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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 topologicalabelian group and if the character group of G is torsion free then G isdivisible. We prove that such claim is false, by presenting a family ofcounterexamples. While other counterexamples are known (see [D. L. Armacost,The structure of locally compact abelian groups, 1981]), we also present afamily of stronger counterexamples, showing that even if one assumes that thecharacter group of G is both torsion free and divisible, it does not followthat G is divisible.

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