Imperfect groups
type: Item
instance of: scholarly article
title: Imperfect groups
zbMATH Open document ID: 0785.20020
DOI: 10.1016/0022-4049(93)90008-H
author: A. J. Berrick
author: Derek J. S. Robinson
published in: Journal of Pure and Applied Algebra
publication date: 1993-11-01T00:00:00Z
review text: A group is imperfect if all its perfect quotient groups are trivial. The authors begin the study of these groups by establishing closure properties of these groups. In particular, an arbitrary group \(H\) is isomorphic with a 2-step subnormal subgroup of an imperfect group \(G\) such that \(| G|= 2| H|^ 2\). Here is given a criterion for a group \(H\) to be isomorphic with a normal subgroup of an imperfect group. For a non-abelian simple group \(H\) this criterion means that \(\text{Out }H\neq 1\). There is given a construction of a countably infinite, locally finite group with minimal condition on subnormal subgroups, which is a union of a chain of imperfect normal subgroups. The imperfect radical and residual of the groups \(GL_ n(\mathbb{Z})\) and \(SL_ n(\mathbb{Z})\) is calculated. Let \(R\) be a ring such that \(sr(R)=1\). If \(n\geq 1\), then \(GL_ n(R)\) is imperfect if and only if \(GL_ n(R/M)\) is imperfect for any maximal ideal \(M\) of \(R\). In particular, let \(R\) be a commutative semilocal ring which does not have the field \(\mathbb{F}_ 2\) as a quotient ring. If \(n>1\), then \(GL_ n(R)\) is imperfect if and only if \(F^*\neq(F^*)^ n\) for every quotient field \(F\) of \(R\).
reviewed by: Vyacheslav A. Artamonov
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20H25
zbMATH DE Number: 427653
zbMATH Keywords: perfect quotient groups
zbMATH Keywords: subnormal subgroup
zbMATH Keywords: imperfect group
zbMATH Keywords: locally finite group
zbMATH Keywords: minimal condition on subnormal subgroups
zbMATH Keywords: chain of imperfect normal subgroups
zbMATH Keywords: imperfect radical
zbMATH Keywords: commutative semilocal ring
Wikidata QID: Q30048369
MaRDI profile type: Publication
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full work available at URL: 0022-4049(93)90008-h
OpenAlex ID: W4210683096
description: scientific article; zbMATH DE number 427653