Densely locally minimal groups

TL;DR

This paper characterizes densely (locally) minimal abelian groups, extending Prodanov's theorem, and identifies conditions under which such groups are either Lie groups or have open subgroups isomorphic to p-adic integers.

Contribution

It extends Prodanov's theorem to densely minimal groups, providing a classification for certain locally compact abelian groups and exploring new examples beyond previous results.

Findings

Densely locally minimal groups are Lie groups or have open subgroups isomorphic to Z_p.

Infinite densely minimal locally compact groups are isomorphic to Z_p.

Existence of a densely minimal, compact, two-step nilpotent group not fitting previous classifications.

Abstract

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups $\mathbb Z_p$ of $p$-adic integers. In [31], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group $G$ is either compact or connected locally compact, then $G$ is densely locally minimal if and only if $G$ either is a Lie group or has an open subgroup isomorphic to $\mathbb Z_p$ for some prime $p$. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to $\mathbb Z_p$. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to $\mathbb Z_p$.