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On some properties of group rings

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

G. Karpilovsky
G. Karpilovsky
Affiliation:
Department of Mathematics La Trobe UniversityBundoora, Victoria, 3083, Australia
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Abstract

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Let Out (RG) be the set of all outer R-automorphisms of a group ring RG of arbitrary group G over a commutative ring R with 1. It is proved that there is a bijective correspondence between the set Out (RG) and a set consisting of R(G × G)-isomorphism classes of R-free R(G × G)-modules of a certain type. For the case when G is finite and R is the ring of algebraic integers of an algebraic number field the above result implies that there are only finitely many conjugacy classes of group bases in RG. A generalization of a result due to R. Sandling is also provided.

MSC classification

Secondary: 20C07: Group rings of infinite groups and their modules 20C05: Group rings of finite groups and their modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 4 , June 1980 , pp. 385 - 392
Copyright
Copyright © Australian Mathematical Society 1980

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