Lemma 3

Let G be a topological group generated by a compact neighborhood of e. Let H be a closed subgroup of G such that the homogeneous space G/H is compact. Then H is generated by a compact neighborhood of e in H.

Let C be a compact set such that G = CH. Enlarging C if necessary, we can suppose that C generates G and that G = ĈH. Then C^2 is compact and is covered by the Ĉs (s ∈ H), which are open. Therefore there exist s_1, ..., s_n in H such that C^2 ⊂ Ĉs_1 ∪ ... ∪ Ĉs_n. Let Γ be the subgroup of H generated by the s_i. Then C^2 ⊂ CΓ. By induction, it follows that C^n ⊂ CΓ for every n, therefore G = CΓ. Every element of H may be put in the form ab with a ∈ C, b ∈ Γ, whence a ∈ H, whence a ∈ C ∩ H. Therefore H is generated by C ∩ H and the s_i, that is, by a compact set.