\(RC_G(R)\)-signalizers in finite groups.
type: Item
instance of: scholarly article
title: \(RC_G(R)\)-signalizers in finite groups.
zbMATH Open document ID: 1205.20020
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: The author considers, for a prime \(r\), a finite \(r'\)-group \(G\) on which an \(r\)-group \(R\) acts as a group of automorphisms. The \(RC_G(R)\)-invariant subgroups of \(G\), called \(RC_G(R)\)-signalizers are studied and proved to have a serious influence on the structure of \(G\). This is a nicely written and rather concentrated paper which establishes a series of results (most of them too technical to present here) which are analogues, in this co-prime action setting, of older results related to \(p\)-local structure. In this reviewer's opinion, Theorem G is a most elegant result about the behavior of the generalized Fitting subgroup \(F^*(G)\): ``Suppose the noncyclic Abelian \(r\)-group \(R\) acts on the \(r'\)-group \(G\), let \(p>3\) be a prime and suppose that \([O^p(F^*(C_G(\alpha))),R]=1\) for all nontrivial elements \(\alpha\in R\). Then \([O^p(F^*(G)),R]=1\).'' The above leads to information on the structure of co-prime automorphism groups acting on simple non-Abelian groups (given that certain technical constraints are satisfied).
reviewed by: Marian Deaconescu
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 5696932
zbMATH Keywords: co-prime actions
zbMATH Keywords: signalizers
zbMATH Keywords: nilpotent actions
zbMATH Keywords: components
author: Paul Flavell
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.01.028
OpenAlex ID: W2104521023
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DOI: 10.1016/J.JALGEBRA.2010.01.028
description: scientific article; zbMATH DE number 5696932