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Inventiones mathematicae

, Volume 170, Issue 1, pp 33–101 | Cite as

A reduction theorem for the McKay conjecture

  • I.M. Isaacs
  • Gunter Malle
  • Gabriel Navarro
Article

Abstract

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 e or q=3 e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J 1, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.

Keywords

Simple Group Maximal Torus Irreducible Character Linear Character Semisimple Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  3. 3.Departament d’Àlgebra, Facultat de MatemàtiquesUniversitat de ValènciaBurjassotSpain

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