is an abstract
cyclic group. Now G contains s cyclic G^li , where r runs through
"271
the series of marks of the GF[s], all conjugate under the trans-
formers S^. Leaving different elements r fixed, they have no sub-
stitution other than the identity in common. Counting also the s 1
substitutions of period p, we have accounted for all the substitutions
of 6r. Besides the cyclic subgroups of G s , G therefore contains no
cyclic subgroups other than the Grf 9 for the various divisors d
of S ~ Among these cyclic groups occurs one whose substitutions
2; 1
may be chosen as the right-hand multipliers in forming the above
array for G'. In fact, within the row of G 1 corresponding to the
generator of the quotient cyclic G d there must exist a substitution A
such that A d j and no lower power, belongs to the group G p m whose
substitutions form the first row. The right-hand multipliers for the
array may thus be chosen to be I, A, A 2 , . . ., A d ~ 1 . Hence G' is
given by the extension of the G p m by a certain G ( d ^\ within which
G p m is self -conjugate. But the largest subgroup of GM($) within
which G p m is self -conjugate is ( 249) the group H of order sK,
K= f ~ j given by the extension of G^ by a cyclic (T^' O) . In
1, 2; 1
particular, d must be a divisor of K, so that d depends upon the G p m.
The cyclic #$?' OJ contains a single cyclic 6rd' 0) . Hence, by trans-
forming G' by a suitably chosen S^, we obtain a group G p m d (con-
jugate with G' under G) given by the extension of G p m by the sub-
group ' 0) of ?' 0) . The substitutions & of G p m transform that
subgroup into p m conjugate cyclic G d ' \ since Si replaces the fixed
elements oo, by elements oo, A. These p m groups together with
G p m contain all the substitutions of G p m d , as shown by simple
272 CHAPTER XII.
enumeration. The largest subgroup of GM(S) transforming G p d into
itself must therefore transform G p m into itself (and thus be a sub-
group of H) and transform the groups of the single set of con-
jugate 6rd ' amongst themselves. Of substitutions of period p y it
must therefore contain only the Si. The required group is thus a
subgroup of the group H 1 of order p m K given by the extension of
G p m by GK' Moreover, it is H' itself since any substitution
of &*' 0) > such that 8 A = A' is of the [Ai, . . ., AJ,
0, a
replaces the elements oo, A by elements oc, A f and consequently trans-
forms OS* 1 J into OS* 1 . Hence the group 6r^m d is one of a system of
(pn -f 1) (pn 1) pn m
(2,1;
conjugate groups. Finally, if the subgroup G r contains no substitu-
tion of period p, it is a cyclic subgroup 6$' x) of one of the cyclic
251. Subgroups of GM( S ) containing operators of period p. - The
substitutions of period p of a subgroup GQ of the GM(S) distribute
themselves over certain s + 1 subgroups G m of the s + 1 con-
jugate G*f* ( 241). By hypothesis at least one of the orders p n 'v
is > 1. By suitable transformation within GM( S ), w e arrange it so
that p m > 1, m = m^ > 0. Under the p m transformers Sp of the G^ ,
the remaining G 2 with m^ > (ft =j= c), if any, arrange them-
selves in sets each consisting of p m conjugate groups. Under the G$
the G m is then one of a set of 1 -f- fp m conjugate groups, f being
(/*)
a positive integer or zero. The G$ contains no group G p m^ (m^ > 0)
other than the 1 + fp m groups of this set. For, any such group
would be one of a set of n^ conjugate groups, where % would
necessarily have at the same time the forms 1 -f- /^"V* and f^p.
Hence: Every G$ which contains operators of period p contains these
(/<)
operators in 1 + fp m groups G p m conjugate under GQ, where for each
G$, f and m are properly determined integers /"> 0, m > 0.
The groups GQ with /"= have been enumerated in 249 250.
Consider the group GQ with f^ 1, m > 0. It contains 1 + fp m groups
(oo)
conjugate with a certain G p m formed of the substitutions ft, where
(J
A ranges over the an additive -group [A 1? . . ., A m ]. The G p m is ( 250)
(oo)
self -conjugate within G-Q under a certain largest subgroup G p m d . Hence
250) Q = (1 + fp m }p m d.
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,pn). 273
(oo)
As in 250, we transform 1 ) by a suitable 8^ and obtain a G p m d
given by the extension of the group G p m of the Si by the cyclic
group G d ' of the substitutions P n contained within the cyclic
group 6r*(2; i) of substitutions P e .. The group G p m d is thus composed
of the substitutions 2 )
Since 17 and -f- ^ l ea( l to the same P n , there are (2; 1t)d marks y,
the distinct powers of a primitive root of rfa = 1. Since each 1? is
an t -, each ^ 2 is a multiplier of the additive -group [A 1; . . .,
To normalize G-Q we transform by P a :
-i /, p
a I -
\y, S
9
, 8 /
-I
Taking 6 = "J/A^" 1 , the transformer P a is a substitution ^'=A^~ 1 ^ of
the 6r(2 ; i)if(,); (?j/(,) is transformed into itself and 6r$ into G'Q. The
new additive -group [Aj, . . ., A^] contains the mark 2 A = 1 and hence
all the marks =j= of its multiplier GF[p k ]. We suppose this trans-
formation to have been made and the primes dropped from G', A}.
(op)
The GQ of order 250) is obtained by extending the G p dj formed
of the V^i, by certain fp m extenders Vj ( j = 1, . . -,fp),
V.
It was shown above that G^Q contains (1 + /j) m ) (j9 m 1) sub-
(oo)
stitutions of period p. Of these p m 1 are the * lying in G p m.
The remaining fp m (p m 1) are substitutions V^iVj satisfying the
necessary and sufficient conditions for period p ( 240),
+ djvr 1 + m* 2 -
Given F^ (% 4= 0) and 7; (77 ={= 0), there are at most (2; 1) values A
satisfying 251). For a given Vj (% =j= 0) there are consequently at
most (2; l)d substitutions V n , i = Vn,i such that V^iVj is of period p
Hence the various Vj lead to at most fp m (2] tyd such substitutions.
1) For pk =3, n/k odd, we have (i = l 7 so that this transformation is here
unnecessary and is reserved for use in 252.
2) The non- fractional substitutions (viz., with y = 0) of GQ are all of the
form Vtj t i. Indeed, they form a group G' leaving the element oo invariant.
/ \
Its substitutions of period p form the subgroup G m which must be self-
(<K)
conjugate under G' . Hence G' = G m d -
DlCKSON, Linear Groups. 18
274 CHAPTER XH.
Comparing this maximum with the actual number fp m (p m l), we
have p m 1^(2 ;!)<#. Since each of the corresponding (2; l)d
marks r\ must be one of the e marks s { of 249, then (2; l)d^e,
where e = (2, 1; 1) (p k 1) Finally ( 70), & is a divisor of w.
Hence
252) i>-l5X2;l)d^(2,l
Since the third number is always <J 2(^* 1) < 2p k 1, we have
^? m <2j9* ; so that m = k, m being a divisor of k. The additive-
group [A 1; . . ., A m ] is therefore its own multiplier G^JPfjp*] and every
4 is zero or a multiplier %.
There .are in all two cases:
[A] w-fc, jp*-l = (2;l)d, Q = (l + /y>*(.P*
[B] m = ~k, p>2, n/k even, p k 1 = d, Q = (1 +. fp k )p k (p k 1),
where for (2 ; 1) we read 2 or 1 according as p > 2 with /& odd
or p = 2.
The following lemma finds repeated application below:
If Vj (yj =f= 0) be of period 2, the ratio ctj/yj differs from the a f -/y,-
of every other V f and so is a characteristic invariant of the V n ^V^.
For i=%=j, ViVj is not of the form V^, since otherwise
V = V ?V
r t - ' /;,/ ' jy
contrary to the choice of the extenders Vi. Hence in V { Vj the term
corresponding to y is =(=0, viz., titfy Jriytfy *$*& Dividing by y t yj
and applying dj = Vj ( Vj being of period 2) we find that
<M- - "tin 4 s -
252. For case [A] with p k > 2, the group GTQ is the group G- M ( P k )
of all linear fractional substitutions of determinant unity in the CrF[p k ].
For p k = 2, (TQ is a dihedron ft (1+27), which for f=l is the 6rj/ (2 ).
For p k > 2, it is shown that every Vj may Jbe chosen so that
a j> fe> Yjy $} a ll belong to the GF[p k ~\. Hence G-Q is a subgroup
of 0- M(l t). But, if f> 1, Q > Jjf (i?*). Hence must f= 1 7 Q = M(p k \
SO that (TQ = G-M(p k )>
For case [A], relations 252) become equalities, so that the earlier
argument shows that, for Vj and i] given (% =)= 0, ^ =f= 0), there
exist exactly (2; 1) marks A of the [^, . . ., AJ which satisfy 251).
The given 17 may be any one of the multipliers %, since the number
(2; !)<# of ?^'s equals the number p k 1 of %'s.
The extender Vj may be replaced by any one of the products
Vtj t iVj and in particular by one of period p y having therefore
K 3 H~ fy = it ^- Changing if necessary the signs of all four coefficients
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP F(2, pn). 275
of Vj, we may take Uj + #y = 2. With this normalized FJ, the condi-
tion 251) becomes (upon setting rj .= x)
253) u f (x - x- 1 ) -f ft-xA = 2- 2x~ l .
For any given j and any given .mark x =j= of the GF[p*] and for
each sign +, this equation must determine a mark A = Ay )X) + of the
GF[p k ]. If p > 2, 253) for ?c = 1 gives
so that % belongs to the G-F[p k ~]. For^ fc > 3, x has a value different
from + 1 and from zero; for such a x 253) requires that ay belong
to the GF[p k ]. Then <?y=2 y belongs to the field. The deter-
minant being unity, ft also belongs to the field.
For p k =%, the non- vanishing marks %-, y may be restricted to
the value + 1. Since a, + <?, = 2 in FJ-, the + <? of V^iVj=V/
has the value #y -^ <Sy + 1 = in the field. Hence V] takes the form
The TF may be taken as extenders in place of the FJ-. The sub-
group Gr p m d is here composed of three substitutions V\,i y >L = 0, 1.
Hence every substitution of 6r^ has as its y and a -f d marks of the
G-F[p k ]> Transforming the group by S ao , where cc Q is a particular a,
each V lh i = S)i is transformed into itself and each W a into W a a -
Hence, in the transformed group each y and a + d belong to the
Among the new extenders W a = W a a occurs W Q . Hence
contains
so that the mark a, being in the position of a y, belongs to the
~
For ^) = 2,^>1, there exist marks ^ different from and
1 (-f 1 = 1); for such a x, 253) shows that c^/%- is a mark Ay of
the 6^[2*]. Since ^ = 2, ay + fy 2 gives a, = dy, and dj/yj = A>.
There are fp k substitutions Vj and |jp* > 2. The product Fl- Fy (f =)=,;)
belongs to 6r^ and is not of the form F Xj ^ since Fi =^= VjV*,\ and
since T^- is of period 2. Hence we may set F;Fj- ?= Fx^Fi. Since
i =j= j, A,- + A ; - 4= (end of 251). We find that .
v/+yft Jii_ J_ -ft
= " i "
Hence every ftyy belongs to the (r^[^]. Then aytfy .ftyy = 1
requires that y| belong to that field and hence also "%, ^> being 2.
Then a^ ft, (Jy belong to the field since their ratios to % do.
