Abstract
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.
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Acknowledgments
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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Bartel, A., Dokchitser, T. Rational representations and permutation representations of finite groups. Math. Ann. 364, 539–558 (2016). https://doi.org/10.1007/s00208-015-1223-y
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DOI: https://doi.org/10.1007/s00208-015-1223-y