Local characterizations of periodic simple groups of finitary transformations
type: Item
instance of: scholarly article
title: Local characterizations of periodic simple groups of finitary transformations
zbMATH Open document ID: 0840.20023
DOI: 10.1007/BF02261690
author: V. V. Belyaev
published in: Algebra and Logic
publication date: 1995-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/187646
review text: A subgroup \(H\) of a group \(G\) is called inert if, for every \(g\in G\), the intersection \(H\cap H^g\) has finite index in \(H\). The group \(G\) itself and any of its finite subgroups are the simplest examples of inert subgroups in \(G\). Below we call these subgroups trivial. It was proved by the author that a countable, locally finite, simple group \(G\) is isomorphic to a group of Lie-type if and only if every inert subgroup in \(G\) is trivial. This result gave rise to the following question: What is the structure of a nontrivial inert subgroup of a simple group? Clearly every proper inert subgroup of a simple group is residually finite. But this property is not the only specific feature of their structure. It was shown that for every proper inert subgroup \(H\) of a locally finite simple group \(G\), one of the following statements is true. 1) The group \(H\) is locally normal; 2) There are no finite nontrivial normal subgroups in \(H\); moreover, there exists a prime number \(p\) such that \(O_p (H)\neq 1\) and the quotient group \(H/ O_p (H)\) is locally normal; 3) There are no finite nontrivial normal subgroups in \(H\) and the locally nilpotent radical of \(H\) is trivial, while the locally solvable radical \(S(H)\) of \(H\) is not; the quotient group \(H/ S(H)\) is locally normal; 4) There are no finite nontrivial normal subgroups in \(H\), and \(S(H)=1\). In this paper we consider simple groups whose proper inert subgroups are all of types 1) or 2). To formulate the results obtained, we need the following definition: a linear transformation \(g\) of a vector space \(V\) is called finitary if \(g\) acts identically on some subspace of finite codimension in \(V\). A group \(G< GL(V)\) is said to be a group of finitary transformations (of \(V\)) if each element in \(G\) is a finitary transformation (of \(V\)). Theorem 1. If \(G\) is a countable, locally finite, simple group, then the following conditions are equivalent: 1) \(G\) is isomorphic to a group…
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 738946
zbMATH Keywords: finite subgroups
zbMATH Keywords: inert subgroups
zbMATH Keywords: countable locally finite simple groups
zbMATH Keywords: groups of Lie-type
zbMATH Keywords: locally nilpotent radical
zbMATH Keywords: locally solvable radical
zbMATH Keywords: group of finitary transformations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 738946