## 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

## 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 $p=3$.## References

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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: guralnic@math.usc.edu
**Gunter Malle**- Affiliation: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany
- MR Author ID: 225462
- Email: malle@mathematik.uni-kassel.de
**Gabriel Navarro**- Affiliation: Departament d’Algebra, Facultat de Matemátiques, Universitat de València, 46100 Burjassot, València, Spain
- MR Author ID: 129760
- Email: Gabriel.Navarro@uv.es
- 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: https://doi.org/10.1090/S0002-9939-03-07161-2
- MathSciNet review: 2045411