Radical subgroups of totally disconnected locally compact groups

TL;DR

This paper explores the structure of totally disconnected locally compact groups by establishing a correspondence between subgroups and normal subgroups, leading to new structure theorems and generalizations of existing results.

Contribution

It introduces a novel correspondence between collections of closed subgroups and normal subgroups, and applies it to derive new structure theorems for specific classes of these groups.

Findings

Established a correspondence between closed subgroups and normal subgroups.

Proved structure theorems for groups with open solvable subgroups.

Provided new proofs and generalizations of existing results on topologically simple groups.

Abstract

We observe a correspondence between collections of closed subgroups and normal subgroups in totally disconnected locally compact groups. This correspondence is applied to prove structure theorems for two classes of totally disconnected locally compact second countable groups: the class of such groups with an open solvable subgroup and the class of such groups with a pro-nilpotent compact open subgroup. As a second application, we give new proofs and generalizations of results of G. Willis and Y. Barnea, M. Ershov, and T. Weigel on totally disconnected locally compact groups which are topologically simple and compactly generated,