Exercise 25, § 4

Let G be a locally compact group, H a closed subgroup of G. Suppose that G/H is compact and that there exists a compact generator set E of G. Now prove that there exists a compact generator set of H. (One may suppose that E is a compact symmetric neighbourhood of e in G; by observing that G is the union of the E^n for $n > 0$, now prove that there exists an integer m such that G = H.E^m; if one sets E^m = C, now prove that the union of H ∩ C and H ∩ CEC^{-1} is a generator set of H; one will first prove that if K is the subgroup of G generated by H ∩ CEC^{-1}, then KC = G). (Cf. A, I, p. 125, Exercise 14).