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Fixed points of the complements of Frobenius groups of automorphisms

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Abstract

Suppose that a finite group G admits a Frobenius group of automorphisms BA with kernel B and complement A. It is proved that if N is a BA-invariant normal subgroup of G such that (|N|, |B|) = 1 and C N (B) = 1 then C G/N (A) = C G (A)N/N. If N = G is a nilpotent group then we give as a corollary some description of the fixed points C L(G)(A) in the associated Lie ring L(G) in terms of C G (A). In particular, this applies to the case where GB is a Frobenius group as well (so that GBA is a 2-Frobenius group, with not necessarily coprime orders of G and A).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    E. I. Khukhro

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  1. E. I. Khukhro
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Correspondence to E. I. Khukhro.

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Original Russian Text Copyright © 2010 Khukhro E. I.

To Yuriĭ Leonidovich Ershov on the occasion of his 70th birthday.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 694–699, May–June, 2010.

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Khukhro, E.I. Fixed points of the complements of Frobenius groups of automorphisms. Sib Math J 51, 552–556 (2010). https://doi.org/10.1007/s11202-010-0057-9

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  • DOI: https://doi.org/10.1007/s11202-010-0057-9

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