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On the length of finite groups and of fixed points

Authors: E. I. Khukhro and P. Shumyatsky
Journal: Proc. Amer. Math. Soc. 143 (2015), 3781-3790
MSC (2000): Primary 20D45
Published electronically: March 4, 2015
MathSciNet review: 3359570
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Abstract: The generalized Fitting height of a finite group $ G$ is the least number $ h=h^*(G)$ such that $ F^*_h(G)=G$, where the $ F^*_i(G)$ is the generalized Fitting series: $ F^*_1(G)=F^*(G)$ and $ F^*_{i+1}(G)$ is the inverse image of $ F^*(G/F^*_{i}(G))$. It is proved that if $ G$ admits a soluble group of automorphisms $ A$ of coprime order, then $ h^*(G)$ is bounded in terms of $ h^* (C_G(A))$, where $ C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $ \vert A\vert$ counting multiplicities. The result follows from the special case when $ A=\langle \varphi \rangle $ is of prime order, where it is proved that $ F^*(C_G(\varphi ))\leqslant F^*_{9}(G)$.

The nonsoluble length $ \lambda (G)$ of a finite group $ G$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors is either soluble or is a direct product of nonabelian simple groups. It is proved that if $ A$ is a group of automorphisms of $ G$ of coprime order, then $ \lambda (G)$ is bounded in terms of $ \lambda (C_G(A))$ and the number of prime factors of $ \vert A\vert$ counting multiplicities.

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Additional Information

E. I. Khukhro
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia – and – University of Lincoln, Lincoln, United Kingdom.

P. Shumyatsky
Affiliation: Department of Mathematics, University of Brasilia, DF 70910-900, Brazil

Keywords: Finite groups, nonsoluble length, generalized Fitting height, coprime automorphism group
Received by editor(s): May 2, 2014
Received by editor(s) in revised form: May 28, 2014
Published electronically: March 4, 2015
Additional Notes: This work was supported by CNPq-Brazil. The first author thanks CNPq-Brazil and the University of Brasilia for support and hospitality that he enjoyed during his visits to Brasilia.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society

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