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Rational irreducible characters and rational conjugacy classes in finite groups

Authors: Gabriel Navarro and Pham Huu Tiep
Journal: Trans. Amer. Math. Soc. 360 (2008), 2443-2465
MSC (2000): Primary 20C15, 20C33, 20E45
Published electronically: November 27, 2007
MathSciNet review: 2373321
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Abstract: We prove that a finite group $ G$ has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

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Additional Information

Gabriel Navarro
Affiliation: Facultat de Matemàtiques, Universitat de València, Burjassot, València 46100, Spain

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611

Keywords: Rational irreducible character, rational conjugacy class
Received by editor(s): February 6, 2006
Published electronically: November 27, 2007
Additional Notes: The first author was partially supported by the Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01.
Part of this work was done while the first author visited the University of Florida in Gainesville, and he would like to thank the Mathematics Department for its hospitality. Special thanks are due to A. Turull. This paper benefited from conversations with M. Isaacs, A. Moretó, A. Turull and B. Wilkens. The authors are grateful to the referee for pointing out some inaccuracies in an earlier version of the paper as well as for helpful comments that greatly improved the exposition of the paper.
The second author gratefully acknowledges the support of the NSA and the NSF
Dedicated: Dedicated to Professor Michel Broué on the occasion of his sixtieth birthday
Article copyright: © Copyright 2007 American Mathematical Society

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