A Nunke type classification in the locally compact 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.