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

  Published:2011-07-09
 Printed: Mar 2013
  • Daniel V. Tausk,
    Departamento de Matemática, Universidade de São Paulo, Brazil
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Abstract

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.
MSC Classifications: 22B05 show english descriptions General properties and structure of LCA groups 22B05 - General properties and structure of LCA groups
 

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