Nilpotency in groups with the minimal condition on centralizers
type: Item
instance of: scholarly article
title: Nilpotency in groups with the minimal condition on centralizers
zbMATH Open document ID: 0932.20027
published in: Journal of Algebra
publication date: 2000-03-09T00:00:00Z
review text: The paper makes a number of substantial contributions to the theory of groups with the minimal condition on centralizers, called \({\mathfrak M}_c\)-groups: these are the groups in which there is no infinite descending (or, equivalently, ascending) chain of subgroups each of which is the centralizer of some subset of the group. Three of the main results are as follows. (1) Suppose that \(G\) is an \({\mathfrak M}_c\)-group which is also a \(p\)-group for some prime \(p\): then \(G\) is locally finite if either \(p=2\) or every pair of elements of \(G\) generates a finite subgroup. The case \(p=2\) answers, positively, a question of John Wilson. (2) Suppose that \(G\) is a periodic \({\mathfrak M}_c\)-group: then the Sylow \(p\)-subgroups of \(G\) are conjugate if either \(p=2\) or every pair of \(p\)-elements of \(G\) generates a finite subgroup. This generalizes a result of the reviewer for locally finite \({\mathfrak M}_c\)-groups [J. Algebra 60, 371-383 (1979; Zbl 0422.20022)]. (3) In any \({\mathfrak M}_c\)-group the bounded left Engel elements form the Fitting subgroup. (An element \(g\) of a group \(G\) is bounded left Engel if there exists a positive integer \(n\) such that \([{x,_n}g]=1\) for all \(x\) in \(G\).) This generalizes a similar result for `substable' groups (see the paper under review for references to this and other related earlier work).
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1340556
zbMATH Keywords: chain conditions
zbMATH Keywords: locally finite groups
zbMATH Keywords: minimal condition on centralizers
zbMATH Keywords: chains of subgroups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: left Engel elements
zbMATH Keywords: Fitting subgroups
author: F. O. Wagner
reviewed by: Roger M. Bryant
MaRDI profile type: Publication
OpenAlex ID: W2004512811
cites work: Q4240512
cites work: Groups with the minimal condition on centralizers
cites work: Periodic locally soluble groups with the minimal condition on centralizers
cites work: Q4381765
cites work: The Engel elements of a soluble group
cites work: Sous-groupes periodiques d'un groupe stable
cites work: Commutator conditions and splitting automorphisms for stable groups
cites work: The Fitting subgroup of a stable group
cites work: Q4355426
DOI: 10.1006/JABR.1998.7824
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description: scientific article; zbMATH DE number 1340556