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Compact and compactly generated subgroups of locally compact groups


Authors: R. W. Bagley, T. S. Wu and J. S. Yang
Journal: Proc. Amer. Math. Soc. 108 (1990), 1085-1094
MSC: Primary 22D05
DOI: https://doi.org/10.1090/S0002-9939-1990-0993738-2
MathSciNet review: 993738
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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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DOI: https://doi.org/10.1090/S0002-9939-1990-0993738-2
Article copyright: © Copyright 1990 American Mathematical Society

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