18*
276 CHAPTER XH.
For p = 2, Tc = 1, the group (TQ of order 2 (1 -f- 2/") is given by
the extension of 6r), formed of the substitutions SQ = ! and 8 lf by
certain 2f extenders Vj (j = 1, . . ., 2f) each of period 2. By 251,
all the substitutions of period 2 in GTQ form one set of 1 + 2f con-
jugate substitutions. Setting F =$ 1? the substitutions of period 2
in G-Q are 7} (j = 0, 1, . . ., 2f) and the remaining substitutions
F FJ- = Z7,- are of period =j= 2. Hence no U is conjugate with a F.
The product Jy'ZTJ cannot be a 7; for the substitution of GQ which
transforms Vf into F transforms the product into FO Ut = Fi-, but
transforms the U into some C7". Hence VfUj is of the form Vj" so
that FO //' Uj = Fo //'. Hence every product Uj' Uj is a Uj". The
substitutions U form a group 6ri . Since Uj = VoVj, we have, for
"tr~ Jt
every ,;,
254) F.-'ffyF.-CT*-
For Z^- and ^ arbitrary, there exists in the (TI a U) such that
T
The group Gri+zf of the C7's is therefore commutative and contains
substitutions of period > 2. By 244, it is a cyclic subgroup of
ft + i. In view of 254) the group G$ is a dihedron #2(1+ a/) based
on the cyclic 6ri + 2/ ( 245). These groups 6r$ have therefore been
enumerated in 246 and may be dropped from further consideration.
253. For case [B], p > 2, n/fc is even and p k \ = d. The
2 c? marks 77 are the square roots of the p k 1 marks K and hence
are the distinct powers of ?7o = )/^o> where x is a primitive root of
the 6r.F[jp*]. In particular, there is a mark 77 = Y~" 1-
Within GTQ there are exactly 1 -f /'^) i groups conjugate with the
G- p k(pk_i). The latter contains p k conjugate cyclic G- p ki and hence
in all p* substitutions T of period 2, each conjugate with
ia /Y=i, o y
\ o, -y^i/
Under 6r Q of order Q = (1 -f- fp k )p k (p k 1), this jT is one of a.
system of (1 + fp k )p k or ~ (1 -f- fp*)p k conjugate substitutions T
according as T Q is within G-Q self -conjugate under the cyclic (r^ALi
or under a dihedron obtained by extending the former by a sub-
stitution TQ which interchanges the elements oo, ( 242, 246). In
the respective cases there would be at least fp 2k or - (fp* T)p k
U
substitutions "V^iVj ( j > 0) of period 2, necessarily satisfying the
relation 251),
a- + + ** = 0.
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,p). 277
For each of the fp k extenders Vj ( j > 0, % =(= 0)> eac ^ value of 77
gives a single value of A, which may or may not belong to the GF[p*\.
Hence there are at most fp k (p k 1} substitutions V^iVj of period 2.
The second alternative therefore holds, so that G-Q contains a sub-
stitution of the form
7S*
Also (1 -f /!**) is an integer so that f is odd.
In case a FJ- (J > 0) gives rise to one or more substitutions
T =~ V n ^Vjj we replace FJ- by one such T, so that the new Vj has
a, + (5} = 0. Let N denote the number of these Vj for which there
exists a product V th iVj distinct from Vj and of period 2. For such
a Vj the equation
= (17 + (>)
will be satisfied by a pair 77, A =|= 1, 0, such that ?? 2 and A belong
to the GF[p k ~]. Hence will
belong to that field. Inversely, if ay/y/ belong to that field, and 77*
be an arbitrary mark =j= of that field, there exists an unique
solution A in the field, so that there will be p k 1 substitutions
F^iFJ- of period 2. By the lemma at the end of 251, the ^sub-
stitutions Vj have distinct values for cty/y/, here shown to belong to
the G-F[p k ]. Hence N^p". Let M denote the number of the Vj
leading to a single V^Vj of period 2. Then M^fp k N. The
total number of the V^iVj ( j > 0) of period 2 is therefore
N(&- 1) + M^ N(p k -l) -f fp k -N.
The second member is greatest when N has its maximum value p k .
By comparing the minimum and maximum numbers for the
r*2F, (j>o)
of period 2 in GQ , we have
255) i (/y - l)i>*^ jp* (^ -1)4- (/--l)^.
Hence must /"= 1 or 3, leading to the two cases:
(/=!) p>2, n/k even, Q = (p* + l)p* ( jp* - 1) = 2 JJf(^)
(/* 8) J) = 3, fc=l, n even, Q = 60.
Consider first the case /"== 1. G^Q contains the transformed of
K by &,
278 CHAPTER XII.
Letting A run through the series of marks of the GF[p k ~], the ratio
a/y = AT/T = A takes p n distinct values. By the lemma at the end
of 251, the. Tl may be chosen as the p k = fp k extenders Vj. For
each Vt the ratios jM ^//X are marks L, Aj' of the G^Ttf*! As
y JI 9 J / r * / */ / y L-i. -J
in case [A] for p = 2, ^ > 1, the ratio /fy /% is a mark A) of the field.
The determinant being unity, yj belongs to the field, so that % is
some yt. Hence
According as % is an even or an odd power of % = y^ > "PJ or
Vij ,oVj has its coefficients in the &F*[|)*J. The one having this
property is denoted by Vj . These p k substitutions Vj serve to extend
the group CPP t of the V x ,i to the group GM^ of all linear
2
fractional substitutions of determinant unity in the GF[p k ~\. It is
transformed into itself by
P* =
* no
whose square P,* = P Xo belongs to 'G-M( P k )> Hence P no extends the
latter to the group G-2M(p k ) of all linear fractional substitutions in
the G-F\_p*~\. The latter is a subgroup of G-Q and is of order Q.
G-Q is therefore identical with the linear fractional group 6r 2 j/( p *).
For the case f = 3, p k = 3, the relation 255) becomes an equality,
so that there are exactly 12 + 3 = 15 substitutions T- of period 2
in 6r 60 . At the beginning of the section, each T was shown to be
self -conjugate within 6r 60 under exactly a dihedron 6r 4 . The 15 sub-
stitutions T are therefore all conjugate under 6r 60 and form 5 con-
jugate four -groups 6r 4 By 251, (r 60 contains one set of 1 -\-fp k = 10
conjugate 6r 3 . Hence, if the 6r 60 exists, it is of the icosahedral type
( 254). For n even, 5=y(3 2 +l) divides ^-(S 271 -!), so that the
existence of icosahedral subgroups 'of G M (^) follows from 259
The question of the conjugacy of the icosahedral subgroups is
answered in that section.
254. A group of order 60 is of the icosahedral type if it contains
exactly ten conjugate 6r 3 and exactly 15 operators of period 2 lying
in 5 conjugate four-groups.
Since there is a complete set of 5 conjugate 6r 4 within the 6r 60 ,
each 6r 4 is self- conjugate under exactly a subgroup 6r 12 . The latter,
is of the tetrahedral type by 247; for if commutative it would
contain a self -conjugate 6r 3 which wpuld be one of a set of at most
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,pn). 279
5 conjugate subgroups of 6r 60 . Hence 6r 60 contains a set of 5 con-
jugate tetrahedral 6r 12 . No two of them are identical since each
contains a single four -group. They have only the identity in common.
Indeed, their common operators form a self -conjugate subgroup of
6r 60 and hence a self -conjugate subgroup of each 6r 12 . Aside from
the identity and 6r 12 itself (cases requiring no further discussion), the
only self -conjugate subgroup of a tetrahedral 6r 12 is its four -group.
But the 5 four -groups are all distinct. Hence the identity is the
only operator of 6r 60 which transforms each 6r 4 into itself. Applied
as transformers, the operators of 6r 60 permute the 5 conjugate 6r 4 ,
so that 6r 60 is holoedrically isomorphic with a substitution -group on
5 letters. Being of order 60, the latter is necessarily the alternating
group on 5 letters. 1 ) Hence the 6r 60 is of the icosahedral type ( 267).
255. It remains to study the conjugacy of the linear fractional
subgroups G M ( P k ) and G^M( P k ) of GM( S )> Within GM(S) the G M ( P k ) is
self -conjugate exactly under G% M ( P k ), G-M^y, Gn( P k ) according as p>2
with n/k even, p > 2 with n/h odd; or p = 2, and hence is one of a
.system of Jf(s)/(2, 1; 1) M(p k ) conjugate groups. In proof, we note
that a substitution V= f^- of &*() transforms ( 240) the sub-
stitutions (.jr-rl and ( r) into respectively
/l-yg, CC*G \ /l+-|Mg, -|3 2 g
V -y a <r, 1 + ^G) \ 8*6, 1-pd
If (? belongs to the G-F[p n ~\, these substitutions belong to that field
if, and only if, a and y are each marks ft of the GF[^p k } or are
each of the form y>Yv, where v is a not -square in the GF[p k ~\,
and /3, d are each marks p or are each of the form ftyV. Since
ccd -- (ly = 1, a, /?, y, d are all of the form ^ or all of the form p
Hence F is either a substitution 5 of G M ( P k ) or else a product
The latter alternative does not occur if p = 2. Also, if p > 2,
belongs to the GF[p n '] if, and only, if n/Jc is even. Hence G
is self -conjugate within 6^jf(,) in a .larger group, viz., 6JW(/), if and
only if p > 2 with w/7c even.
Within G M ( S ) the G% M (p k ), when existent, is self -conjugate only
under itself. For any substitution of the former which transforms
the latter into itself must transform its self -conjugate subgroup
1) If a G^ contained odd substitutions, it would have a subgroup
of even substitutions. The latter would be of index two under the alternating
group G ( j$ and hence self - conjugate under it, whereas it is simple.
280 CHAPTER XII.
into itself and hence belong to 6r 2 j/(/). The latter thus forms one
of a system of M(s)/2M(p k ) conjugate subgroups.
It remains to determine the number of systems of conjugate
subgroups of these two types; indeed, in 251, there entered the
transformer Py^i which belongs to G M (s) if and only if A is a
square in the G-F[p n ~]. For p = 2, A is necessarily a square; for
p > 2, n/k odd, A may be chosen as a square, since every additive-
group [A x , . . ., AJ with the multiplier G-F[p k ~] has half of its non-
vanishing marks squares in the GF[p n ~\. In these two cases there
is evidently but one system of conjugate subgroups G-M( P *) of GM()-
For p > 2, nfk even, all the marks of [A 1; . . ., A*] are squares or all
are not -squares in the G-F[p n ^ indeed, they are all obtained from a
single one by multiplication by the p n marks of the multiplier GrF[p Ji ]
and the latter are all squares in the G-F\_p n ]. In this case there are
consequently two systems of conjugate subgroups GM( P *-) and two
systems of conjugate CrzM(p k )j the systems of each type being inter-
changed upon transformation by Py^ f belonging to 6r 2 jf(), where v
is any not- square in the 6rjF[# n ]. Hence there are (2, 1; 1) systems
of conjugate G M ( P k ) and (2, 0;0) systems of conjugate 6IW(p*) within
256. Subgroups of GM(S) containing no operators of period p.
Every substitution of such a subgroup GQ lies in and determines a
largest cyclic subgroup G d of G-Q ( 242 243). Two such groups
G d have only the identity in common. According as G d is self-
conjugate within 6rg only under itself or under a dihedron G$ d based
on G dj it is one of a system of Q/d or Q/2d conjugate subgroups
of (TQ. Let r denote the number of such systems. The enumeration
of the substitutions of G leads to the relations
256) Q = 1 +(^-1)- (fi- 1 or 2)
i=l
* *
257) $>fidi (; = 1, 2, ...,r).
