Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-06-08T20:59:29.675Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Bergman, G. M. and Dicks, W. (1975). ‘On universal derivations’, J. Algebra 36, 193–211.CrossRefGoogle Scholar
Berman, S. D. and Rossa, A. R. (1966), ‘Integral group rings of finite and periodic groups’, Algebra and Math. Logic, Izdat Kiev, Univ. Kiev, pp. 44–53.Google Scholar
Curtis, C. W. and Reiner, I. (1962), Representation theory of finite groups and associative algebras (Interscience, New York and London).Google Scholar
Fröhlich, A. (1973), ‘The Picard group of noncommutative rings, in particular of orders’, Trans. Am. Math. Soc. 180, 1–46.CrossRefGoogle Scholar
Hughes, I. and Pearson, K. R. (1972), ‘The group of units of the integral group ring ZS3’ Canad. Math. Bull. 15, 529–534.CrossRefGoogle Scholar
Nagao, H. (1957), ‘On the groups with the same table of characters as symmetric groups’ J. Inst. Polytech. Osaka City Univ. (Ser. A8), 1–8.Google Scholar
Peterson, G. (1976), ‘Automorphisms of the integral group ring of Sn’, Proc. Amer. Math. Soc. 59, 14–18.Google Scholar
Saksonov, A. I. (1966), ‘Certain integer valued rings associated with a finite group’, Dokl. Akad. Nauk SSSR 171, 529–532.Google Scholar
Sandling, R. (1972), ‘Note on the integral group ring problem’, Math. Z. 124, 255–258.CrossRefGoogle Scholar
Sehgal, S. K. (1978), Topics in group rings (Marcel Dekker, Inc., New York and Basel).Google Scholar
Whitcomb, A. (1968), The group ring problem (PhD thesis, University of Chicago).Google Scholar