Finite soluble groups with metabelian centralizers.
type: Item
instance of: scholarly article
title: Finite soluble groups with metabelian centralizers.
zbMATH Open document ID: 1308.20014
author: Carlo Casolo
author: Enrico Jabara
published in: Journal of Algebra
publication date: 2014-11-17T00:00:00Z
review text: Let \(G\) be a finite solvable group. The authors analyze what can be said about \(G\) when we know that the centralizer in \(G\) of every non-trivial element of \(G\) satisfies some given property. More specifically, they define \(G\) to be in \(CA_d\) if the derived length of the centralizer of every non-identity element of \(G\) is at most \(d\). In general, they prove that any group in \(CA_d\) has derived length at most \(d(3d+2)\) and Fitting height at most \(3d+2\). When \(d=2\) then the derived length of \(G\) is at most \(4\). When the centralizer of every non-identity element of \(G\) is such that its derived subgroup is nilpotent then \(G^{(3)}\) is nilpotent.
reviewed by: Alexandre Turull
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6370255
zbMATH Keywords: finite soluble groups
zbMATH Keywords: centralizers
zbMATH Keywords: derived lengths
zbMATH Keywords: Fitting heights
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.09.032
OpenAlex ID: W1969883486
cites work: On the Fitting height of factorised soluble groups.
cites work: Solvable groups contain large centralizers
cites work: The theory of finite groups. An introduction.
cites work: Q4273959
cites work: Fitting height of groups and of fixed points
DOI: 10.1016/J.JALGEBRA.2014.09.032
description: scientific article; zbMATH DE number 6370255