If two non- conjugate cyclic G d ., G d . of odd order are present
in 6rQ, there are at least dj groups in the system determined by G di9
viz., the transformed of the latter by the operators of G d ., and vice
versa, so that
258) Q :> d t (dj - 1) + dj (d { - 1) + 1.
Solving 256) for 1/Q, we get
259) 1
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,p). 281
Since f { = 1 or 2, the least value of (d { 1) lf { di is 1/4. Since 259)
must be positive, there can be at most three terms in the sum,
whence r < 3.
For r=l, the reciprocal of 259) is not an integer if /i = 2.
For /i = 1, Q = d lf and the (TQ is a cyclic group considered in
242 243.
For r = 2, we have
I..JL. .JLfi.-^l 4- -iYl - -iV
a 5ft St; /Ti U
If f 1= = f z = 1 ? the left member is < 1 and the right member is > 1.
I* /I == /2 == " > 9 1 i o 1
Hence these two cases are to be excluded. The case f = 2, f^= 1
differs only in notation from the case /j = 1, f% = 2. In the latter case,
_LJ_J_ JL = JL _L
"Q" :S ^""2^"2 < ^d 1 "4 ?
so that ^ < 4. For ^ = 2, Q = 2(^ 2 , so that 6r$ is a dihedron 6r 2 </ a
with d 2 odd ( 245) yielding a group considered in 246. For d t = 3,
d% must be 2, whence Q = 12. The operator of period d% = 2 is
self -conjugate within 6r 12 under exactly a dihedron 6r 4 , so that 6r 12
is not a commutative group. Since the operators of period 2 fall
into a single set of 3 conjugate operators, there is a single sub-
group 6r 4 , so that it is self - conjugate under 6r 12 . By 247, the 6r 12
is a tetrahedral group.
For r = 3, then /i = = /* 3 = 2. For if 1, for example,
259) becomes
l_ (d, - 1) fo - 1) == 1 J_^0
^ " ft4i f 8 ^ ^-A 4 4 ^
Setting each f f = 2, equation 259) may be written
1 -i- A JLj. -l-i.
' ft " ^ " ' <? 2 " ' d s
If every c? t - jj> 3, the right member would be ^ 1. Setting (? 3 = 2,
JL 4. A JL L
2 ' " 2 ' " d t ' d^
If either d^ or c? 2 is 2, we may take <? 2 2, whence Q = 2^ and
GQ is a dihedron 6r 2 dl with ^ even ( 245) yielding a group consid-
ered in 246. In the contrary case, d l > 2, d z > 2. Then both
do not exceed 3, since otherwise the right member would be at most
- -f - = _ . Taking d% = 3, we have
i . , A . . JL
6 Q " " d t
282 CHAPTEE XII.
Hence d < 6. For d = 3, 4, 5 we find Q = 12, 24, 60 respectively.
But ^ =3, d 2 = 3, dg = 2, Q = 12 is excluded by 258). For d = 4,
e? 2 = 3, t2 3 = 2, the 6r 24 is of the octahedral type ( 248). For ^ = 5,
d 2 = 3, fl? 3 = 2, the 6r 60 is of the icosahedral type ( 254).
257. The tetrahedral and octahedral subgroups of the G M ( S -). A
group of either type must contain a self - conjugate four- group. For
p > 2, the desired groups are therefore given by the theorem at the
end of 246. For p = 2, they contain operators of period 2 and
are therefore to be sought among the subgroups determined in
250 253. But for p = 2, the dihedron (r 2 (i+2/) and the G M ( P k )
are neither of the tetrahedral and neither of the octahedral type.
There remain for consideration only the subgroups of the 6rl*!_i) of
250. There is no octahedral subgroup of (T% i) since the sub-
stitutions of period p = 2 in the latter are all commutative. In a
tetrahedral group the three substitutions of period 2 are all commu-
tative. Hence if there be a tetrahedral subgroup of (r^s i), P == %>
then must 2 m = 4, d = 3 and n even (since 3 must divide 2" 1).
Inversely, if m = 2, p = 2, n even, there exists a subgroup G#n d = 6r 12
of 6ri$_i). The 6r 12 is not commutative, since it would then contain
only operators of period p = 2 ( 241), and therefore 6r 12 has the
.tetrahedral type ( 247). We may state the complete theorems:
For s =p n = Sh + 1, the 6rj/(,) contains two systems each of M(s)/24t
conjugate octahedral groups 6r 24 and two systems each of Jf(s)/24 con-
jugate tetrahedral groups 6r 12 . Every 6r 12 is self -conjugate under a 6r 24 .
The two systems are conjugate under 6r 2 ^( a ).
For s = Sh 3 or s = 2*, w evew, $e GWw contains no octahedral
6r 24 W contains one system of M(s)/\2 conjugate tetrahedral 6r 12 .
For p > 2, #&e G$M(S) contains one system of M(s)/l2 conjugate octa-
hedral 6r 24 each containing one 6r 12 . .For s = 2 n , n odd, GTM() contains
no octahedral and no tetrahedral groups.
258. Icosahedral subgroups of G- M ( S ) for p = 5. An icosahedral
6r 60 is generated by two - operators F 5 , F 2 different from the identity
and subject to the generational relations ( 267)
y 5 __ y 2 __ j r Y V V = J
* ' K '"* * ' - *-'
contains 4 (5 -f 1) = 24 substitutions of period 5 and each
is conjugate within 6^5) with one of the substitutions ( 241)
^ =^ (mod 5).
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP iF(2,jp). 283
The only substitutions F 2 of period 2 of GTMW which satisfy the
condition 1 ) (F 5 F 2 ) 3 = J are seen to be the following five
a
_ . - (ft ^ j 2 3 4)
*
Hence GM(S) is an icosahedral group 2 ) and contains 24 5 = 120 pairs
of generators F 5 , F 2 . By 255, Gru(t n ) contains Jf(5 n )/60 icosahedral
subgroups forming two systems or one system of conjugate groups accord-
ing as n is even or odd.
259. Icosahedral subgroups of GM() for jp=j=^- ^ e or( ler
p(jp2-l)/(2; 1) of #*(,) is divisible by 60 if, and only if, p* n -l
be divisible by 5 and hence either p n -\- 1 or p n 1 divisible by 5.
In either case GTM^ contains cyclic subgroups 6r 5 all of which are
'conjugate ( 242, 243).
(i) Let p n 1 be divisible by 5 and set A='(# n 1)75. Let (>
be a primitive root of the G-F\_p n ~\, so that p* is of period 5. Setting
<- 260) - 2
we seek the conditions under which the product
shall be of period 3. The necessary and sufficient condition is
a(0* f-*) 1'.
The upper sign may be chosen, changing if necessary the signs of a,
fi, y in F 2 . Hence a is determined uniquely. Combining with 260),
Indeed, if the second member vanish, p 4 * 2 *-f !_== 0, so that
06* _|_ i __ Q an( j therefore p 2 *= -f 1, whereas (>^ is of period 5. Hence
to each of the p n \ values =j= of /3 corresponds a single value
1
of y. But 6rj/(,) contains ( 242) exactly -^p n (p n + 1) distinct cyclic 6r 5 .
Hence there are 2p n (p 2n T) pairs of generators F 5 , F 2 of icosahedral
subgroups.
1) It is readily verified that a substitution f** 1 " |
period 3 if, and only if , a -f ^ = + 1. ^ y ' d '
of determinant unity
1 .r ,V /
is of
2) Cf. 280.
284 CHAPTER XII.
(ii) For p n -f 1 divisible by 5, let g = (p n + I)/ 5 and set ( 243)
0,
The condition (F 5 F 2 ) 3 = J is satisfied if, and only if,
The A thus determined satisfies the condition A = A. Then must
The last term is a mark f* =|= of the GF [p n ] . Hence B p ' ~ l = p
has a solution .Z? in the GF[p 2n ~\ and consequently p* + 1 distinct
solutions JB^, JB <7, J5 J 2 , . . ., B J P . But 6rj/( s) contains exactly
_^n (^ _ i) conjugate cyclic 6r 5 ( 243). 5ewce there are 2p n (p* n 1)
_2?airs of generators F 5 , F 2 o/" icosahedral subgroups.
Since each icosahedral group contains ( 258) exactly 120 pairs
of generators F 5 , F 2 , it follows that, for p* n 1 divisible by 5,
Cr M ( P n ) contains in all ^) n (^ 2n 1)/60 icosahedral subgroups.
For p = 2, 2 2 1 is divisible by 5 = 2 2 + 1 if and only if n
be even. If n be even, G-M(% n ) contains a single system of Jf(s)/60
subgroups G- M (v) ( 255), the latter being icosahedral by case (ii).
Hence 6rj/( 2 ) contains no icosahedral groups if n be odd, but, for n
even, contains 2 rt (2 2n 1)/60 icosahedral groups forming a single
system of conjugate groups.
To determine, for p > 2, the distribution of the icosahedral sub-
groups into sets of conjugates within G M ( S ) and within 6r 2 j/(,), consider
first the case (i) and set 2 =Q, so that only the even powers of s
belong to the G-F[p n ]. Then will
transform F 5 into itself, but transforms F 2 into
f <*, (V
Hence the groups 6r 60 are all conjugate under G%M(S) and form at
most two systems of conjugate subgroups under GM( S ). But if there
were a single system, their number would be at most Jf(s)/60,
whereas it is Jf(s)/30. Hence there are two systems each of M(s)/60
conjugate icosahedral groups within GM( S ) ^nd each is self -conjugate
only under itself.
SUBGROUPS OF THE LINEAR FRACTIONAL GROUP XF(2,jpn). 285
For case 1 ) (ii), let JE7 2 = J, so that EE = J(p n +V/*= - 1. Then
transforms F 5 into itself and transforms V 2 into
/ A, BJ e \
\-BJ~* -A'
Taking e 0, 1, . . ., p n , we reach the various p n -f 1 substitutions F 2 .
If e be even, the transformer belongs to the hyperorthogonal group
since J = J~ . For e odd, it may be given the hyperorthogonal
form with determinant a not -square. In fact, there exist in the
~] solutions of X^" 1 **-!, so that X=--X. Then
/#, \ = (E, ON = /XE, \ = /X#, _0_ \
Vo, J0- 1 / ' " \o, ~E) \ o, -XE) ' \ o, x^y
of determinant X 2 . ifewce ^ groups 6r 60 are aZ? conjugate within.
form two systems of conjugates mthin G-M( 8 ).
260. Summary of the subgroups of 6rj/( s ), s=p n :
s H- 1 conjugate commutative groups of order s ;
1 S ~T~ 1
s(sl) conjugate cyclic groups of order "f ? 2; 1 according
as #> 2; ^) = 2;
Y(sl) conjugate cyclic G- d - for every divisor d+ of ^" ;
M(s)/2d+ conjugate dihedron (r 2d -, for d+ odd;
two systems each of M(s)/4*d+ conjugate dihedron G^ d -j for d+ even
and >2;
for _p n = 87& 3, one set of Jf(s)/12 conjugate four -groups;
for p n =8h 1, two sets each of M(s)/24 conjugate four -groups 2 );
...- -
pnp) (pm pmr-1) (2, 1 ; 1) (#* 1)
commutative groups of order jp m , where (2, 1; 1) is read 2, 1,
or 1 according as p > 2 with w/& an even integer, p > 2
with w/A; an odd integer, or p = 2 with n/Jfc an integer, and
where & is a divisor of m depending on the particular Gy;
1) This case may be made to depend on (i) since 5 divides p 2 1. Hence
each 6? 60 is self -conjugate only under itself within the group Crif(f) and so
within its subgroup 6r2J/(). Hence each <r 60 is one of a system of 2M(s)/60
conjugate groups within Gr2M(), so that the icosahedral subgroups all form a
single system of conjugates within 6r2if(*). They fall into two systems in GM().
