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Frobenius groups as groups of automorphisms


Authors: N. Yu. Makarenko and Pavel Shumyatsky
Journal: Proc. Amer. Math. Soc. 138 (2010), 3425-3436
MSC (2010): Primary 20D45, 17B70
DOI: https://doi.org/10.1090/S0002-9939-2010-10494-X
Published electronically: May 20, 2010
MathSciNet review: 2661543
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Abstract: We show that if $ GFH$ is a double Frobenius group with ``upper'' complement $ H$ of order $ q$ such that $ C_G(H)$ is nilpotent of class $ c$, then $ G$ is nilpotent of $ (c,q)$-bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook. The proof is based on an analogous result on Lie rings: if a finite Frobenius group $ FH$ with kernel $ F$ of prime order and complement $ H$ of order $ q$ acts on a Lie ring $ K$ in such a way that $ C_K(F)=0$ and $ C_K(H)$ is nilpotent of class $ c$, then $ K$ is nilpotent of $ (c,q)$-bounded class.


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  • 1. Chong-Yun Chao, Some characterizations of nilpotent Lie algebras, Math. Z. 103 (1968), 40-42. MR 0223415 (36:6463)
  • 2. D. Gorenstein, Finite groups, Harper and Row, New York, 1968. MR 0231903 (38:229)
  • 3. P. Hall, Some sufficient conditions for a group to be nilpotent. Ill. J. Math. 2 (1958), 787-801. MR 0105441 (21:4183)
  • 4. G. Higman, Groups and rings which have automorphisms without non-trivial fixed elements. J. London Math. Soc. (2) 32 (1957), 321-334. MR 0089204 (19:633c)
  • 5. E. I. Khukhro, Fixed points of $ p$-automorphisms of finite $ p$-groups. Algebra i Logika 14 (1975), 697-703; English transl., Algebra and Logic 14 (1976), 417-420. MR 0427472 (55:504)
  • 6. E. I. Khukhro, Nilpotent groups and their automorphisms, Walter de Gruyter, Berlin, New York, 1993. MR 1224233 (94g:20046)
  • 7. E. I. Khukhro, Graded Lie rings with many commuting components and an application to $ 2$-Frobenius groups, Bull. London Math. Soc. 40 (2008), 907-912. MR 2439656 (2009m:17029)
  • 8. E. I. Khukhro and V. D. Mazurov (Eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, no. 17, Institute of Mathematics, Novosibirsk, 2010.
  • 9. V. A. Kreknin, The solvability of Lie algebras with regular automorphisms of finite period, Math. USSR Doklady 4 (1963), 683-685. MR 0157990 (28:1218)
  • 10. V. A. Kreknin and A. I. Kostrikin, Lie algebras with regular automorphisms, Math. USSR Doklady 4 (1963), 355-358. MR 0146230 (26:3752)
  • 11. V. D. Mazurov, Recognition of the finite simple groups $ S_4(q)$ by their element orders, Algebra and Logic 41 (2002), 93-110. MR 1922988 (2003e:20018)
  • 12. A. Shalev, Automorphisms of finite groups of bounded rank, Israel J. Math. 82 (1993), 395-404. MR 1239058 (95b:20038)
  • 13. J. G. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. USA 45 (1959), 578-581. MR 0104731 (21:3484)

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Additional Information

N. Yu. Makarenko
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
Address at time of publication: Laboratoire de Mathématiques, Informatique et Application, Université de Haute Alsace, Mulhouse, 68093, France
Email: natalia_makarenko@yahoo.fr

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

DOI: https://doi.org/10.1090/S0002-9939-2010-10494-X
Keywords: Automorphisms, centralizers, associated Lie rings
Received by editor(s): November 13, 2009
Published electronically: May 20, 2010
Additional Notes: The first author was supported in part by the Programme of Support of Leading Scientific Schools of the Russian Federation.
The second author was supported by CNPq-Brazil.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society

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