Dense normal subgroups and chief factors in locally compact groups

TL;DR

This paper extends the concept of chief series to locally compact groups, classifies chief factors into seven types, and explores their structure and decomposition, revealing how properties like amenability are preserved.

Contribution

It introduces a comprehensive classification of chief factors in locally compact groups and analyzes their structure and preservation of properties under normal compressions.

Findings

Chief factors exist in all sufficiently rich locally compact groups.

Chief factors are classified into seven types with specific structural decompositions.

Normal compressions preserve properties like amenability and elementary decomposition rank.

Abstract

In 'The essentially chief series of a compactly generated locally compact group', an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors necessarily exist in all locally compact groups with sufficiently rich topological structure. We also show that chief factors have one of seven types, and for all but one of these types, there is a decomposition into discrete groups, compact groups, and topologically simple groups. Our results for chief factors require exploring the theory developed in 'Chief factors in Polish groups' in the setting of locally compact groups. In this context, we obtain tighter restrictions on the factorization of normal compressions and the structure of quasi-products. Consequently, both (non-)amenability and elementary decomposition rank are preserved by normal compressions.