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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Self-normalizing Sylow subgroups
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by Robert M. Guralnick, Gunter Malle and Gabriel Navarro PDF
Proc. Amer. Math. Soc. 132 (2004), 973-979 Request permission


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 $p=3$.
  • 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 10.1007/BF02808112
  • 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, DOI 10.1007/978-3-642-64981-3
  • 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
  • Email:
  • Gunter Malle
  • Affiliation: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany
  • MR Author ID: 225462
  • Email:
  • Gabriel Navarro
  • Affiliation: Departament d’Algebra, Facultat de Matemátiques, Universitat de València, 46100 Burjassot, València,  Spain
  • MR Author ID: 129760
  • Email:
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 973-979
  • MSC (2000): Primary 20D20
  • DOI:
  • MathSciNet review: 2045411