Compact and compactly generated subgroups of locally compact groups

Bagley, R. W.; Wu, T. S.; Yang, J. S.

doi:10.1090/S0002-9939-1990-0993738-2

Compact and compactly generated subgroups of locally compact groups
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by R. W. Bagley, T. S. Wu and J. S. Yang PDF

Proc. Amer. Math. Soc. 108 (1990), 1085-1094 Request permission

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.

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