A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible

http://dx.doi.org/10.4153/CMB-2011-146-4
Canad. Math. Bull. 56(2013), 213-217

  Published:2011-07-09
 Printed: Mar 2013

It was claimed by Halmos in 1944 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 a claim is false by presenting a family of counterexamples. While other counterexamples are known, 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.