2) For p 2, the four-groups occur among the groups of order pm= 2*
given later.
286 CHAPTER XII. SUBGROUPS OF THE LINEAR FRACTIONAL etc.
f Aj2 n _ 1") v)n -m
certain sets of / 1W . 77- conjugate G- p m d _, where & and <L_
l*i *; -v IJP '"" *J
depend on m;
(2, 1; 1) sets each of Jf(s)/(2, 1; 1) Jf(^) conjugate G M ( P k ^ k a
divisor of w, each group being isomorphic with the group
of linear fractional substitutions of determinant unity in the
two systems each of M(s)/2M(p*) conjugate GW(p*), i? > 2, nfk an
even integer, each group isomorphic with the linear fractional
group in the 6rF[jp 1 ];
for s = Sh.ly two sets each of M(s)/24t conjugate octahedral 6r 24 ;
for s Sh 1, two sets each of Jf(s)/24 conjugate tetrahedral 6r 12 ;
for s = Sh 3 or s == 2% w even, Jf{s)/12 conjugate tetrahedral 6r 12 j
for s = 10Z 1, two sets each of _M"(s)/60 conjugate icosahedral ^Q. 1 )
261. Theorem. If p n > 3, ^ Zmear fractional group GM^ is
simple.
Indeed, the only cases in which the number of groups in a set
of conjugate subgroups is unity are the following two:
p n = 2, d+ = 3^ M(s)/2d+ = 1, when the 6r 6 has a self - conjugate G- 3 -
jp n =3, -M"(s)/12 = l, when the ^ 12 has a self -conjugate four-group.
262. Theorem. 2 ) The group G-y( S ) always has subgroups of
index s + 1? but has subgroups of lower index only when
s = 2, 3, 5, 7, 3 2 , 11.
Every subgroup of GM() -is contained in one of the following:
G S ( S VJ dihedron 6r, + i (p > 2), G M ( P k ) (n/Jc an odd integer i
(p > 2, w/& an even integer), 6r 12 (s = 8fe 3), 6r 24 (s = 8^ 1),
^o ( s = 10? 1). The first group is always of order greater than
the G-M(p lt ) and ft^p*)*, indeed, since &<#/ 2,
Also s (s - 1) /(2; 1) > s + 1 > s - 1 if s > 3 and s (s -I)/ (2; 1) > 60
if s ]> 11. Hence G s ( s \) of index s -f- 1 has the maximum order
2;l
if s > 11. The same result holds for s = 2 3 since the G M (y) == G 6Q
is then not a subgroup; likewise for s = 2 2 since it is ( 257) then
1) For p = 2 or p = 5 the icosahedral subgroups are of the type
or 6rjf(5) given earlier.
2) For n 1, this is the celebrated theorem stated without proof by Galois
in the letter to his friend Auguste Chevalier written before the fatal duel. For
references to the proofs by Betti, Gierster, etc., see Klein, Math. Ann., vol. 14.
CHAPTER XIII. AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 287
the tetrahedral 6r 12 . For s = 1 1, 3 2 , 7, 5, the subgroups of maximum
order are 6r 60 , 6r 60 , 6r 24 , 6r 12 respectively, the index under G- M(t ) being
11, 6, 7, 5 and hence < s + 1. For s = 2, 3 the G M () is a dihedron 6r 6 ,
a tetrahedron 6r 12 , respectively, and has a subgroup of maximum
order 6r 3 , 6r 4 respectively.
263. A simple group can be represented as a transitive sub-
stitution-group on N letters if, and only if, it contains a complete
system of N conjugate subgroups. 1 ) For s > 3, G M ( t ) is simple ( 261).
Hence G M () can be represented as a transitive group on < s -f 1 letters
only when s = 5, 7, 3 2 , 11. For s = 2, 3 it can be represented as a
transitive group on 3, 4 letters respectively, but on no fewer, being
of order 6r 6 , 6r 12 . If a simple group be represented as an intransitive
substitution -group on D letters, D must equal the sum of the degrees
of two or more transitive representations; for 6rj/< s ) we have always
D > s 4- 1. Hence the linear fractional group GM( S ) may be represented
as a substitution -group on s -f 1 letters but on no fewer number except
when s = 5, 7, 9, 11, for which the minimum number of letters is 5, 7,
6, 11 respectively.
CHAPTER XHL
AUXILIARY THEOREMS ON ABSTRACT GROUPS. ABSTRACT
FORMS OF VARIOUS LINEAR GROUPS. 2 )
264. Theorem. The symmetric substitution- group on k letters
is holoedricallt/ isomorphic with the abstract group G-Qc) generated by
the operators B lf B%, . . ., JB^i with the generational relations
261) Bl = B\ = ... - = _! = 7,
262) BtBj^BjBi ( = 1, 2, . . ., fc-3;j = ^ + 2, i+3, . . .,&-!),
263) SjS j+l B j = S j+l S j B j+l (j = 1, 2, . . ., fc- 2).
The symmetric group 6r*? on the letters Z x , ? 2 , . . ., Ik may be
generated by the transpositions
which satisfy the relations 261), 262), 263) prescribed for the
generators B d of the abstract group G(ti) and conceivably also other
1) For a proof of this theorem due to Dyck see Burnside , The Theory of
Groups, 123.
2) The theorems of 264, 265 are due to Professor Moore, Proceed. Land.
Math. Soc., vol. XXVIII, pp. 357 366. The proofs given in 264, 266 are due
to the author; for that in 264 see Proceed. Lond. Math. Soc., vol. XXXI,
351353; for that in 266 see Math. Ann., vol.. 54, pp. 564 569.
288 CHAPTER
relations not derivable therefrom. The order 0(ti) of 6r(fc) is
therefore > Jc\
Denote by G the subgroup Gr(k 1) generated by J5^ .Z? 2 , . . .,
-Z?_ 2 and consider the following sets of operators 1 ) of
It will be shown that these sets of operators are merely permuted
amongst themselves upon applying as right-hand multipliers the
generators B r (r = 1, . . ., k 1). Since 2?j? = I, we have
ft-Br = GrBki . . . -Z?,-^/- EE 0r.fl.
If i > r H- 1, we find, on applying 262) to move B r to the left of
i . . . BiB f = GB r Bici . . . Bi= GB^i . . . BI= 0,-.
If i < r, we find, on moving B r to the left of B i9 J5,-+i, . . ., ^ r
By 263), we may replace B r B r iB r by B r iB r B r i. We then
move the first B r i to the left of -R-fi, . . ., JR_i and merge it
into 6r and get
O t JB r = GB k i . . . B r +iB r B r iB r 2 . . . Bi : = 0,-.
Hence the right-hand multiplier JB r gives rise to the transposition
(O r O r _)_i) on the k sets 1; . . ., O k . It follows that the product of
any operator of these & sets by an arbitrary operator of GQc) is an
operator belonging to these sets. Taking for the former operator
the identity, we see that these h sets include all the operators of the
group 6r(&). The number of operators in G-Qc) is therefore at most
~k times the number in G(k 1). Hence
0(&)^fc - OQc-1) <;<;!
Combining this result with the earlier one, we have 0(ti) = Jcl
The proof of the holoedric isomorphism of G(K) and G ( i\ is there-
fore complete.
The relations 261), 262), 263) may be combined into the formulae
264) I^Bt^
1) It turns out that these sets form a rectangular table for G(k] with the
operators of G- in the first line.
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 289
265. Theorem. - - The alternating group on k letters is holoedrically
isomorphic with the abstract group G-(k] generated by the operators
E!, E 2 , . . ., E k 2 subject to the generational relations
265) J=^=^H-i=(^^+i) 3 =(^^) 2 ftj=l,'.,*--2;j>+l).
The abstract symmetric group 6r(&) may be generated by S and
266) E d = # d +i-Bi (d = 1, 2, . . ., ft - 2).
From the relations 264) we readily derive 265) together with
267) Bl = I, E d S 1 = S i E^ 1 (d = l, 2, . . ., ft- 2).
Inversely, from 265) and 267), we can easily get relations 264).
Hence S 19 E lf E 2 , . . ., Ek*, subject only to the relations 265) and
267), generate the abstract group 6r(&). Upon extending 6r{&} by
the operator S subject to the relations 267), we obtain a group
whose operators are of the form E or ES lf E being derived from
E ly E%, . . ., E k -z, and hence of order 20 {&}. But the extended
group was shown to be G(k). Hence Gr{k] is a subgroup of
of order --&! It is readily shown to be the abstract alternating
group 6ri . Since the generational relations 264) involve the
T* 1
generators Bi evenly, the various expressions for an operator of 6r(&)
in terms of its generators involve all an even or all an odd number
of the generators, so that its operators may be classed into even and
odd operators. By 266), the operators of the subgroup G(k] are all
even, so that it is a subgroup of G\ . Since its order is &!, it
is identical with the latter.
266. The last theorem may be readily proved by the direct
method of 264. The generational relations 265) are seen to be
satisfied by the substitutions
A - (I 7 } (I I } = S S (d = 1 Tf, 2^
which generate the alternating group on l , Z 2 , . . ., fa* Hence
The theorem being evident if k = 3, we take & ^> 4. Denote by f
the subgroup G-{k 1} generated by E f E%, . . ., E k 3 and consider
the following sets of operators of Cr{Jc}:
3,
The reader may readily verify, as in 264, that E 1 and E r (r > 1),
when applied as right-hand multipliers to the above sets, give rise
DlCKSON, Linear Groups. *9
290 CHAPTER XIII.
to the permutations (EiEkEz) and (_R r jR r+ i) (BiR k ) respectively. The
sets R!, . . ., R k therefore include all the operators of Gr(k}, so that
-! 4- 03=
Combining this result with the earlier one, 0{&}= -^-fc!
267. Theorem. The abstract alternating group G-i may be
Y 51
generated by two operators V and W subject to the generational relations
268)
For & = 5, the relations 265) denning 6ri may be written
5t
269)
The group contains two operators V^E 1 E 2 E 37 W=E S such that
W 2 = I, (VW)*=(EiEtf = L To prove that V 5 = I, we apply 269)
and find that
Inversely, if V, W satisfy 268) and we set 1 )
the relations 269) will follow. We have at once Ej = I, Ej = I,
^Etf = I, (E E,Y = I. Also (E, E^ = I and El = I. In fact
V~WV 2
= V- VWV 2 WV< V 2 WV~ i WV 2 =V 2 WV 2 WV s - VWV> V 2
vwv> v 3 =v
268. Theorem. 2 ) The general linear homogeneous group GLH(4, 2)
is holoedrically isomorphic with the alternating group on 8 letters.
1) The later reductions depend upon the formulae
2) Jordan, Traits' des substitutions, No. 516; Moore, Math. Annalen, vol. 51,
pp. 417 444; Dickson, ibid vol. 54, pp. 564 569.
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc.
291
The following substitutions of GLH(, 2)
"iiin 'o i o n roiii
0001 0010 0101
1100' E * = 0100' E * = 1100
010 l) v l 1 0, V 1
1010^ r o o i o ^ 'oiii
0100 0101 0010
0010' ^ 5= 1000' EG= 0100
010 1 1 loooij 1110
/ \ / \ f
satisfy the relations 265) for k = 8 and therefore generate a sub-
group L which is isomorphic with the alternating group on the
letters 1, 2, . . ., 8. The latter group being simple, the isomorphism
is holoedric. Since the order of JGrLH(, 2) equals SI by 99, it
coincides with its subgroup L. The correspondence of generators of
L=GLH(4:,2) and 6ri is as follows:
Y 8
JEi~(23)(12),
J5
(M) (12),
(67)(12),
(45) (12) ,
(78) (12).
