A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups

TL;DR

This paper classifies abelian, topologically characteristically simple, locally compact second-countable groups, revealing their role as monoliths in certain soluble groups and detailing their isomorphism types within the broader structure theory.

Contribution

It provides a classification of abelian chief factors of l.c.s.c. groups, especially those occurring as monoliths in soluble groups of derived length at most 3.

Findings

All such groups occur as monoliths of soluble groups of derived length ≤ 3.

Most can be realized in compactly generated groups, except for certain cases involving nd dual groups.

The classification advances understanding of the structure of locally compact groups.

Abstract

We classify the locally compact second-countable (l.c.s.c.) groups $A$ that are abelian and topologically characteristically simple. All such groups $A$ occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.