Artinian modules over hyperfinite groups
type: Item
instance of: scholarly article
title: Artinian modules over hyperfinite groups
zbMATH Open document ID: 0960.20020
author: V. Pereyra
published in: Southeast Asian Bulletin of Mathematics
publication date: 2001-02-08T00:00:00Z
review text: A group \(G\) is said to be hyperfinite if it has an ascending normal series with finite factors. The structure of Artinian modules over hyperfinite locally soluble groups is described in this article. In particular, the following result is proved. Let \(F\) be a field, \(G\) a hyperfinite locally soluble group and \(A\) a monolithic Artinian \(FG\)-module such that \(C_G(A)=\{1\}\). (1) If \(F\) has characteristic \(0\), then the socle \(S\) of \(G\) contains a subgroup \(J\) such that \(S/J\) is locally cyclic and the core of \(J\) in \(G\) is trivial. (2) If \(F\) has prime characteristic \(p\), then the subgroup \(P=O_p(G)\) is nilpotent with finite exponent and there exists a normal subgroup \(H\) of \(G\) containing \(P\) such that \(H/P\) is finite and the socle \(\overline S\) of \(\overline G=G/J\) is a \(p'\)-group containing a subgroup \(\overline J\) such that \(\overline S/\overline J\) is locally cyclic and \(\overline J\) has trivial core in \(\overline G\).
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F22
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 1353543
zbMATH Keywords: hyperfinite groups
zbMATH Keywords: Artinian modules
zbMATH Keywords: ascending normal series
zbMATH Keywords: locally soluble groups
zbMATH Keywords: socles
zbMATH Keywords: locally cyclic groups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1353543