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58 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA

PROBLEMS

1. (a) Let G be a topological group, K its identity component, H a subgroup of G
contained in K. Show that the connected components of the space G/H are the images
of the connected components of G under the canonical mapping TT : G -»-G/H. Show
that K is the smallest of the subgroups L of G such that G/L is totally disconnected.

(b) Let G be the additive subgroup of the Banach space <^_R(N) (7.1.3) consisting of
the mappings n\-*f(ri) such that/(tf) e Q for all n e N and lim /(/?) exists in R. The

«-* 00

distance induced on G by that on ^_R(N) defines a topology compatible with the group
structure of G, and with respect to this topology G is totally disconnected. Let H be the
subgroup of G consisting of all /e G such that lim f(n) = 0. Show that H is closed

«~foo

in G and that G/H is isomorphic to R, and therefore connected.

(c) Let G be a locally compact metrizable group, H a closed subgroup of G, and
TT : G ~> G/H the canonical mapping. Show that the connected components of G/H
are the closures of the images under TT of the connected components of G. (Reduce
to the case where G is totally disconnected, and use Problem 3 of Section 12.9.)

2. In the additive group R, let H be the subgroup Z and let A be the subgroup 0Z, where
6 is an irrational number. Show that the canonical bijection A/(A n H) -> (A + H)/H
is not an isomorphism of topological groups.

3. Let/? be a prime number and let (G_n) be an infinite sequence of topological groups all
equal to the discrete group Z/p²Z. Let H_n be the subgroup pZ/p²Z of G_n. Let G be the
subgroup of the product group fj G_n consisting of all x = (x_n) such that x_n e H_n for all

but a finite set of values of n. Let S3 be the set of neighborhoods of 0 in the product
group fj H_n = H c G. Show that S3 is a fundamental system of neighborhoods of 0

for a topology on G compatible with the group structure, and that in this topology G
is metrizable and locally compact and G/H is discrete. Show that the homomorphism
u: x\-+px of G into G is not a strict rnorphism of G into G, and that u(G) is not closed
inG.

4. Let G be a metrizable group, K a closed normal subgroup of G. If K and G/K are
complete, show that G is complete.

5. (a) Let G be a topological group, K a normal subgroup of G. Show that if K and
G/K have no small subgroups (Section 12.9, Problem 6), then G has no small subgroups,
(b) Deduce from (a) that if HI, H₂ are two normal subgroups of a topological group
G, such that G/H! and G/H₂ have no small subgroups, then G/(Hi n H₂) has no small
subgroups.

6. Let G be a connected, locally compact, metrizable group, K a compact normal sub-
group of G, and N a closed normal subgroup of K, such that K/N has no small sub-
groups (Section 12.9, Problem 6). Show that N is a normal subgroup of G. (Observe
that the hypothesis on K/N implies the existence of a neighborhood U of e in G such
that xNx"¹ «= N for all x e U.)

7. Let G be a connected metrizable group and H a compact normal commutative sub-
group of G, with no small subgroups. Show that H is contained in the center of G.



	58 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA

	PROBLEMS 1. (a) Let G be a topological group, K its identity component, H a subgroup of G contained in K. Show that the connected components of the space G/H are the images of the connected components of G under the canonical mapping TT : G -»-G/H. Show that K is the smallest of the subgroups L of G such that G/L is totally disconnected. (b) Let G be the additive subgroup of the Banach space <^_R(N) (7.1.3) consisting of the mappings n\-f(ri)* such that/(tf) e Q for all n e N and lim /(/?) exists in R. The «-* 00 distance induced on G by that on ^_R(N) defines a topology compatible with the group structure of G, and with respect to this topology G is totally disconnected. Let H be the subgroup of G consisting of all /e G such that lim f(n) = 0. Show that H is closed «~foo in G and that G/H is isomorphic to R, and therefore connected. (c) Let G be a locally compact metrizable group, H a closed subgroup of G, and TT : G ~> G/H the canonical mapping. Show that the connected components of G/H are the closures of the images under TT of the connected components of G. (Reduce to the case where G is totally disconnected, and use Problem 3 of Section 12.9.) 2. In the additive group R, let H be the subgroup Z and let A be the subgroup 0Z, where 6 is an irrational number. Show that the canonical bijection A/(A n H) -> (A + H)/H is not an isomorphism of topological groups. 3. Let/? be a prime number and let (G_n) be an infinite sequence of topological groups all equal to the discrete group Z/p²Z. Let H_n be the subgroup pZ/p²Z of G_n. Let G be the subgroup of the product group fj G_n consisting of all x = (x_n) such that x_n e H_n for all n but a finite set of values of n. Let S3 be the set of neighborhoods of 0 in the product group fj H_n = H c G. Show that S3 is a fundamental system of neighborhoods of 0 n for a topology on G compatible with the group structure, and that in this topology G is metrizable and locally compact and G/H is discrete. Show that the homomorphism u: x\-+px of G into G is not a strict rnorphism of G into G, and that u(G) is not closed inG. 4. Let G be a metrizable group, K a closed normal subgroup of G. If K and G/K are complete, show that G is complete. 5. (a) Let G be a topological group, K a normal subgroup of G. Show that if K and G/K have no small subgroups (Section 12.9, Problem 6), then G has no small subgroups, (b) Deduce from (a) that if HI, H₂ are two normal subgroups of a topological group G, such that G/H! and G/H₂ have no small subgroups, then G/(Hi n H₂) has no small subgroups. 6. Let G be a connected, locally compact, metrizable group, K a compact normal sub- group of G, and N a closed normal subgroup of K, such that K/N has no small sub- groups (Section 12.9, Problem 6). Show that N is a normal subgroup of G. (Observe that the hypothesis on K/N implies the existence of a neighborhood U of e in G such that xNx"¹ «= N for all x e U.) 7. Let G be a connected metrizable group and H a compact normal commutative sub- group of G, with no small subgroups. Show that H is contained in the center of G.