Commutator conditions and splitting automorphisms for stable groups
type: Item
instance of: scholarly article
title: Commutator conditions and splitting automorphisms for stable groups
zbMATH Open document ID: 0782.03017
DOI: 10.1007/BF01375553
author: F. O. Wagner
published in: Archive for Mathematical Logic
publication date: 1994-03-10T00:00:00Z
review text: There are various results in group theory of the following type: a group in a certain class is nilpotent provided it satisfies a certain commutator condition. The author proves a series of results of this type for stable groups -- an object that is extensively studied in model theoretic algebra. He shows that a stable soluble group satisfying some left normalized commutator condition \([x_{i_ 0},x_{i_ 1},\dots,x_{i_ n}]= 1\) with \(i_ 0\neq i_ 1\) is nilpotent. A stable group which is generically left (right) Engel is nilpotent. (A stable group is called generically left (right) Engel if its generic elements are left (right) Engel.) A stable soluble group with a generically splitting automorphism of prime order is nilpotent-by-finite. (An automorphism \(\varphi\) of order \(n\) of a stable group \(G\) is called generically splitting if \(xx^ \varphi\cdots x^{\varphi^{n-1}}=1\), for any generic element \(x\) of any type-definable characteristic subgroup of \(G\).) As a corollary, a stable soluble group with a generic element of prime order is nilpotent- by-finite.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F45
zbMATH DE Number: 438618
zbMATH Keywords: nilpotent group
zbMATH Keywords: commutator condition
zbMATH Keywords: stable groups
reviewed by: Oleg Belegradek
MaRDI profile type: Publication
cites work: Q3983708
cites work: Groups with the minimal condition on centralizers
cites work: Q3263741
cites work: À propos d'équations génériques
full work available at URL: https://doi.org/10.1007/bf01375553
OpenAlex ID: W2024785885
description: scientific article; zbMATH DE number 438618