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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9939-2010-10494-X
PII: S 0002-9939(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