Maximal Compact Normal Subgroups and Pro-Lie Groups

Abstract

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.