Stable actions of torsion groups and stable soluble groups
type: Item
instance of: scholarly article
title: Stable actions of torsion groups and stable soluble groups
zbMATH Open document ID: 0813.20037
DOI: 10.1016/0021-8693(92)90165-I
author: Andreas Baudisch
author: John S. Wilson
published in: Journal of Algebra
publication date: 1993-03-10T00:00:00Z
review text: If \(G\) is a stable group (that is, a group whose elementary theory is stable) then, by the Baldwin-Saxl condition, there is a finite upper bound on the lengths of chains of centralizers \(C_ G(X_ 1) < C_ G(X_ 2) < \dots < C_ G(X_ r)\) where each \(X_ i\) is a subset of \(G\). The authors study such groups by means of the following key result. Let \(A\) be an abelian group (written additively) and \(B\) an abelian \(p\)- group (where \(p\) is a prime) which acts on \(A\) in such a way that there is an integer \(n\) such that every chain of centralizers \(C_ B(a_ 1) < C_ B(a_ 2) < \dots < C_ B(a_ r)\) with \(a_ 1,\dots,a_ r \in A\) satisfies \(r \leq n\). Then \(B/C_ B(p^{n-1}A)\) has Prüfer rank at most \(n -1\). This result is applied to prove that if \(G\) is a soluble stable group having a nilpotent normal subgroup \(N_ 1\) such that \(G/N_ 1\) is a torsion group then \(G\) has a nilpotent normal subgroup \(N\) such that \(G/N\) is an abelian-by-finite torsion group of finite Prüfer rank. The case where \(N_ 1 = 1\) follows from a result of the reviewer and \textit{B. Hartley} [J. Algebra 61, 328-334 (1979; Zbl 0427.20028)].
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 03C45
zbMATH DE Number: 130531
zbMATH Keywords: elementary theory
zbMATH Keywords: chains of centralizers
zbMATH Keywords: soluble stable group
zbMATH Keywords: nilpotent normal subgroup
zbMATH Keywords: abelian-by-finite torsion group
zbMATH Keywords: finite Prüfer rank
reviewed by: Roger M. Bryant
MaRDI profile type: Publication
cites work: Periodic locally soluble groups with the minimal condition on centralizers
cites work: Strong finiteness conditions in locally finite groups
cites work: Solvable groups of finite Morley rank
cites work: Q3028990
cites work: Q3926557
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description: scientific article; zbMATH DE number 130531