269. To effect the inversion of 270), so that we shall be able
to pass readily from an arbitrary substitution of L to the correspond-
ing substitution of 6ri , we begin with the simple identities,
Y 8!
Since these relations can be solved for E 5 , E, E 2 , E 6 , E ly E 5 in
order, their left members may be chosen as generators of L. By 270),
we have
(U s )l? 24 ~(67)(12), (y 4 )-B 31 ~ (57) (12), (S,&)(8,W-
BuJBi, ~(34)(78), (| 5 yB,,- (23) (78),
From these generators of L, we obtain in succession the substitutions
~
. .
' l 2 4 3y V2 4y -31 Vl 2
) -#12 = -#12 -#21 = (^3 SJ -#12 ' (^3 4
(
= (IkUBa-.&Uft, BuB*,
19*
292 CHAPTER XIII.
These results lead at once to the following correspondences:
(|,y^~(265)(347), (y 2 | 4 | 3 )~(27)(3645), (i 3 | 4 )B 12 ~(24)(17),
(U 3 ):B 12 ~(187)(243), (6A)(t,SO ~ (18)(34), (^)B ~(187)(234),
B 42 ~(16)(25)(34)(78), 32 ~(23)(45)(67)(18), (| 2 y~(18)(27)(35)(46).
By simple transformations, we complete the proof of the
Theorem. The correspondences 270) give reciprocally
(t) ~ (13) (27) (48) (56), fcf,) ~ (16)(27)(34)(58),
(y 4 )~(18)(27)(36)(45),
({&) ~ (18)(27)(35)(46), (| 2 | 4 ) ~ (15)(27)(34)(68),
(1,6.) ~ (14)(27)(38)(56),
5 12 ~ (12)(38)(47)(56), B sl ~ (17)(25)(34)(68),
JB 32 rv (18)(23)(46)(67),
14 ~ (18) (23) (46) (57), 2? 24 ~ (17) (26) (34) (58),
^-(12) (37) (48) (56).
By 100, these relations enable us to pass from an arbitrary sub-
stitution of the linear group on 4 indices modulo 2 to the corresponding
even substitution on 8 letters.
Abstract form of the simple group F0(5, 3) 1 ), 270274
270. By the notation of 194, F0(5, 3) denotes the group 0J(5, 3).
By 189 and 181, it is of order 25920 and is generated by the
substitutions 2 )
QiC h (&&)(&&), WEEETF 1234 (*,j,&, Z = l, ..., 5).
It has a commutative subgroup Z 16 composed of the substitutions
^ ^lQ; ^lQ> ^1^4; ^Mi Q^37 ^2^4; ^2^5? ^3^4? ^3^5? ^4^
QQC 8 C 4 , CiC;C 8 C7 B , QC 8 C 4 C 6 and C 2 C,C&. _ The (fefeKS*60 generate
a subgroup Z 60 of the even linear substitutions on | 1; . . ., | 5 . The
groups L 16 and Z 60 are commutative with each other and have only
the identity in common; hence they generate a subgroup A^ &Q of
FO (5, 3). We readily determine the abstract forms of these sub-
groups. By 265, we have the theorem:
1) Taken from the author's papers, Comptes Rendus, vol. 128, pp. 873 875;
Proceed. Lond. Math. Soc., vol. 32, pp. 3 10. In the earlier paper, Proceed.
Lond. Math. Soc., vol. 31, pp. 30 68, another set of generators was determined
by a more complicated analysis.
2) For pn = 3, Of/ is either the identity, C-C y , (!J y )C, or (M y )0,, the
first two alone being of the form Q r . Here (^-f-) denotes the linear substitu-
tion i' = |., i'- = i r They are to be compounded as linear substitutions; for
example, (ils)(ii^) a *(&si8)- Also C f denotes the substitution changing the
sign of the index |..
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 293
The abstract group 6r 60 generated by E lf E 2) E B subject to the relations
271) El-El-El-I, (E 1 E^^(E,E 3 Y=(E i E^ = I
is put into holoedric isomorphism with L &0 by the correspondences
272) .^-feUe), ^-ay&is), J5 3 ~(i 4 y(y 2 ).
The following theorem is quite evident:
The abstract group 6r 16 generated ~by B if B 2 , B B , B subject to
the relations
273) Bf-I, BtB^BjBi (i, j = 1, 2, 3, 4)
is put into holoedric isomorphism with L i6 by the correspondences
274) B^C^C,, Bz~C,C B , B B ~C 3 C, # 4 ~C 4 C 5 .
If we impose the relations 275) below, the two groups 6r 60 and
6r 16 will be permutable. Writing the analogous relations between
the corresponding orthogonal substitutions 272), 274), we readily see
that they are satisfied. We have therefore the theorem:
The abstract group generated by E lf E 2 , E B , S 19 B%, B B , JB 4 subject
to the generational relations 271), 273), and
275)
1 1 ~T> TTT T~>
E~ B^ 2 = B , E~ B 2 E% = B B^B S , E%
3 BiE B = B i} E B B 2 E B = B B 2 , E B B B E B =
E~ B^E B = B^
is of order 960 and is holoedrically isomorphic with the linear group A$ M .
271. Theorem. The abstract group 6r 960 of 270 may be
generated by the operators E ly E 2 , E B , B 1 subject to the generational
relations
E* = Ej = E = Sf = I, (E.E,) 3 = (E,E,Y = (1W = /,
] (E 1 E 3 Y=(B 1 E,Y=(B 1 E i )^L I 1
These relations follow immediately from 271), 273), 275), with
the exception of (B 1 E i ) s = I > which is derived from the first two of 275):
together with Ef = B = I. Furthermore, we have by 275),
277) BI=E
294 CHAPTER xin.
Inversely, if B 2 , 2? 3 , B be defined by 277), the relations 271),
273), 275) all foUow from 276). Since B^ is of period 3,
B 3 = E*B& - B 3 = E&E& EfB&E, E*
= EfB 1 - E^E^Es - BiE^E*
= E*E 2 B 1 E 1 B i E 2 E i E 2 E* (interchanging^
= E?E 2 El^El E*EtE> - E^E.E.E.B.E^
= E 2 Ej&i B E 2 E*E 2 = E 2 E i B 1 E*E 2 = E E 2 B 2 E
Upon setting B 1 E 2 = E 2 B i , B i E^B 1 E 1 =E 1 B l E^ y we find that
= E 2 E^E.B.E.E.E, = E.E^B.E.E, = E~B,
Since E 2 E L E 2 E* =E 1 E 2 E?E 2 , we get .
= E&E* (E.B^E^E? = B 3 .
Since E E 2 E?E 2 E 3 E^E 2 E* = E 2 E E 2 E* E B EE 2 E?
= E 2 E E 2 E E 3 E 2 E* = E*E 2 E 3 E 2 E? = EfE 3 E 2 E 3
= E E E E E =E E E 2 E E
-L/ 3 J^iJ 2 J^i^J-j^ J^/ 3 J-J 2 J^i^ -t>2 -L^z >
we find by 277) that
B.B^E^E^B, E 3 E 2 E?E 2 E 3 B L E,E 2 EIE 3 E 2
== EE 2 EE 3 E% B l E^B i E 2 E 3 E t E 2 E 3 E E 2
-^1-^3 -^2 -^1 (-^3 -^1-^) -^I 2j ^2 -^3 -^l
= E 3 - E 1 E t E^B L
upon setting E 3 B E 3 = B { = E 2 B E 2 , E^E 2 E E 2 = E t E 2 E^ and
applying also the equation given by taking the reciprocals of the
last substitutions. Using 277) and the last result,
~ l
E 2 BE 2 = E 3 B 3 E 3 =
In order to prove that E~B i E 1 = B 4c , we note that
-^T (E^B 3 E B )E i ^E i (B 3 B^)E i)
or -i
^3 ^1^3 ^1^3 = ^2 ^3 * ^ ^4 -^1
But the left member equals B 2 B 3 B 4 . Indeed, by the earlier results,
E~ B 1 B 2 B 3 E 1 = B 1 B 2 B l - B 2 B 3 = (E?B 1 E 1 } 2 B 3 = B 3 .
ft f T10P
BA BS y ES ^B 2 B 3 E 3 = B i - B^ 2 B 3 J? 4 .
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 295
Finally,
77* T) 77* 77* 77* 77* T? 77* 77* 77* 77* 77* f 77* ~D 77* \ 77* 77* ~D
-&3 ^^S = A &* ^3 -#3 ^3 Aj Al A A I A A A) A -^ 2 = - tf i'
We have now derived from 276) all of the relations 275). It remains
to derive 273). Since B 2 , B 3 , B are conjugate with B by 277),
they are of period 2. By 275), B^B Z is conjugate with S 19 B 2 B 3
with B 3 , B 3 B with B 3 . Hence they are of period 2 and therefore
B B is commutative with B 2 and J5 4 , B with J5 2 . Since E~ 1 B 2 E 2
is its own reciprocal, we have
so that B 1 B 3 = B 3 B i . Since B 2 B 3 B was shown to be the trans-
formed of B 3 by E*E 3 , we have
Hence B 2 is commutative with B. Since B^ is commutative with
jB 3 , E 2 and E B , it is commutative with J3 4 by 277).
272. Theorem. - - Every substitution of FO (5, 3) is given once
and l)ut once by the following 27 sets, in which A denotes the sub-
s== ly ^5 2 = 0, 1, 2
\
2 5 <
Since w is not in A, a substitution of R t belongs to E t if and only
if t = r. If a substitution of R t belong to JR^tr, the product
must belong to A, whereas it replaces | 5 by a linear function of
i> 62 > s> 4? everv coefficient being + 1.
If a substitution of R S i t belong to R a jt> the product
must belong to .A. Supposing first that t -- r =f= 0, we show that S
replaces | 6 by a function involving more than one index and there-
fore does not belong to A. In fact, w~ s S replaces 5 by a function
of the form fc fc fc
/ ^ = 5a fe6 c ?5;
where a, &, c are three of the integers 1, 2, 3, 4. Then iv* replaces f
by /i ? 5 , where is a linear function of | u |j, | 3 , | 4 with coefficients
not all :E (mod 3). Hence /S^ replaces | 5 by /i | 5 , involving two
or more indices. Suppose, however, that t = x. If then i=%=j,
S replaces 5 by a linear function of 17 1 2 , | 3 , | 4 with coefficients + 1.
If i=j, S==w s ~ a , which belongs to A only if s = a. But in the
latter case, the two sets P S it and E jt are themselves identical.
296 CHAPTER XIII.
273. Theorem. The abstract group generated by the operators 1 )
E 19 E 2 , E 3 , .Z?!, W subject to the generational relations 271) and
w*=i, W-^W^B^, w~ l E 2 w=B 1 E 2 ,
W-^W^B.E.B,,
279) TFJ5 4 W= B^E^E, E 2 ,
280) (WE.E.E.W^E^E^E^E^WE^E.W^
B 2 and B being defined by 277), is holoedrically isomorphic with
Writing these relations for the corresponding orthogonal sub-
stitutions as defined by 272), 274) and W<^>w, we obtain relations
which reduce to identities modulo 3. The order Q of is therefore
> 25920. The holoedric isomorphism will be established when it is
shown that Q < 25920. To prove this statement, consider the
following 27 sets 2 ) of operators of 0, those of the first set being
the operators of G = 6r 960 :
/* = 0,1,2\
\s=l,2 J
It is shown in the next section that the generators E ly E 2 , E s , W>
and therefore an arbitrary operator a of the group 0, gives rise to
a mere interchange of the above 27 sets when applied as a right-
hand multipliers. Since the first set G contains the identity 1, the
product la = a lies in one of the 27 sets. Hence contains at
most 27 960 = 25920 operators. In particular, it follows that the
27 sets form a rectangular table for with the operators 6r 960 in
the first row.
