Fixed points of the complements of Frobenius groups of automorphisms.
type: Item
instance of: scholarly article
title: Fixed points of the complements of Frobenius groups of automorphisms.
zbMATH Open document ID: 1208.20026
published in: Siberian Mathematical Journal
publication date: 2010-11-15T00:00:00Z
review text: Let \(G\) be a finite group, let \(A\leqslant\Aut(G)\) and let \(N\) be a normal \(A\)-invariant subgroup of \(G\). The author is interested in finding sufficient conditions for the equality \(G_{G/N}(A)=C_G(A)N/N\) to hold. This is shown to happen (Theorem 1) when \(A=KC\) is a Frobenius group with kernel \(K\) and complement \(C\) such that \((|N|,|K|)=1\) and \(C_N(K)=1\). The author mentions that P. Shumyatsky noticed that the coprimeness condition can be dropped. As a direct consequence, it is shown in Theorem 2 that if, moreover, the group \(G\) in Theorem 1 is nilpotent, the fixed point subring of the action induced by \(A\) on the associated Lie ring \(L(G)\) can be nicely described as a direct sum of the fixed point subrings induced by \(A\) on the factors of the lower central series.
reviewed by: Marian Deaconescu
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5816270
zbMATH Keywords: Frobenius groups
zbMATH Keywords: automorphisms
zbMATH Keywords: nilpotent groups
zbMATH Keywords: associated Lie rings
zbMATH Keywords: fixed point subrings
author: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: s11202-010-0057-9
OpenAlex ID: W1975902633
cites work: Fixed points of p-groups acting on p-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: Frobenius groups as groups of automorphisms
cites work: Q3084921
cites work: Q3941589
cites work: Finite groups admitting a fixed-point-free automorphism group
cites work: Q5340151
DOI: 10.1007/S11202-010-0057-9
description: scientific article; zbMATH DE number 5816270