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A cancellation theorem for modules over integral group rings

Part of: Whitehead groups and $K_1$ Low-dimensional topology Representation theory of groups

Published online by Cambridge University Press:  27 November 2020

JOHN NICHOLSON
JOHN NICHOLSON*
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT. e-mail: j.k.nicholson@ucl.ac.uk
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Abstract

A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups G for which the integral group ring ℤG has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤG has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝG. This generalises the Eichler condition in the case of integral group rings.

MSC classification

Primary: 20C05: Group rings of finite groups and their modules
Secondary: 19B28: $K_1$ of group rings and orders 57M60: Group actions in low dimensions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 317 - 327
Copyright
© Cambridge Philosophical Society 2020

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