Exercise 1, § 4

a) A subgroup $H$ of a topological group $G$ operates properly on $G$ according to the external law $(s, x) \to sxs^{-1}$ if and only if $G$ is Hausdorff and $H$ is compact (use Propositions 2 and 4).

b) Let $G$ be a topological group and let $H$ be a subgroup of $G$. Then the mapping $(x, y) \to xy$ of $H \times G$ into $G$ is proper if and only if $H$ is quasi-compact (consider the inverse image of $e$ under this mapping).