On the structure of locally compact topological groups.

*(English)*Zbl 0784.22002This paper is a continuation of the work by the authors on the maximal compact subgroups and compactly generated subgroups of locally compact topological groups. It is shown that the normal nilpotent subgroups of certain solvable groups are compactly generated. A solvable group which satisfies a maximal condition on closed normal subgroups is shown to be compactly generated. Several results on the existence of maximal compact normal subgroups of locally compact groups are also obtained. For example, if \(G\) has a uniform solvable subgroup \(H\) which has compactly generated derived subgroups, then \(G\) has maximal compact subgroups and the resulting maximal compact normal subgroup of \(G\) has a Lie factor. If \(P(G)\) denotes the subset of \(G\) of elements which are contained in compact subgroups, and if \(G\) has a closed normal solvable subgroup \(F\) such that \(P(F) = F\) and \(P(G/F) = G/F\), then it is shown that \(P(G) = G\). Finally, the authors also show that if \(G\) is a compactly generated locally compact solvable group and \(P(G) = G\), then \(G\) is compact.

Reviewer: R.W.Bagley (Miami)

##### MSC:

22D05 | General properties and structure of locally compact groups |

20E07 | Subgroup theorems; subgroup growth |

20E28 | Maximal subgroups |