Hall universal group as a direct limit of algebraic groups
type: Item
instance of: scholarly article
title: Hall universal group as a direct limit of algebraic groups
zbMATH Open document ID: 0873.20032
published in: Journal of Algebra
publication date: 1997-11-02T00:00:00Z
review text: The Hall universal group \(U\), discovered by Philip Hall some forty years ago, is the unique countable locally finite group that contains copies of every finite group such that any two copies of the same finite group in \(U\) are conjugate. \(U\) is usually constructed as a direct limit of symmetric groups. The authors discuss ways in which \(U\) is a direct limit of locally finite (possibly finite) simple groups. Let \(F\) be a locally finite (possibly finite) field and \(G_n\) a classical simple group over \(F\) of rank \(n\). The authors prove that any infinite subsequence of \(\{G_n\}\) contains a subsequence whose direct limit is \(U\). (The embeddings \(G_r\to G_s\) here are not the natural embeddings.) The authors also show that \(U\) is not a direct limit of finitary nonlinear simple locally finite groups. In particular \(U\) is not a direct limit of infinite finitary alternating groups.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20E18
zbMATH DE Number: 1026081
zbMATH Keywords: direct limits of locally finite simple groups
zbMATH Keywords: Hall universal group
zbMATH Keywords: countable locally finite groups
zbMATH Keywords: direct limits
zbMATH Keywords: embeddings
author: Mahmut Kuzucuoğlu
author: A. Zalesski
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: jabr.1996.6982
OpenAlex ID: W2005114845
cites work: Some groups generated by transvection subgroups
cites work: Infinitesimal Invariants of Algebraic Groups
cites work: Infinite alternating groups as finitary linear transformation groups
cites work: Some Constructions for Locally Finite Groups
cites work: Universal Locally Finite Central Extensions of Groups
cites work: Lokale Systeme in universellen Gruppen
cites work: Strong finiteness conditions in locally finite groups
cites work: The structure of finitary permutation groups
DOI: 10.1006/JABR.1996.6982
description: scientific article; zbMATH DE number 1026081