A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible
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.
22B05 - General properties and structure of LCA groups