A Nunke type classification in the locally compact setting

TL;DR

This paper characterizes lcH-slender groups as torsion-free groups excluding Q and p-adic integers, extending Nunke's classical characterization to the locally compact setting.

Contribution

It provides a necessary and sufficient condition for lcH-slender groups in the locally compact setting, generalizing Nunke's characterization.

Findings

A group G is lcH-slender iff it is torsion-free and excludes Q and Zp for all primes p.

The result extends Nunke's classical characterization to a broader topological context.

The paper establishes an equivalence between algebraic and topological properties of groups in this setting.

Abstract

In this short note we prove that a group G is lcH-slender -- that is, every abstract group homomorphism from a locally compact Hausdorff topological group to G has an open kernel -- if and only if G is torsion-free and does not include Q or the p-adic integers Zp for any prime p. This mirrors a classical characterization given by Nunke for slender abelian groups.