We make use of the formulae derived' from 271), 278), 279),
280), 277):
= ^2 ^3 ^2 ^1 2 7 ^3 ^2 ^1^3 = ^1^2 ^3 -
281)
E 2 E i W=WE 2 E 1 .
1) For simplicity S l is retained. It may "be dropped since
#! = W-^E^WE^^ w~ l E z wE^ 1 .
2) They correspond in F0(5, 3) with the 27 rows of the rectangular table.
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 297
274. Theorem. When applied as right-hand multipliers to the
above 27 sets > the generators W, E , E 2 , E B give rise to the respective
permutations :
[TF]: (.Bo-Z^I^) (ftioft/iftva)?
^\ I (ft 10 ft 30 ft 2o) (Us 21 ft 81 E 2 9 4l) (ft 22 ft 5 12 ft 82) ,
I j g 10 S 2o s 30 j*4o s 22 2 6' 12 g 33 S
I (E s E s 40) (ft 10 ft 20) (-RjU -^242) (^221 Asi) (-^112 ^222) ( -^122 ft 4l)
where i = l, 2, 3, 4 and s = l, 2, w;M6 ^Ae first subscript 2s i's
l)e reduced modulo 3.
The form of [TF] is evident. Consider the multiplier E 2 .
= ftu
[by 281)].
=E tZi
[by 281)].
by 279), since
Next, E s2i E l = GW s E B E 2 E l B z E^W
= GW s E 1 E B E 2 W = E,3i, upon applying 281).
E S11 E L =GW*E B E 2 E* S B E 2 E,W= GWE B E 2 E? - E 2 E^W
[by 281)]
The remaining cases follow immediately.
298
CHAPTER XIII.
For the right-hand multiplier E 3 , the calculations are not so
simple.
E E = GWE E E WE = GWE E E W= E [by 280)1.
^221^3 = GW(E 2 E 2 E 3 E 2 E, WE 3 E 2 E, W)
= GWE 3 E 2 E 1 WE 3 E 2 E 1 W.
= GWE 3 E 2 E 1 W'E 2 E 1 W=GWE 3 E 2 E 1 -E 2 E i W 2 = E 132 .
? 1 -E 3 = GW 2 E 3 E 2 E WE 3 B E 2
= G WE 3 E 2 - B^
2 W 2 =GW 2 E 3 W 2 = J!^.
= GW 2 E 3 E 2 E 2 -
75 7^ (~1
= E 132 E 3 E 2 = E 22l E = jR 141 .
7? 77 (rW 2 E E ~R W 2 E E =
JL(/onc) JLL/o \J rf J-JoJJa -*^1 ' ' J ^1 ~^S
7? 7T 7 772 7? 7^2 7?
= -fl232 -C/3 J-J-i == -^2B2 i == 112*
= GWE S E,E, WE& [by 280)]
. W= G W*E S W= R^
[by 281)].
* W 2
W 2 E 3 E 2
G WE S E 2
! B 3 E 2 W= G WE 3 E 2 E E 2 W= E in [by 281)].
G WE 3 W 2 E 5 = G WE 3 E 2
t !32
275. Theorem. The simple group HA(^ 2 2 ) is put into holoedric
isomorphism with the abstract group by the correspondences of
generators
'0110^ [10 'I 2 ^ '00101
1110 00 I 2 ! 2 0011
1 ? E ~ 1000 ' E ^ 1000
v l 1 1 , (llllj (llOO
W
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc.
299
r o 1 / 2
/ 2 I
/Oil
/ / 1 1
ri o i 11
0100
0110
1 1
11001
0100
0001
0010
1 / 2 /
0100
0/10
0/01
ri i r / 2 i
0100
0/1
0/0
1
where / is a root of the irreducible congruence x 2 = x + 1 (mod 2).
Indeed, it may he verified that these correspondences preserve
the generational relations ( 273) prescribed for the generators of 0.
Furthermore, by 132 the order of HA(4, 2 2 ) is 25920, so that the
isomorphism is holoedric.
276. The correspondences established in the last section enable
us to pass readily from any orthogonal substitution S to the correspond-
ing substitution of HA(A, 2 2 ). In fact, we have only to express S
in terms of the simple generators w, (li 2 3 ) ? (tli)'(ti4)> (&A&)(3if&
CiC,, C,C 3 , C 3 <7 4 , C,C 5 of *0(5,3).
It is not difficult to invert these correspondences and obtain the
orthogonal substitutions which correspond to the simplest set of
generators of HA(4:, 2 2 ) ? viz.:
10112
02000
10121
10211
V 2 1 1 lj
> 2,1
'20000^
01112
01121
01211
02111
, 1,1
ro 2 1 1 1
20111
11202
11022
[11220;
Here J denotes the hyperabelian substitution of period 3:
277. By 189, the orthogonal group F0(5, 3) is holoedrically
isomorphic with the Abelian group A(4, 3). Given an arbitrary
Abelian substitution, the process of forming the second compound
and a subsequent transformation of indices ( 189) enables us to find
quite readily the corresponding orthogonal substitution. The inverse
problem is solved by employing the set *) of Abelian substitutions which
correspond to the simplest orthogonal generators w, (li^s)
1) Transact. Amer. Math. Soc., July, 1900, p. 366.
300 CHAPTER XIII.
278. Theorem. 1 ) The special linear homogeneous group SLH(2,p n }
of binary linear substitutions of determinant unity in the G-F[p n ] is
holoedrically isomorphic with the abstract group L generated by the
operators T and Si, where I runs through the series of p n marks of the
field, subject to the generational relations
a ) S = I, SiS /Ll = Si+ /u , (&> P any marks)
b) T* = I, SiT 2 =T*Si,
c) & TSp TSii TS (i/ui) TSiui T=I (^,, ^ any marks, A ^ =1=1).
Since the relations a), b), c) are satisfied by the substitutions
T - ( * ~~ M <? (1> ^\
vi, oJ' = (o, i)
which ( 100, Cor. II) serve to generate SLH(2,p n ), the order I of
the abstract group is at least p n (p 2n 1). We proceed to prove that
I is at most p n (p* n -V). Then will SLH(2,p n ) and L be of equal
order and so holoedrically isomorphic.
Consider the following sets of operators of L
S a TS a TS-i, S a TS a TS t T (a,a,r arbitrary, + 0).
At most p n (p n 1) -\-p 2n (p n 1) = (p n V)p n (p n +V) of them are
distinct. If it be shown that every operator of L occurs in these
sets, it will follow that I <^p n (p 2n 1). The proof consists in
showing that the product of any operator of the sets by T or by
any Si equals an operator of the sets. Since an arbitrary operator X
of L is derived from T t and Si, it will follow that JX = X belongs
to the sets.
In view of a) the reciprocal of Si is Si. For A = 1, ^ =|= l r
c) gives
d)
Applying T as a right-hand multiplier, the product of any
operator of the first set by T gives one of the second set. We
next show that
alT ' T = SoZa
Applying a) and b) the condition for this identity is seen to be
e) TS a TS 2a -iTS a TS 2a -iT* = I.
For p = 2, it reduces to an identity. For p > 2, we have by c)
From this e) follows upon replacing S-iTS-^TS-i by T 3 as
allowed by d).
1) Due to Professor Moore, who gave a different proof.
AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 301
For operators of the second set with a =^= 0, r =j= a ~ S we prove that
S a TS a TS t T. T=S 0l TS ai TS tl T,
where <? 1 , j, ^ are suitably chosen marks, c^ =j= 0. The equivalent
condition T 7
ao-L = J-
may be satisfied by c) by proper choice of r 19 cc lf <5 lf with
a t = ax 1 =)= 0.
We next apply S Q as a right-hand multiplier. SaTS a TS a iS Q
will be of the form 8 ^T8 ttl l8^T^ and consequently belong to the
sets by the previous proof, if we have
SaTSa^-^-qTS^TSa^TSa^aT^ I.
Since ^(a^ + p) = 1 -f- p =(= 1, this condition is of the form c) if
a i> G D r i ^ e suitably chosen. If Q = a/(ar 1), so that at =j= 1,
we have, by c),
o fji ci nn cf rn o cf rno T'O
D a J. > jt O t Jl O = O i_tr J.iJat1-Lb 1
s a+
1 at 1 at 1
For the case J. = 0(T l)=f=0, we prove that
If ar =|= 1, we replace TS a TS t T by its equivalent derived from c)
and find that condition f) becomes
1 rrr ci rr r* ^PC T T
llJL&Al J.iJ(gii)-L OatA*- !-*- = -Lj
at 1 at 1
and hence is satisfied from c). If, however, r = l, so that A=u,
then f) takes the simpler form
f ') ' TS a TS a -iTS Q = Sa-^TSvTSa-iT.
If also Q =|= a, we replace TS a iTS Q by its equivalent derived from c)
and find the condition, where v = a~ 1 ^ 1
This reduces to the identity c) for A = ~~ 2 (>, /u = , whence Af*=j= 1.
In particular, f ') is true if Q = a + x 9 H =)= 0, so that
(TS a TS a -i
The products in the parentheses are identical and so f) is true for
Q = a, if the following condition be true for any particular mark ^ =[= 0,
The latter is of the form f ') for Q = x and hence is true if H =J= a<
But marks x 4= > " exist if jp" > 2. For #" = 2, a = 1, so that f ')
is true for any Q by d).
302 CHAPTER XIII. AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc.
Corollary. The quotient -group LF(2,p n ') is lioloedrically iso-
morphic with the abstract group F generated by the operators T and S%
subject to the relations T 2 = I together with a) and c).
279. For A = or 1 or for ^ = or 1, relations c) always
reduce to d) upon applying a) and b). For the group LF(2,p n \
d) becomes
D) (^T) 3 = J.
If neither A nor ^ is or 1, the product of any two consecutive
subscripts in c) is not unity, the first subscript A being regarded as
consecutive with the last subscript (p l)/(Afi 1). Using any two
consecutive subscripts as the initial A, ft, the resulting identity c) is
seen to be an immediate consequence of the given identity c). Taking
for A any one of the p n 2 marks =)= 0, 1 and for [i any of the p n 3
marks =[= 0, 1, A 1 , the remaining subscripts in c) are different from
and 1. Hence those identities c) which do not reduce to D) are
equivalent in sets of five, an exception being those with all subscripts
equal to A, where A 2 -j-A=l. If the latter has a solutions in the
) n ] ; it follows that there are exactly
distinct identities c) not immediately reducible to D) . For p = 2,
= or 2 according as n is odd or even; for p = 5, <?=!; for
p =j= 2, =(=5, tf = or 2 according as p n =5k2 or p n =5Jc + l.
280. For the group L F(2, 5) of order 60, the ^V= 2 relations c) are
(S 2 T) 5 = 7, S 2 TS TS 5 TS S TS^ 1=1.
These may both be derived from a), D) and T 2 = J, so that LF(2, 5)
is generated by A ~ S 1 , B = T subject to the relations
282) A 5 = I, JS 2 = J, (AB)* = I.
In proof, we apply D) repeatedly and find that
( 2 T) 5 = (& TS-! TS T) 2 S 2 T=S TS^TS^ TS 2 T
= S^TSi S.TS.TS^ S 1 T=S 1 TS 1 - T S,T=I.
