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Self-normalizing Sylow subgroups


Authors: Robert M. Guralnick, Gunter Malle and Gabriel Navarro
Journal: Proc. Amer. Math. Soc. 132 (2004), 973-979
MSC (2000): Primary 20D20
DOI: https://doi.org/10.1090/S0002-9939-03-07161-2
Published electronically: August 7, 2003
MathSciNet review: 2045411
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Abstract: Using the classification of finite simple groups we prove the following statement: Let $p>3$ be a prime, $Q$ a group of automorphisms of $p$-power order of a finite group $G$, and $P$ a $Q$-invariant Sylow $p$-subgroup of $G$. If $\mathbf{C}_{\mathbf{N}_G(P)/P}(Q)$ is trivial, then $G$ is solvable. An equivalent formulation is that if $G$ has a self-normalizing Sylow $p$-subgroup with $p >3$ a prime, then $G$ is solvable. We also investigate the possibilities when $p=3$.


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Additional Information

Robert M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
Email: guralnic@math.usc.edu

Gunter Malle
Affiliation: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany
Email: malle@mathematik.uni-kassel.de

Gabriel Navarro
Affiliation: Departament d’Algebra, Facultat de Matemátiques, Universitat de València, 46100 Burjassot, València, Spain
Email: Gabriel.Navarro@uv.es

DOI: https://doi.org/10.1090/S0002-9939-03-07161-2
Received by editor(s): June 6, 2002
Received by editor(s) in revised form: November 29, 2002
Published electronically: August 7, 2003
Additional Notes: The first author was partially supported by NSF Grant DMS 0140578. He would like to thank George Glauberman for some helpful conversations
The third author was supported by the Ministerio de Ciencia y Tecnologia Grant BFM 2001-1667-C03-02
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society