Minimality of the Semidirect Product

TL;DR

This paper characterizes when semidirect products of compact topological groups with subgroups of automorphisms are minimal, providing conditions, examples, and counterexamples that advance understanding of minimal topological groups.

Contribution

It establishes necessary and sufficient conditions for the minimality of semidirect products involving compact groups and automorphism subgroups, including new examples and counterexamples.

Findings

Semidirect product is minimal for closed automorphism subgroups.

For abelian groups, minimality holds for all automorphism subgroups.

Existence of non-minimal semidirect products with certain nilpotent groups.

Abstract

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product $G\leftthreetimes P,$ where $G$ is a compact topological group and $P$ is a topological subgroup of $Aut(G)$. We prove that $G\leftthreetimes P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In case $G$ is abelian, the same is true for every subgroup $P \subseteq Aut(G)$. We show, in contrast, that there exist a compact two-step nilpotent group $G$ and a subgroup $P$ of $Aut(G)$ such that $G\leftthreetimes P$ is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik.