A locally compact non divisible abelian group whose character group is torsion free and divisible

TL;DR

This paper disproves a long-standing claim by providing counterexamples of locally compact abelian groups with torsion free and divisible character groups that are not divisible themselves.

Contribution

It constructs new counterexamples showing that a locally compact abelian group can have a torsion free, divisible character group without being divisible, refuting previous assumptions.

Findings

Counterexamples of non-divisible groups with torsion free, divisible character groups

Disproof of Halmos's claim about the divisibility of G

Extension of known counterexamples to stronger conditions

Abstract

It has been claimed by Halmos in [Comment on the real line, Bull. Amer. Math. Soc., 50 (1944), 877-878] that if G is a Hausdorff locally compact topological abelian group and if the character group of G is torsion free then G is divisible. We prove that such claim is false, by presenting a family of counterexamples. While other counterexamples are known (see [D. L. Armacost, The structure of locally compact abelian groups, 1981]), we also present a family of stronger counterexamples, showing that even if one assumes that the character group of G is both torsion free and divisible, it does not follow that G is divisible.