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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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$p$-rational characters and self-normalizing Sylow $p$-subgroups
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by Gabriel Navarro, Pham Huu Tiep and Alexandre Turull PDF
Represent. Theory 11 (2007), 84-94 Request permission

Abstract:

Let $G$ be a finite group, $p$ a prime, and $P$ a Sylow $p$-subgroup of $G$. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of $p’$-degree of $G$ and the irreducible characters of $p’$-degree of $\mathbf {N}_G(P)$, which preserves field of values of correspondent characters (over the $p$-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If $p>2$, then $G$ has no non-trivial $p’$-degree $p$-rational irreducible characters if and only if $\mathbf {N}_G(P)=P$.
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Additional Information
  • Gabriel Navarro
  • Affiliation: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, Spain
  • MR Author ID: 129760
  • Email: gabriel@uv.es
  • Pham Huu Tiep
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
  • MR Author ID: 230310
  • Email: tiep@math.ufl.edu
  • Alexandre Turull
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
  • Email: turull@math.ufl.edu
  • Received by editor(s): November 23, 2004
  • Published electronically: April 19, 2007
  • Additional Notes: The first author was partially supported by the Ministerio de Educación y Ciencia
    The second author acknowledges the support of the NSA (grant H98230-04-0066) and the NSF (grant DMS-0600967)
    The third author acknowledges the support of the NSA
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 11 (2007), 84-94
  • MSC (2000): Primary 20C15; Secondary 20C33
  • DOI: https://doi.org/10.1090/S1088-4165-07-00263-4
  • MathSciNet review: 2306612