Exercise 11, § 4

Let $G$ be a topological group and let $H, K$ be two closed subgroups of $G$.

a) Show that the following three conditions are equivalent: (i) $H \times K$ operates properly on $G$ according to the external law $((h, k), s) \to hsk^{-1}$; (ii) $H$ operates properly on the homogeneous space $G/K$; (iii) $K$ operates properly on the homogeneous space $G/H$.

b) If $G$ is locally compact, then the (equivalent) conditions of a) are also equivalent to the following: for every pair of compact subsets $A, B$ of $G$, the intersection $HA \cap BK$ is compact. In particular these conditions are satisfied if one of the subgroups $H, K$ is compact.

c) Suppose that $G$ is locally compact, and show that the following conditions are equivalent: (i) for each $x \in G$ the mapping $h \to h.xK$ of $H$ into $G/K$ is proper; (ii) for each $x \in G$ the mapping $k \to k.xH$ of $K$ into $G/H$ is proper; (iii) for each $x \in G$ and each compact subset $A$ of $G$, $Hx \cap AK$ is compact; (iv) for each $x \in G$ and each compact subset $A$ of $G$, $Kx \cap AH$ is compact. If one of the subgroups $H, K$ is normal or if the left and right uniformities on $G$ coincide (§ 3, Exercise 3), show that these conditions imply those of a).