Locally finite groups containing a finite inseparable subgroup
type: Item
instance of: scholarly article
title: Locally finite groups containing a finite inseparable subgroup
zbMATH Open document ID: 0836.20051
DOI: 10.1007/BF00975160
author: V. V. Belyaev
published in: Siberian Mathematical Journal
publication date: 1996-01-15T00:00:00Z
review text: Let \(G\) be an infinite locally finite group and let \(H_0\) be an arbitrary finite subgroup in \(G\). Choose a finite nontrivial \(H_0\)-invariant subgroup \(H_1 < G\) with \(H_1 \cap H_0 = 1\). For the subgroup \(\langle H_0, H_1\rangle\), find again a finite nontrivial \(\langle H_0, H_1\rangle\)-invariant subgroup \(H_2\) with \(\langle H_0, H_1\rangle \cap H_2 =1\). Continuing this process further, we thus construct an infinite subgroup \(\langle H_0, H_1, H_2, \dots \rangle\) which is obviously a residually finite group. Executing such a procedure in an arbitrary locally finite group, one sometimes faces the impossibility of choosing the subgroup \(H_i\) at some step of the construction. For instance, the procedure is obviously impossible in the case of a quasi-cyclic group or, more generally, a Chernikov group. It is interesting to classify locally finite groups in which constructing infinite subgroups causes none of the difficulties indicated. To specify the question, we give the following definition. A subgroup \(H\) of a group \(G\) will be called separable in \(G\), if there is a nontrivial \(H\)- invariant subgroup \(K < G\) such that \(H \cap K = 1\). Otherwise, we call \(H\) inseparable. Thus, the main purpose of the present article is to investigate the structure of locally finite groups containing a finite inseparable subgroup.
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 825552
zbMATH Keywords: residually finite groups
zbMATH Keywords: locally finite groups
zbMATH Keywords: infinite subgroups
zbMATH Keywords: finite inseparable subgroups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 825552