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ISSN 1088-4165

 

 

$ p$-rational characters and self-normalizing Sylow $ p$-subgroups


Authors: Gabriel Navarro, Pham Huu Tiep and Alexandre Turull
Journal: Represent. Theory 11 (2007), 84-94
MSC (2000): Primary 20C15; Secondary 20C33
Published electronically: April 19, 2007
MathSciNet review: 2306612
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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
Email: gabriel@uv.es

Pham Huu Tiep
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: tiep@math.ufl.edu

Alexandre Turull
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: turull@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S1088-4165-07-00263-4
Keywords: McKay conjecture, $p$-rational characters, self-normalizing Sylow $p$-subgroups
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.