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Finite groups and the fixed points of coprime automorphisms


Author: Pavel Shumyatsky
Journal: Proc. Amer. Math. Soc. 129 (2001), 3479-3484
MSC (1991): Primary 20D45
DOI: https://doi.org/10.1090/S0002-9939-01-06125-1
Published electronically: April 25, 2001
MathSciNet review: 1860479
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Abstract: Let $p$ be a prime, and let $G$ be a finite $p'$-group acted on by an elementary abelian $p$-group $A$. The following results are proved:

1. If $\vert A\vert\ge p^3$ and $C_G(a)$ is nilpotent of class at most $c$ for any $a\in A^\char93 $, then the group $G$ is nilpotent of $\{c,p\}$-bounded class.

2. If $\vert A\vert\ge p^4$ and $C_G(a)'$ is nilpotent of class at most $c$ for any $a\in A^\char93 $, then the derived group $G'$is nilpotent of $\{c,p\}$-bounded class.


References [Enhancements On Off] (What's this?)

  • 1. G. Glauberman, On solvable signalizer functors in finite groups, Proc. London Math. Soc. 33 (1976), 1-27. MR 54:5341
  • 2. D. Gorenstein, Finite groups, New York, Evanston, London: Harper and Row, 1968. MR 38:229
  • 3. R. Guralnick and P. Shumyatsky, Derived subgroups of fixed points, preprint, 2000.
  • 4. E. I. Khukhro and P. Shumyatsky, On fixed points of automorphisms of Lie rings and locally finite groups, Algebra and Logic 34 (1995), 395-405. MR 97c:20059
  • 5. V.A. Kreknin, Solvability of Lie algebras with a regular automorphism of finite period, Soviet Math. Dokl. 4 (1963), 683-685.
  • 6. P. Shumyatsky, On periodic soluble groups and the fixed point groups of operators, Comm. Algebra 20(10) (1992), 2815-2820. MR 93i:20038
  • 7. J.N. Ward, On finite groups admitting automorphisms with nilpotent fixed-point group, Bull. Austral. Math. Soc. 5 (1971), 281-282. MR 45:5224
  • 8. J.N. Ward, On finite soluble groups and the fixed-point groups of automorphisms, Bull. Austral. Math. Soc. 5 (1971), 375-378. MR 45:3572

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Additional Information

Pavel Shumyatsky
Affiliation: Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil
Email: pavel@ipe.mat.unb.br

DOI: https://doi.org/10.1090/S0002-9939-01-06125-1
Keywords: Automorphisms, centralizers, associated Lie rings
Received by editor(s): April 26, 2000
Published electronically: April 25, 2001
Additional Notes: The author was supported by CNPq-Brazil
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2001 American Mathematical Society

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