Let $G$ be a locally compact topological group, that is also Hausdorff and second countable. Let $S$ be a compact subset that generates $G$ as a group, which contains the identity and is closed under taking inverses. Let $T$ be any compact subset of $G$. Is it true that $T \subseteq S^k$ for some $k$?
What if the assumption of second countability is dropped?