Exponent of a finite group with a fixed-point-free four-group of automorphisms.
type: Item
instance of: scholarly article
title: Exponent of a finite group with a fixed-point-free four-group of automorphisms.
zbMATH Open document ID: 1275.20021
author: Pavel Shumyatsky
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: Let \(G\) be a finite group acted on by the four-group \(V\simeq C_2\times C_2\) in such a manner that \(C_G(V)=1\) (in particular \(G\) has odd order). An earlier theorem of the author [Monatsh. Math. 168, No. 1, 113-124 (2012; Zbl 1270.20026), Theorem 1.5] states that if the centralizers \(C_G(v_1)\) and \(C_G(v_2)\) of two distinct elements of \(V\) have exponent \(e\), then the exponent of \(G'\) is bounded in terms of \(e\) only. In this paper it is proved that if \(C_G(v)\) has exponent \(e\) for some \(v\in V\), then the exponent of \(G''\) is bounded in terms of \(e\) only.
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20F40
zbMATH DE Number: 6176745
zbMATH Keywords: exponents of finite groups
zbMATH Keywords: fixed-point-free automorphisms
zbMATH Keywords: four-groups of automorphisms
zbMATH Keywords: Lie ring methods
MaRDI profile type: Publication
full work available at URL: s11856-012-0094-0
OpenAlex ID: W1967228015
cites work: Fixed points of coprime operator groups.
cites work: Identities of graded algebras
cites work: The Klein group as an automorphism group without fixed point
cites work: Q3218303
cites work: Q3997907
cites work: Solvability of groups of odd order
cites work: Q5577154
cites work: Derived subgroups of fixed points.
cites work: On the p -Length of p -Soluble Groups and Reduction Theorems for Burnside's Problem
cites work: PERIODIC LOCALLY SOLUBLE GROUPS CONTAINING AN ELEMENT OF PRIME ORDER WITH ČERNIKOV CENTRALIZER
cites work: Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements
cites work: Nilpotent groups and their automorphisms
cites work: Fixed points of Frobenius groups of automorphisms.
cites work: Bounding the exponent of a finite group with automorphisms
cites work: Sur les groupes nilpotents et les anneaux de Lie
cites work: \(p\)-adic analytic groups
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms
cites work: On locally finite groups with a small centralizer of a four-subgroup.
cites work: Groups with regular elementary 2-groups of automorphisms
cites work: Involutory automorphisms of finite groups and their centralizers
cites work: Nilpotency of Some Lie Algebras Associated with p-Groups
cites work: Centralizers of involutory automorphisms of groups of odd order.
cites work: On groups admitting a fixed-point-free four-group of automorphisms
cites work: On the exponent of a finite group with an automorphism group of order twelve.
cites work: FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER
cites work: Q4296491
cites work: SOLUTION OF THE RESTRICTED BURNSIDE PROBLEM FOR GROUPS OF ODD EXPONENT
cites work: Q3362124
DOI: 10.1007/S11856-012-0094-0
Recommended article: Exponent of a finite group with a four-group of automorphisms.
Recommended article: On the exponent of a finite group admitting a fixed-point-free four-group of automorphisms
Recommended article: Q3371028
Recommended article: Groups of exponent 4 as automorphism groups
Recommended article: Finite Groups Admitting a Fixed-Point-Free Automorphism of Order 4
Recommended article: On groups admitting a fixed-point-free four-group of automorphisms
Recommended article: Exponent of a finite group with an involutory automorphism
Recommended article: Involutory automorphisms of finite Abelian groups with exponent 4
Recommended article: Group and lie algebras with a fixed-point-free four-group of automorphisms
Recommended article: Exponent of a finite group admitting a coprime automorphism
description: scientific article; zbMATH DE number 6176745