A group theoretical characterization of simple, locally finite, finitary linear groups
type: Item
instance of: scholarly article
title: A group theoretical characterization of simple, locally finite, finitary linear groups
zbMATH Open document ID: 0791.20039
DOI: 10.1007/BF01199095
published in: Archiv der Mathematik
publication date: 1994-07-13T00:00:00Z
review text: As the title implies, the aim of the paper is to obtain a straightforward group theoretical property that singles out the finitary linear groups from among the simple locally finite groups. A finitary linear group is by definition a group that can operate faithfully on a vector space in such a way that each element operates trivially on a subspace of finite codimension. The property obtained is as follows. Theorem 1. Let \(G\) be a simple non-abelian locally finite group. Then \(G\) is finitary linear if and only if there exists an involution \(x \in G\) and an integer \(d \geq 1\) such that if \(S\) is any 2-subgroup \(G\) containing \(x\), then the normal closure of \(x\) in \(S\) is soluble of derived length at most \(d\). The fact that this condition holds if \(G\) is finitary linear is a recent result of \textit{U. Meierfrankenfeld, R. E. Phillips} and \textit{O. Puglisi} [J. Lond. Math. Soc., II. Ser. 47, 31-40 (1993; Zbl 0738.20043)]. The converse depends on the classification of finite simple groups. Theorem 1 breaks down as it stands if 2 is replaced by an odd prime. In fact, for any odd prime \(p\), there is a simple locally finite group that is not finitary linear, but nevertheless contains an element \(x\) of order \(p\) whose normal closure, in any \(p\)-subgroup containing it, is abelian. Two questions about analogous characterizations of simple locally finite linear groups are raised in the paper. They have been answered affirmatively in a remarkable preprint of \textit{U. Meierfrankenfeld} [Michigan State Univ.] which contains other very interesting results about simple locally finite groups.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 571904
zbMATH Keywords: finitary linear groups
zbMATH Keywords: simple locally finite groups
zbMATH Keywords: involution
zbMATH Keywords: 2- subgroup
zbMATH Keywords: normal closure
zbMATH Keywords: derived length
zbMATH Keywords: classification of finite simple groups
author: J. I. Hall
author: Brian Hartley
reviewed by: Brian Hartley
MaRDI profile type: Publication
cites work: Q5662096
cites work: Q3228640
cites work: Infinite alternating groups as finitary linear transformation groups
cites work: MONOMORPHISMS AND DIRECT LIMITS OF FINITE GROUPS OF LIE TYPE
cites work: Über einfache, lokal endliche Gruppen
cites work: Q3361039
cites work: The structure of groups of finitary transformations
cites work: The classification of the simple periodic linear groups
cites work: Sylow p-Subgroups of the Classical Groups Over Finite Fields with Characteristic Prime to p
full work available at URL: https://doi.org/10.1007/bf01199095
OpenAlex ID: W2060575446
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description: scientific article; zbMATH DE number 571904