Groups admitting no non-discrete locally minimal group topologies

Abstract

In this note we show that if G is a countably infinite abelian group such that $nG=0$ for some integer n, then the only locally minimal group topology on G is the discrete one. This answers a question posed by D. Dikranjan and M. Megrelishvili in [7], in particular, a question posed by L. Außenhofer, M. Jesús Chasco, D. Dikranjan and X. Domínguez in [1]. Moreover, we give a necessary and sufficient condition for a bounded abelian group admits a non-discrete locally minimal group topology.

Introduction

A Hausdorff topological group G is called minimal, if G admits no properly coarser Hausdorff group topology, equivalently, G satisfies the open mapping theorem with respect to continuous isomorphisms with domain G. The study of minimal topological groups was inspired by a challenging problem set by G. Choquet at the ICM in Nice 1970; it was quite intensive for almost five decades (see [5], [6], [7], [9]). It was shown that the notion has deep roots in Analysis, Algebra and Number theory.

A Hausdorff topological group $(G,\tau )$ is called locally minimal if there exists a neighbourhood V of the identity of G such that for every Hausdorff group topology $\sigma \le \tau$, if V is a σ-neighbourhood of the identity, then σ coincides with τ. The notion of locally minimal groups was introduced by Morris and Pestov [11] in 1998 (see also T. Banakh [4]). Later, locally minimal groups have been further studied by many authors, for example [1], [2], [8], [13].

In [1], Außenhofer et al. posed the following question:

Question

Does the group ${⨁}_{\omega }\mathbb{Z}(2)$ admit a non-discrete locally minimal group topology? Where $\mathbb{Z}(2)$ denotes the group of integers mod 2.

In [7], Dikranjan and Megrelishvili extended this question to a more general case by replacing 2 with any prime p.

In this note we will show that if G is a countably infinite abelian group such that $nG=0$ for some integer n, then the only locally minimal group topology on G is the discrete one. This answers the question in negative.

Recall an abelian group is called bounded if it is of finite exponent. A torsion group is a group such that every element is of finite order. So it is evident that a bounded abelian group is torsion. For an abelian group G and a positive integer n, we denote by $G[n]$ the subgroup $\{x\in G:nx=0\}$. The multiplicative subgroup $\{z\in \mathbb{C}:|z|=1\}$ of the complex plane $\mathbb{C}$ is called the circle group or torus and denoted by $\mathbb{T}$. We denote by $\mathbb{P}$ the set of all prime numbers. The socle $Soc(G)$ of G is the subgroup ${⨁}_{p\in \mathbb{P}}G[p]$. An abelian group G is called reduced if the maximal divisible subgroup of G is trivial.