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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On isomorphisms between centers of integral group rings of finite groups


Author: Martin Hertweck
Journal: Proc. Amer. Math. Soc. 136 (2008), 1539-1547
MSC (2000): Primary 20C05, 20C15; Secondary 13F99
Published electronically: January 8, 2008
MathSciNet review: 2373581
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Abstract: For finite nilpotent groups $ G$ and $ G^{\prime}$, and a $ G$-adapted ring $ S$ (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings $ SG$ and $ SG^{\prime}$ is monomial, i.e., maps class sums in $ SG$ to class sums in $ SG^{\prime}$ up to multiplication with roots of unity. As a consequence, $ G$ and $ G^{\prime}$ have identical character tables if and only if the centers of their integral group rings $ \mathbb{Z} G$ and $ \mathbb{Z} G^{\prime}$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.


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Additional Information

Martin Hertweck
Affiliation: Universität Stuttgart, Fachbereich Mathematik, IGT, 70569 Stuttgart, Germany
Email: hertweck@mathematik.uni-stuttgart.de

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09252-6
PII: S 0002-9939(08)09252-6
Keywords: $p$-group, integral group ring, class sum correspondence
Received by editor(s): December 7, 2006
Published electronically: January 8, 2008
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.