Fitting height of a finite group with a Frobenius group of automorphisms.
type: Item
instance of: scholarly article
title: Fitting height of a finite group with a Frobenius group of automorphisms.
zbMATH Open document ID: 1270.20024
published in: Journal of Algebra
publication date: 2013-03-04T00:00:00Z
review text: Let \(G\) be a finite group admitting a Frobenius group of automorphisms \(FH\) (here \(F\) is the kernel, \(H\) is the complement) such that \(C_G(F)=1\). A theorem of \textit{V. V. Belyaev} and \textit{B. Hartley} [Algebra Logic 35, No. 4, 217-228 (1996); translation from Algebra Logika 35, No. 4, 389-410 (1996; Zbl 0870.20025)] ensures that \(G\) is solvable. The author of the paper under review is interested in deriving information on the structure of \(G\) given information on the structure of \(C_G(H)\), which is the subgroup of \(G\) fixed by the complement \(H\). The main Theorem 2.1 states that: a) \(F_i(C_G(H))=F_i(G)\cap C_G(H)\) for every term \(F_i\) of the Fitting series of \(G\), and b) the Fitting heights of \(G\) and \(C_H(G)\) are equal. A similar result holds for the \(\pi\)-lengths of \(G\) and \(C_G(H)\). -- The bulk of the proof is to establish the nontrivial inclusion in part a) of Theorem 2.1 for the most important case \(i=1\). For this, the author uses a large number of results on \(FH\)-modules which have independent interest.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6140809
zbMATH Keywords: finite groups
zbMATH Keywords: Frobenius groups of automorphisms
zbMATH Keywords: fixed points
zbMATH Keywords: Fitting lengths
zbMATH Keywords: Fitting heights
zbMATH Keywords: Fitting series
author: Evgenii I. Khukhro
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.05.011
OpenAlex ID: W2010612776
cites work: On a conjecture of V. D. Mazurov
cites work: Centralizers of finite nilpotent subgroups in locally finite groups
cites work: Q5340151
cites work: Carter subgroups and Fitting heights of finite solvable groups
cites work: On a theorem of Thompson on fixed points of groups acting on p-groups
cites work: Fixed points of \(p\)-automorphisms of finite \(p\)-groups.
cites work: Graded Lie rings with many commuting components and an application to 2-Frobenius groups
cites work: Fixed points of the complements of Frobenius groups of automorphisms.
cites work: Nilpotent length of a finite group admitting a Frobenius group of automorphisms with fixed-point-free kernel.
cites work: Fixed points of Frobenius groups of automorphisms.
cites work: \(p\)-Automorphismen von auflösbaren \(p\)-Gruppen
cites work: Frobenius groups as groups of automorphisms
cites work: Q3941589
cites work: On the exponent of a finite group with an automorphism group of order twelve.
cites work: Positive laws in fixed points of automorphisms of finite groups.
cites work: Automorphisms of solvable groups
cites work: Fixed points of p-groups acting on p-groups
cites work: Fitting height of groups and of fixed points
cites work: Q3084921
DOI: 10.1016/J.JALGEBRA.2012.05.011
description: scientific article; zbMATH DE number 6140809