Perfect radicals and homology of group extensions
type: Item
instance of: scholarly article
title: Perfect radicals and homology of group extensions
zbMATH Open document ID: 0618.20023
DOI: 10.1016/0166-8641(87)90010-1
author: A. J. Berrick
author: Brian Hartley
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: If G is a group, \({\mathcal P}G\) denotes the perfect radical of G. An epimorphism \(\phi\) : \(G\to Q\) with \(\phi\) (\({\mathcal P}G)={\mathcal P}Q\) is called an epimorphism preserving perfect radicals \((EP^ 2R)\). The motivation for study is topological, namely certain questions about the plus construction. It is easy to see that \(\phi\) is \(EP^ 2R\) (with \(N=Ker \phi)\) if one of the following holds: (i) G splits over N, (ii) \(G^{(m)}\leq N({\mathcal P}G)\) for some finite m, (iii) \(N^{(m)}\leq {\mathcal P}G\) for some finite m. However the property fails if G is free and Q is perfect. The authors give a number of examples disproving conjectures about \(EP^ 2R\) (and thereby certain topological conjectures). For example, they construct a group G with \(N\triangleleft G\) such that \({\mathcal P}G=1\), \(Q=G/N\) is perfect and non-trivial and elements of G induce inner automorphisms in N. (Thus Q operates trivially on the integral homology of N.) The group G is an interdirect product of nilpotent groups.
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 55R20
zbMATH DE Number: 4002371
zbMATH Keywords: perfect radical
zbMATH Keywords: epimorphism preserving perfect radicals
zbMATH Keywords: plus construction
zbMATH Keywords: inner automorphisms
zbMATH Keywords: integral homology
reviewed by: Derek J. S. Robinson
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90010-1
OpenAlex ID: W2079278904
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description: scientific article; zbMATH DE number 4002371