Stable groups, mostly of finite exponent
type: Item
instance of: scholarly article
title: Stable groups, mostly of finite exponent
zbMATH Open document ID: 0782.03018
published in: Notre Dame Journal of Formal Logic
publication date: 1993-12-06T00:00:00Z
review text: The author studies stable groups and their subgroups (not necessarily definable). Let \(G\) be an \(\mathfrak R\)-group of finite exponent. (The class of \(\mathfrak R\)-groups was introduced by the author in J. Symb. Log. 56, 1026-1037 (1991; Zbl 0743.03028); it is a subclass of the class of stable groups and contains all superstable and small stable groups.) Then \(G\) has a normal nilpotent 2-subgroup \(N\) such that \(G/N\) has finite 2-Sylow subgroups. If the exponent of \(G\) is \(3\cdot 2^ n\) then \(G\) is nilpotent-by-finite. (The latter result was proven by B. Poizat for groups of finite Morley rank; the case of arbitrary stable groups is still open.) It is shown that a locally finite subgroup of finite exponent in a stable groups is nilpotent-by-finite. Among other general results, the author shows that in a stable group any family of pairwise normalizing \(k\)-nilpotent (\(k\)-soluble) subgroups generates a nilpotent (soluble) subgroup. He proves that, under certain circumstances, a stable group interprets an algebraically closed field of characteristic 2.
reviewed by: Oleg Belegradek
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 469214
zbMATH Keywords: nilpotent group
zbMATH Keywords: stable groups
zbMATH Keywords: locally finite subgroup
zbMATH Keywords: finite exponent
author: F. O. Wagner
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093634651
OpenAlex ID: W2036577062
DOI: 10.1305/NDJFL/1093634651
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description: scientific article; zbMATH DE number 469214