The whole group is covered by compact translating of subgroups

Question

$G$ is a locally compact (may not necessarily Hausdorff) group, $H$ is a subgroup in $G$, $G/H$ is compact as a quotient space , then there exist a compact subset $K$ such that $G=KH$(or $G=HK$).

Answer 1

It seems to me that what you want follows from the result of the following paper:

http://www.mscand.dk/article/viewFile/12073/10089

The above paper show the existence of an interesting cross section $q$ of the fibration $\pi: G \to G/H$. Namely, a map $q: G/H \to G$ such that $\pi \circ q = id_{G/H}$, with the additional property that $q(C)$ is a relatively compact subset of $G$ for any compact subset $C \subset G/H$. So, if you assume $G/H$ to be compact then by setting $$ K := \overline{q(G/H)}$$ you get a compact subset $K \subset G$ such that $G = K H$ as the OP asked.