ON THE STRUCTURE OF LOCALLY COMPACT TOPOLOGICAL-GROUPS

We show 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 compactly generated. We obtain several results on the existence of maximal compact normal subgroups of locally compact groups. 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, we show that if G has a closed normal solvable subgroup F such that P(F) = F and P(G/F) = G/F, then P(G) = G. On the other hand, we also show that if G is a compactly generated locally compact solvable group and P(G) = G, then G is compact.