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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
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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  • Michael D. Fried, Robert Guralnick, and Jan Saxl, Schur covers and Carlitz’s conjecture, Israel J. Math. 82 (1993), no. 1-3, 157–225. MR 1239049, DOI
  • J. Humphreys, Linear Algebraic Groups, 2nd ed., Springer-Verlag, Berlin, New York, 1981.
  • B. Huppert, Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
  • Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
  • F. Menegazzo and M. C. Tamburini, A property of Sylow $p$-normalizers in simple groups, Quaderni del Seminario Matematico di Brescia, n. 45/02 (2002).
  • T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
  • Yan Ming Wang and Zhong Mu Chen, Solubility of finite groups admitting a coprime order operator group, Boll. Un. Mat. Ital. A (7) 7 (1993), no. 3, 325–331 (English, with Italian summary). MR 1249108

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Additional Information

Robert M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113
MR Author ID: 78455

Gunter Malle
Affiliation: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany
MR Author ID: 225462

Gabriel Navarro
Affiliation: Departament d’Algebra, Facultat de Matemátiques, Universitat de València, 46100 Burjassot, València,  Spain
MR Author ID: 129760

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