Hence also (T$ 3 ) 5 = J, so that the second relation becomes
S 2 TS L (S 9 TY TS, T=Ss TS L TS 2 TS, T = S, (S L T^S, (S, T) 2 = L
281. The group LF(2, 2 2 ) of order 60 may be generated by
A=TSt and B = S^ subject to the relations 282), where i and i 2
are the roots of x 2 + x = 1 (mod 2). Indeed, the N=6 = 2
relations c) to be considered in addition to D) are
CHAPTER XIV. GROUP OF THE EQUATION FOR THE 27 STR. LINES etc. 303
The latter only serves to define the operator 1 in terms of A and B:
The resulting expressions for $,-, S 19 S? are seen to be commutative
and of period 2, so that relations a) follow from 282),
282. The group LF(2, 7) of order 168 is defined by relations
a), D), I 2 = I, together with the following N = 4 relations
T I, $ 3 " TS 3 TS% TS 6
Applying a), D) and T 2 = 1, the second and third relations become
3 T8 B TS, -S-ilSt J5_i 8, T = (S 5 T) 4 = J,
= TS S IS 3 .
The first relation may be written S^TS^TS^, 4 T# 4 T5 4 -S l l =
The fourth relation becomes an identity if we replace S$TS 6 1 by
IS^TS^TS^ as derived from the first relation. Hence the 6r 168 may
be generated by $ A and T subject only to the generational relations 1 )
283) T 2 = J, SI = I, (^T) 8 = I, (S?Ty = I.
Corollary. - The group LF(3, 2) of order 168 25 isomorphic
with LF(2, 7). In fact, the relations 283) are satisfied by the sub-
stitutions
T =
1 .0'
'111 1
010
/ 1
101
.0 1 1.
.1 0,
CHAPTER XIV.
GROUP OF THE EQUATION FOR THE 27 STRAIGHT LINES
ON A GENERAL SURFACE OF THE THIRD ORDER. 2 )
283. A general cubic surface contains 27 straight lines such that 3 )
1. Any one of the lines A meets ten others which intersect
two by two, forming with A five triangles. The total number of
euch triangles on the cubic surface is 5 27/3 = 45.
1) Dyck, Math. Ann., vol. 20, p. 41; Burnside, The Theory of Groups, p. 305.
2) Compare Jordan, Traits', pp. 316 329, 365 369; Dickson, Comptes
Rendus, vol. 128, pp. 873 875.
3) Steiner, Crelle, vol. 53.
304 CHAPTER XIV.
2. Any two triangles ABC and A'B'C' having no side in
common determine uniquely a third triangle A" B" C" such that the
corresponding sides of the three triangles intersect and form three
new triangles A A' A", BB'B", CC'C". The former set of three
triangles is said to constitute a trieder, which will be designated
[ABC,A'B'C',A"B"C"-].
These two properties completely define the configuration of the
45 triangles formed by the 27 lines on the cubic surface.
Denoting the lines by B t , E sit (s = 1, 2; i = 1, 2, 3, 4; t = 0, 1, 2),
it will be shown that the 45 triangles are given by the notation 1 )
B () B i B 9 y B s v)B s iiB s i2 [s = 1, 2]
B t B iit B, it = 0,1,2; * = 1,2,3,4]
R.2<fl.8ilWi I> = 1,2; * = 0,1, 2 (mod 3)]
B slt B sj t -,B 2sjt+1 [s = 1, 2; j = 2, 3, 4; * = 0, 1, 2]
where the subscript 2s is to be replaced by 1 when s = 2.
Each element B lies in exactly five of these sets. Thus B t lies
in the sets B B 1 B. 2) B t Bi it B^it (i = 1, 2, 3, 4); B sit lies in the 5 sets
B a wB s iiB s i2, BtBiitBzit, BguBsjtiBzsjt+i (j = 2, 3, 4);
finally, B sjt lies in the following 5 sets, in the last two of which r
is to be suitably chosen modulo 3:
Hence each element can be associated with exactly ten other elements
to determine a set. Property 1 thus holds for the 45 sets.
The set B Q B i B z lies in exactly the following sixteen trieders:
uBj, -^110^111-^112^ -^210 -^211 ^212 J>
where j = 2, 3 or 4, tf = 0, 1 or 2 (mod 3). Property 2 therefore
holds for the set B Q B^B^ in- conjunction with any set no one of
whose elements is B , B L or B^. It is next shown that the property
holds for an arbitrary pair of sets ABC, A'B'C' which have no
element in common. By the next section the 45 sets are merely
permuted by the substitutions [TF], [-EJ, [E 2 ], [E B ~] given in 274:
The latter generate a substitution -group [0] holoedrically isomorphic
1) The connection with the 27 sets of orthogonal substitutions exhibited
in 272 will be shown in the sequel.
GROUP OF THE EQUATION FOR THE 27 STRAIGHT LINES etc. 305
with the abstract simple group of 273. From its origin [0] is
transitive and hence contains a substitution S which replaces E by
an arbitrary element A. We proceed to prove that [0] contains a
substitution S l which leaves E Q fixed and replaces E L by an arbitrary
one of the ten elements B lf E%, EUQ, HZM (* = 1, 2, 3, 4) which lie
in sets with E Q . The substitutions [J57J, [J? 3 ][J 2 ], [J? 8 ] [.E 2 ] [.EJ,
[^[EgH-EJ 2 replace ^ by E^, E 180 , E 120 , JR 110 respectively, without
altering E . The transformed of [.EJ by [W] gives the substitution
( JR S ll JR 31 ES 2l) (Its 22 -K 32 E 2 s 42J (-R* 20 E 2 s 10 E a 3o)
which replaces E 120 by E 210 , E 110 by E 230 . Then [.EJ and [E 2 ] replace
^230 by ^220 an l -^240 respectively. Finally, [E s ] replaces E 240 by J? 2 .
It follows that [0] contains a substitution 5^$ which replaces the
set E^Ej^E^ by a set ABC in which ^i is any one of the 27 elements
and IB any of the 10 elements which lie in sets with A. Hence [0]
contains a substitution Z replacing the set E E 1 E 2 by an arbitrary
one of the 45 sets. Then Z" 1 replaces the given pair ABC, A'B f C'
by a pair E E 1 E 2f A^B^C^ having no elements in common. The
latter sets determine a trieder by the earlier proof. Applying to it
the substitution Z, which was derived from ["FT] and [Ei] and there-
fore replaces sets by sets, we obtain a trieder containing ABC,
A'B'C' and determined by them. Hence the above distribution of
the 27 elements E into 45 sets is a suitable notation for the con-
figuration of the 45 triangles formed by the 27 lines on a general
cubic surface.
284. The next step is to verify that the substitutions [TT], [J^J,
[.Eg] and [E 3 ] of 274 permute amongst themselves the 45 triangles.
\W"\ gives rise to the following even substitution:
a 20 s 31-*42; a 21 S ^ S 40, 22-*30s4l
l s 2 Q E s 8 2 E s 41, JR21^30^*42, E s 22 It, 31 E 8 49)
where ^ = 1,2,3,4; j = 2,3, 4; s = l,2.
[.E 2 ] gives rise to the even substitution on the 45 triangles:
(^0-^110^210* -^0-^120-^220) C^* 10 It s 11 -B* 12; -^3 20-^511-^2 5 22)
(-^0 -^130 --^230 J -^0 -^140 -^24o) (-^ 10 -R 22 -?^2 21 ; ^ 20 -K 2 5 12 ^2 2l)
(JR 2 Ji 112 JR 212 , E 2 E 1 ^E 2 22) \E S 2Q EssiEs^, E s ioE2 S siE s s2)
l 20 -R* 32 -^ 41 > -K* 10 -R 42 E% s 41)
^g 12-Rs40-??2*4l) (-K*22-^*81-^40> ^2* 12-^25 31-^* So).
Similarly [J^] and [_E 3 ] give rise to even permutations of the
45 triangles.
DlCKSON, Linear Groups. 20
306 CHAPTER XIV. GROUP OF THE EQUATION FOR THE 27 STR. LINES etc.
285. Theorem. The group G of the equation for the 27 lines
on a general cubic surface is of order 51840 and has a subgroup of
index 2 holoedrically isomorphic with the abstract group 0.
The group G is formed of the substitutions on the 27 elements R
which permute the 45 triangles. These substitutions can replace R Q
by at most 27 elements. Those leaving R Q fixed can replace R L by
no element other than the ten lying with jR in some triangle;
namely, R t , R%, RUO, Rzio (i = 1, 2, 3, 4). The substitutions leaving
R and R! fixed and consequently the triangle E Q E^E^ cannot alter
J?2 and must replace JR 130 by one of the 8 elements
B, i9 = 1,2; * = 1,2, 3,4)
which enter the four remaining triangles containing JR . The sub-
stitutions leaving R , R lf JR 130 fixed cannot alter R% or I^ 30 , and
must permute amongst themselves the triangles which contain R 1 and
likewise the triangles which contain -R 130 . Hence they must permute
the pairs JR m , -R 211 ; JR 121 , I2 221 ; ^m? ^2325 -^ui? ^24i5 &&& likewise
permute the pairs R 1Uf R^j ^212? ^231 5 ^122? -^ui"? ^m > ^142- Hence
the elements J? m , -^121? -^231? -^141 common to the two sets must be
permuted amongst themselves, which can be done in at most 24 ways.
Finally, a substitution of 6r which leaves fixed R Q , E ly J^ 7
R m , JR 122 and R i42 and therefore must leave fixed the third element
in each of the triangles J^n^u^o, ^241^232^220; ^221-^122^110?
-^210-^121-^222? -^230-^131-^112? ^"230-^221 -^242? -^131-^142-^120? -"* 22 -ttg 31 -tig 40?
and -Ri 2 i A4o^i32' Such a substitution therefore leaves fixed every
element and is therefore the identity. The order of G is therefore
at most 27 - 10 - 8 24 = 51840.
But G contains the subgroup [0] of order 25920 whose sub-
stitutions permute the 45 triangles evenly. Also G contains
T~-n(R s2t R 8 ^ = 1,2; # = 0,1,2)
which gives rise to the following odd substitution on the triangles:
14 t l
tl> Rs2 tlRsltRsS t+l)
containing 3 + 34-3 + 6 = 15 transpositions. The order of G is
therefore at least 2 25920. The order is consequently 51840.
286. Certain subgroups of the abstract group of order 25920
appear at once by considering the various isomorphic linear groups.
CHAPT.XV. SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS. 307
By 118 and 133, the simple group HA(4, 2 2 ), which is isomorphic
with by 275, has a complete set of 36 conjugate subgroups
-4(4, 2) holoedrically isomorphic with the symmetric group on 6 letters.
By 136, HA(k, 2 2 ) has a complete set of 216 conjugate subgroups
LF(2, 2 2 ), holoedrically isomorphic with the alternating group on
5 letters. By 270274, has a subgroup 960 of index 27.
The quotient -group -4(4, 3) of the special Abelian group SA(4, 3)
is ( 189) holoedrically isomorphic with F0(5, 3) and therefore with
the abstract group 0. By 114, SA(4,S) contains 3 s (3 2 - 1)3
substitutions which leave ^ fixed, so that .4(4, 3) contains a sub-
group of index 25920 -f- 8 3 4 = 40. By 121, SA(, 3) contains
exactly (3 2 +l)3 2 substitutions conjugate with T^I. But the latter
is conjugate with jF 2) _i, the two being identical in the quotient-
group ^.(4, 3). Hence A(4, 3) has a subgroup of index 45. Hence
the simple group has subgroups of indices 27, 36, 40, 45, 216. By
a lengthy analysis 1 ), it has been shown that contains no subgroup
of index < 27. The problem of the determination of the 27 straight
lines on a general cubic surface has therefore resolvent equations of
degrees 27, 36, 40, 45 but none of degree < 27.
