Frobenius groups as groups of automorphisms
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- by N. Yu. Makarenko and Pavel Shumyatsky PDF
- Proc. Amer. Math. Soc. 138 (2010), 3425-3436 Request permission
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.References
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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
- MR Author ID: 250501
- Email: pavel@mat.unb.br
- 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
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2661543