Compact and Compactly Generated Subgroups of Locally Compact Groups

Abstract

Our main interest is the existence of maximal compact normal subgroups of locally compact topological groups and its relation to compactly generated subgroups. If a topological group $G$ has a compact normal subgroup $K$ such that $G/K$ is a Lie group and every closed subgroup of $G$ is compactly generated, we call $G$ an $\mathcal {H}(c)$-group. If $G$ has a maximal compact normal subgroup $K$ such that $G/K$ is a Lie group, we call $G$ an $\mathcal {H}$-group. If $G$ is an $\mathcal {H}(c)$-group, then $G$ is a hereditary $\mathcal {H}$-group in the sense that every closed subgroup is an $\mathcal {H}$-group. If $H$ is a closed normal subgroup of $G$ and both $H,G/H$ are $\mathcal {H}(c)$-groups, then $G$ is an $\mathcal {H}(c)$-group. A corollary of this is that a compactly generated solvable group whose characteristic open subgroups are compactly generated is an $\mathcal {H}$-group. If $G$ has a compactly generated closed normal subgroup $F$ such that both $F/{F_0}G/F$ are $\mathcal {H}$-groups, then $G$ is an $\mathcal {H}$-group.