C-Minimal topological groups

TL;DR

This paper investigates topological groups with all closed subgroups being minimal, establishing conditions under which such groups are compact, especially in the context of Lie groups and solvable groups, and providing counterexamples.

Contribution

It characterizes c-minimal and c-totally minimal groups, proves compactness results for certain classes, and constructs examples of non-compact totally minimal solvable Lie groups.

Findings

Locally compact c-minimal connected groups are compact.

C-minimal locally solvable Lie groups are compact.

Existence of non-compact totally minimal solvable Lie groups.

Abstract

We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group $G$ contains a compact open normal subgroup $N$, then $G$ is c-totally minimal if and only if $G/N$ is hereditarily non-topologizable. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering a question by Dikranjan and Megrelishvili we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact. We also prove that the group $A\times F$ is c-(totally) minimal for every (respectively, totally) minimal abelian group $A$ and every finite group $F.$