On the torsion of group extensions and group actions

Abstract

In this note, we study the torsion of extensions of finitely generated abelian by elementary abelian groups. When the action is trivial ( mod p) , we make a specific choice of a 1-cochain for a vanishing multiple of the cohomology class defining the extension and use it to completely describe the torsion of central extensions. As an application, one gets that, under the assumption of trivial action on homology, Z p r may act freely on ( S 1) k if and only if r⩽ k, providing an alternative proof of the main theorem in [Trans. Amer. Math. Soc. 352 (6) (2000) 2689–2700] for central extensions.