Existentially closed CSA-groups.
type: Item
instance of: scholarly article
title: Existentially closed CSA-groups.
zbMATH Open document ID: 1080.20026
author: Eric Jaligot
author: Abderezak Ould Houcine
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: A CSA-group (conjugately separated Abelian) is a group \(G\) in which every maximal Abelian subgroup \(H\) is malnormal, that is, \(H\cap H^x=1\) for every \(x\in G\setminus H\). Clearly, every Abelian group is a CSA-group; free groups are examples of non-Abelian CSA-groups. It is known that a non-Abelian CSA-group has no involutions. The class of CSA-groups without involutions is inductive; therefore any member of the class embeds into an existentially closed one. The authors show (answering B.~Poizat's question) that the groups existentially closed in the class of CSA-groups without involutions share many properties of the so called bad groups of finite Morley rank (the existence of which is a major open problem). In particular, they are simple and divisible, their maximal Abelian subgroups are conjugate and cover the group. However, they are not even \(\omega\)-stable. This is proven using the following embedding theorem: every countable CSA-group without involutions embeds into a finitely generated one, which has, up to conjugacy, the same maximal Abelian subgroups, except maybe the infinite cyclic subgroups.
reviewed by: Oleg Belegradek
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 2136748
zbMATH Keywords: CSA-groups
zbMATH Keywords: existentially closed groups
zbMATH Keywords: bad groups
zbMATH Keywords: conjugately separated Abelian groups
zbMATH Keywords: malnormal subgroups
zbMATH Keywords: maximal subgroups
zbMATH Keywords: finitely generated groups
zbMATH Keywords: simple groups
zbMATH Keywords: first-order theories
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.06.007
OpenAlex ID: W2067410209
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DOI: 10.1016/J.JALGEBRA.2004.06.007
description: scientific article; zbMATH DE number 2136748