Fixed points of Frobenius groups of automorphisms.
type: Item
instance of: scholarly article
title: Fixed points of Frobenius groups of automorphisms.
zbMATH Open document ID: 1238.20038
DOI: 10.1134/S1064562411020050
author: N. Yu. Makarenko
author: Evgenii I. Khukhro
author: Pavel Shumyatsky
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: Let \(G\) be a finite group admitting a Frobenius group of automorphisms \(FH\) with kernel \(F\) and complement \(H\) such that \(C_G(F)=1\). The main results described in this paper are: (a) \(|G|=|C_G(H)|^{|H|}\); (b) the rank of \(G\) is bounded by a function of \(|H|\) and the rank of \(C_G(H)\); (c) if \(C_G(H)\) is nilpotent then \(G\) is nilpotent; (d) if \(F\) is cyclic, then the exponent of \(G\) is bounded in terms of the exponent of \(C_G(H)\) and \(|FH|\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5998351
zbMATH Keywords: Frobenius groups of automorphisms
zbMATH Keywords: rank of finite groups
zbMATH Keywords: exponents
zbMATH Keywords: nilpotency classes
zbMATH Keywords: Lie algebras
MaRDI profile type: Publication
cites work: Centralizers of finite nilpotent subgroups in locally finite groups
cites work: Graded Lie rings with many commuting components and an application to 2-Frobenius groups
cites work: Bounding the exponent of a finite group with automorphisms
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: Frobenius groups as groups of automorphisms
cites work: Automorphisms of finite groups of bounded rank
cites work: Q3084921
description: scientific article; zbMATH DE number 5998351