Mathematische Annalen

, Volume 364, Issue 1–2, pp 539–558 | Cite as

Rational representations and permutation representations of finite groups



We investigate the question which \(\mathbb {Q}\)-valued characters and characters of \(\mathbb {Q}\)-representations of finite groups are \(\mathbb {Z}\)-linear combinations of permutation characters. This question is known to reduce to that for quasi-elementary groups, and we give a solution in that case. As one of the applications, we exhibit a family of simple groups with rational representations whose smallest multiple that is a permutation representation can be arbitrarily large.



The first author is supported by a Research Fellowship from the Royal Commission for the Exhibition of 1851, and the second author is supported by a Royal Society University Research Fellowship. We would like to thank Alexandre Turull for his help with Corollary 6.6. We are grateful to an anonymous referee for a careful reading of the manuscript and many helpful comments.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Mathematics Institute, University of WarwickCoventryUK
  2. 2.Department of MathematicsUniversity WalkBristolUK

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