Since is isomorphic with A(4; 3), our problem is identical
with the problem of the trisection of the periods of hyperelliptic
functions with four periods. 2 )
CHAPTEE XV.
SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS.
287. In the preceding chapters were derived the following systems
of simple groups, with the specified restrictions upon the prime
number p and the positive integers m and w 8 ):
LF(m,p n ): ^(p nm l)p n ( m -V(p n ( m -V V)p*(-V . . . (p* n l)p n
where p n > 3 if m = 2, and d is the greatest common divisor of m
and p n 1.
H0(m, p 2n )l [p nm ( l)w]p(m-l) j-^(m-l) _ (_ -^ m _rj^tt( m _2)
[p 2n l~\p n
where p n >3 if m = 2, p n > 2 if m = 3, and g is the greatest
common divisor of m and p n -f- 1.
1) Jordan, Traite, pp. 319 329.
2) Jordan, pp. 354 369.
3) The notations were introduced in 108, 119, 148, 194 and end of 209.
308 CHAPTER XV.
A(2m, p n y. (p n & m )l)p n( * m -V(p n(2m - 2} l)p n(2m -v...
where p K > 3 if m 1, p n > 2 if m = 2, and a = 1 if p = 2, a = 2
if # > 2.
-f- 1, _P W ): (p n ( 2m ) 1) p(2wz 1) (^n(2m 2) _ n(2n 8)
where jp > 2 and, for w = 1, # w > 3.
l
where p> 2 and m > 2, while = + 1 according as ^) n = 4Z + 1.
:
where ^9 > 2 and m > 1, = 1 according as p n = 4? 1.
FH(2m, 2): (2 nm 1
(2 2n l)2 2w
where m > 2.
SH(2m, 2"): (2 rem + 1) (2 2 *^- 1 ) - 1) 2 2w ("'- 1 ) . . . (2 2 - 1) 2 2 ", m > 1.
In addition to these systems may be added the cyclic groups of
prime order and the alternating group on n > 4 letters.
288. Between certain of the above groups there exists holoedric
isomorphism, a relation indicated by the symbol ~. For p > 2, the
following isomorphisms were established in 178, 187 190,
197198:
.F0(3, p n ) - LF(2,p n *)i FO(G,p n ) - LF(4,p*), for p n = 4Z + 1;
n ]] S0(6,p n ) - LF(4,p n ), for jp= 4Z + 3;
the latter holding also for p n = 3, a case not treated in 197198.
For any p,
LF(2, p n ) - A(2, p n ) - #0(2, i> 2 ).
For j) = 2, it was shown in 198, 206, 207 that
By chapter XIII,
.F0(5, 3) - #0(4, 2 2 ), iJP(4, 2)
SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS. 309
289. Theorem. 1 ) The simple groups A(2m,p n ) and
F0(2m + l,p n ), p>2,
of equal order are not isomorphic if m > 2.
The proof consists in showing that the orthogonal group contains
a greater number of sets of conjugate operators of period two than
the Abelian group. By 122, A(2m, p n ), p> 2, has exactly
(m + 2) or (m + 1) distinct sets of conjugate operators of period
two according as m is even or odd. But FO(2m-\- l,p n ) contains
the following m distinct substitutions of period two,
C 1 C 1 C 1 C* C C 1 C C* C C 1 C C*
having the respective characteristic determinants,
(1 + JT) 2 (1 - JT) 2 - 1 , (1 + K)* (1 - K)* m ~ s , . . ., (1 + j~) 2wi (1 - J).
By 102, no two of these m substitutions are conjugate under linear
transformation.
For m = 1 or for m = 2, the corresponding groups are iso-
morphic ( 288).
290. The following table gives the 53 known simple groups of
composite order less than one million. The alternating group on
n letters is designated by its order n\ The isomorphisms indicated
a
in 288 are not given in the table.
LF(2, 23)
, . ZF(2,5 2 )
P7QOA Group on 9 letters 2 )
LF(2, 3 3 )
LF(2, 29)
LF(2, 31)
Z^(4,2)~y8!
LF(3, 2 2 )
LF(2, 37)
J.(4, 3) ~ #0(4, 2 2 )
LF(2, 2 5 )
60
LF(2, 5) -
168
LF(2, 7) r
360
LF(2, 3 2 )
504
LF(2, 2 3 )
660
LF(2, 11)
1092
LF(2, 13)
2448
LF(2, 17)
2520
7'
2
3420
LF(2, 19)
4080
LF(2, 2 4 )
5616
L JP(3, 3)
6048
50(3, 3 2 )
in 238.
LF(2, 43)
1) The existence of two non- isomorphic groups of order --8! was noted
2) Cole, Quart. Journ. of Math., vol. 27, p. 48, foot-note.
310 CHAPT.XV. SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS.
51888
58800
62400
74412
95040
102 660
113460
126000
150348
178 920
181440
194472
246480
262080
LF(2, 47)
LF(2, 7 2 )
#0(3, 2 4 )
LF(2, 53)
Group on 12 letters 1 )
LF(2, 59)
LF(2, 61)
#0(3, 5 2 )
LF(2, 67)
I, 71)
,79)
2,2 6 )
265 680
285 852
352440
372000
443 520
456288
515 100
546 312
612468
647460
721392
885 720
976 500
979 200
LF(2, 3 4 )
LF(2, 83)
iJ?(2, 89)
LF(3, 5)
Group on 22 letters 2 )
LF(2, 97)
LF(2, 101)
LF(2, 103)
LF(2, 107)
iJF(2, 109)
LW(2, 113)
LF(2, II 2 )
LF(2, 5 3 )
, 2 2 ) ;
Aside from the simple groups LF(2, p n ), the known simple
groups of composite orders between one million and one billion are
the following:
1 10!
1451520 4(6,2)
1814400
1876896 LF(3, 7)
3265920 #0(4,3 2 )
4680000 4(4,5)
5515776 #0(3,2 6 )
5663616 #0(3, 7 2 )
6065280 JLF(4,3)
9 999 360
10200960
13685760
16482816
19958400
LJF(5, 2)
Group on 23 letters 2 )
42456960
42573600
70915680
138 297 600
174182400
197 406 720
212427600
239500800
244823040
270178272
811273008
987 033 600
,3 2 )
#0(3, 3 4 )
#0(3, II 2 )
4(4, 7)
^#(8, 2)
8H(S, 2)
LF(3, 11)
-i-12!
Group on 24 letters 2 )
.F(3, 13)
#0(3, 13 2 )
,2 2 )
1) Mathieu, Journal de Mathematiques , 1861, p. 270; proof of simplicity
by Miller.
2) Miller, Bull. Soc. Math, de France, vol. 28, p. 266 (1900).
INDEX OF SUBJECTS.
311
INDEX OF SUBJECTS,
(The numbers refer to pages; # or G denotes group.)
Abelian g, 89, 110, 115, 117, 151, 179,
200, 201, 299, 309.
abstract field, 9, 13.
abstract g, 287, 289, 292, 300.
additive -field, 5.
additive -g, 49, 269.
alternating g, 4 letters, 269.
5 letters, 279, 290.
8 letters, 259, 290.
It letters, 289.
basis -system, 49.
Betti-Mathieu g, 64, 67, 69.
canonical form, 221, 237, 244.
characteristic determinant, 80.
equation, 222.
class of quantic, 29.
residue, 3, 6, 7.
commutative g, 262, 265.
substitution, 193, 229.
compound of 0, 145.
configuration 27 lines, 303.
congruent, 3.
conjugate, 52, 100, 236.
cubic surface, 303, 306.
cyclic base, 266.
dihedron g, 265.
doubly -transitive, 248, 261.
exercises, 19, 42, 70, 216.
existence of Galois F, 14, 19.
exponent of mark, 11.
of function, 19.
factors of composition, 81, 91, 94, 191,
192.
Fermat's theorem, 4, 11.
field, 5.
first hypoabelian, 201, 208.
first orthogonal, 131, 159, 191, 292,
299, 309.
four- group, 267.
Galois Field, 6, 14.
general linear homogeneous #, 69, 75,
77, 124, 146, 147, 236, 290.
group, 65; G 168 , 303; 20160 , 259;
25920 , 293, 296; 61840 , 306;
see alternating, icosahedral,
dihedron, tetrahedral, octahedral,
symmetric, linear, general,
special, simple.
Hermite's theorem, 59.
homogeneous, see general, special.
hyperabelian g, 115, 183, 209, 298.
hyperelliptic, 307.
hyperorthogonal g, 131, 264.
hypoabelian, see first, second.
icosahedral, 278, 283, 302.
index of subgroup, 286, 307.
infinity (mark), 260.
invariant, quadratic, 144, 153, 156,
191, 194, 197, 206.
of degree 2, 126, 218.
irreducible, 10, 15, 44.
isomorphic, 99, 164, 174, 183, 194,
208, 209, 287, 298, 308.
linear independence, 10, 52.
linear fractional g, 87, 126, 132, 164,
174, 179, 193, 194, 208, 242,
259, 260, 286, 302, 303.
mark, 9.
modulus, 3, 6.
multiplier Galois F, 51, 270.
Newton's identities, 53.
non- isomorphic, 260, 309.
not -square, 44, 48.
octahedral g, 269, 282.
order of field, 5, 10.
orthogonal, see first, second.
period of mark, 11.
Pfaffian, 147, 172.
primitive root, 13, 36.
irreducible quantic, 21, 35, 44.
quadratic equation, 46.
, see invariant.
rank, 49.
reduced quantic, 63.
312
INDEX OF SUBJECTS. ERRATA.
representation of substitutions, 55.
residue, 3, 6.
self- conjugate, 82, 117, 279.
second hypoabelian, 201, 209.
orthogonal, 159, 191, 194.
simple g, 87, 97, 100, 120, 138, 152,
191, 212, 260, 286, 307, 309.
special linear homogeneous g, 82, 125,
147, 151, 153, 300.
squares, 44, 48.
substitution -quantic, 55, 63.
surface third order, 303.
symmetric g, 6 letters, 99.
k letters, 287.
tetrahedral g, 268, 282.
transformation of indices, 80.
transformed subst., 81, 288.
transitive, 248, 261.
trieder, 304.
Page 14,
11 * i
20,
11 71,
,i 78,
93,
11 95,
i, 102,
113,
132,
139,
152,
11 172,
11 189,
,, 209,
,i 221,
i, 227,
,, 267,
i, 272,
300,
ERRATA.
line 12, read GF[p m ] for GF[p n ].
n mi ., ** Wi
31, read y = x p for y = x p 1.
21, read q. for qi.
2 of 67, read number of squares.
5 of Ex. 6, read JT 1 ^ for IT 1 L
15, read B r 9 ^ for B rjf ^.
6, read yJ 2 , a^ m ; line 2, read Jfj for Mj
30, read o: 21 for a 21 .
17, read T~^_ 1>6 ; line 16, read T n
3, readjj for j.
28 and line 33, read for -
a lm
8, read a
5, read 139) for 139,.
16, p. 175, 1. 14, read ] 2 for 6? 4> ,.
3 of 192, delete comma before "are".
1, for hyperabelian read hypoabelian.
14, for \L] read {j?}-
read Y = y + ifK + & i + ' ' +
line 10, for G 6 _ read G d _.
3 from bottom, delete "an".
16, for S- 1 read S
a _ r
Si**
PT. JUN3
QA Dickson, Leonard Eugene
171 Linear groups
D53
Set
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