Infinite alternating groups as finitary linear transformation groups
type: Item
instance of: scholarly article
title: Infinite alternating groups as finitary linear transformation groups
zbMATH Open document ID: 0665.20017
DOI: 10.1016/0021-8693(88)90064-6
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let V be a vector space over a field k. A finitary linear transformation of V is an invertible linear map g: \(V\to V\) such that V(g-1) is finite- dimensional, or equivalently, such that g acts as the identity on a subspace of finite codimension. The set of all such g is a group FGL(V) under composition of maps, the finitary general linear group on V. This interesting paper launches a program to find all infinite simple locally finite groups that can be embedded in some FGL(V). (Note that an easy extension of Schur's well known result proves that every periodic subgroup of FGL(V) is locally finite.) The case when V is finite dimensional was dealt with by a number of authors (see the paper for details), using the classification of finite simple groups. A natural example requiring an infinite dimensional space is the alternating group Alt(\(\Omega)\) on an infinite set \(\Omega\), acting on the space \(k\Omega\). The author's main theorem is that in characteristic 0, this is the only possibility: Let G be an infinite, simple, locally finite subgroup of FGL(V), where V is a vector space over a field of characteristic zero. Then V is isomorphic to an alternating group. It was suggested by \textit{A. Zalesskij} that such a result might be true [Dokl. Akad. Nauk BSSR 13, 485-488 (1969; Zbl 0223.20046)], though the author was evidently unaware of this at the time the paper was written. This beautiful result also depends on the classification of finite simple groups, as well as a substantial amount of information about the structure and properties of these groups. The possible finitary representations of Alt(\(\Omega)\) are also considered. Let V(k\(\Omega)\) denote the ``sum zero'' submodule of \(k\Omega\). Theorem 8.1 asserts that if V is a vector space over a field of characteristic zero, \(G\leq FGL(V)\) and \(G\cong Alt(\Omega)\) for an infinite set \(\Omega\), then [V,G] is a direct sum of a finite number of copies of V(k\(\Omega)\). Other interesting facts about Alt…
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 4087913
zbMATH Keywords: finitary linear transformation
zbMATH Keywords: finitary general linear group
zbMATH Keywords: infinite simple locally finite groups
zbMATH Keywords: periodic subgroup
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: infinite, simple, locally finite subgroup of FGL(V)
zbMATH Keywords: alternating group
zbMATH Keywords: finitary representations
zbMATH Keywords: infinite simple groups
zbMATH Keywords: sectional cover
zbMATH Keywords: faithful projective representation
author: J. I. Hall
reviewed by: Brian Hartley
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90064-6
OpenAlex ID: W2017092545
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description: scientific article; zbMATH DE number 4087913