Decomposing locally compact groups into simple pieces

TL;DR

This paper advances the structure theory of locally compact groups by classifying compactly generated groups without infinite discrete quotients, revealing their subgroups and quotient structures, and exploring quasi-product concepts.

Contribution

It introduces a detailed classification of such groups, including the existence of characteristic cocompact subgroups and descriptions of characteristically simple groups.

Findings

Existence of characteristic cocompact subgroups that are either connected or have simple quotients.

Characterization of groups with all proper quotients compact.

Subnormal series for Noetherian groups with specific subquotients.

Abstract

We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product.