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Finite $ p$-groups with a Frobenius group of automorphisms whose kernel is a cyclic $ p$-group


Authors: E. I. Khukhro and N. Yu. Makarenko
Journal: Proc. Amer. Math. Soc. 143 (2015), 1837-1848
MSC (2010): Primary 20D45; Secondary 17B40, 17B70, 20D15
DOI: https://doi.org/10.1090/S0002-9939-2015-12287-3
Published electronically: January 22, 2015
MathSciNet review: 3314095
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Abstract: Suppose that a finite $ p$-group $ P$ admits a Frobenius group of automorphisms $ FH$ with kernel $ F$ that is a cyclic $ p$-group and with complement $ H$. It is proved that if the fixed-point subgroup $ C_P(H)$ of the complement is nilpotent of class $ c$, then $ P$ has a characteristic subgroup of index bounded in terms of $ c$, $ \vert C_P(F)\vert$, and $ \vert F\vert$ whose nilpotency class is bounded in terms of $ c$ and $ \vert H\vert$ only. Examples show that the condition of $ F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $ FH$. It is also proved that $ P$ has a characteristic subgroup of $ (\vert C_P(F)\vert, \vert F\vert)$-bounded index whose order and rank are bounded in terms of $ \vert H\vert$ and the order and rank of $ C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $ C_P(H)$.


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Additional Information

E. I. Khukhro
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
Email: khukhro@yahoo.co.uk

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

DOI: https://doi.org/10.1090/S0002-9939-2015-12287-3
Keywords: Finite $p$-group, Frobenius group, automorphism, nilpotency class, Lie ring
Received by editor(s): February 14, 2013
Received by editor(s) in revised form: May 29, 2013
Published electronically: January 22, 2015
Additional Notes: The first author was supported by the Russian Science Foundation, project no. 14-21-00065
The second author was supported in part by the Russian Foundation for Basic Research, project no. 13-01-00505
Dedicated: Dedicated to Victor Danilovich Mazurov on the occasion of his 70th birthday
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society