Exercise 22, § 4

a) Let G be a Hausdorff topological group. Show that every stable subset S of G which is either compact, or open and relatively compact, is a subgroup of G [use Exercise 21 above, and Exercise 28 a) of § 2]. Deduce that a compact commutative group is a radical group [§ 2, Exercise 28 d)].

b) Deduce from a) that every locally compact stable subset of a compact group K is a subgroup of K.

c) Let G be a Hausdorff topological group and let S be a stable closed subset of G with the property that there is a subset K of G which is precompact with respect to the right uniformity of G, and such that SK = G. Show that S is a subgroup of G. (Given $s \in S$, define recursively a sequence of elements $x_i \in K$ and a sequence of elements $s_i \in S$ such that $s^{-1} x_i = s_{i+1} x_{i+1}$; observe that if $i < j$ then $s^{-1} x_i x_j^{-1} \in S$, and that for every neighbourhood V of e in G there is a pair of elements $x_i, x_j$ such that $i < j$ and $x_i x_j^{-1} \in V$).