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 | References | Similar Articles | Additional Information

Abstract: We show that if is a double Frobenius group with ``upper'' complement of order such that is nilpotent of class , then is nilpotent of -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 with kernel of prime order and complement of order acts on a Lie ring in such a way that and is nilpotent of class , then is nilpotent of -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:
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