Group extensions and their trivialisation
type: Item
instance of: scholarly article
title: Group extensions and their trivialisation
zbMATH Open document ID: 0584.20023
author: A. J. Berrick
published in: L'Enseignement Mathématique. 2e Série
publication date: 1985-01-01T00:00:00Z
review text: Given groups \(K\) and \(Q\), let \(\text{Ext}(Q,K)\) denote the set of all equivalence classes of extensions \(K\rightarrowtail G\twoheadrightarrow Q\). The author studies the set \(\text{Ext}(Q,K)\) with ``emphasis on the topological approach''. The set \(\text{Ext}(Q,K)\) is embedded in a 4-term exact sequence of pointed sets (Proposition 1.1). As a consequence of a more general (and also technical) result it is proved that if \(K\) is nilpotent and \(Q\) is perfect then the set of equivalence classes of extensions with kernel \(K\) in the hypercentre and with quotient \(Q\) is in 1-1 correspondence with \(H^ 2(Q,Z(K))\), where \(Z(K)\) denotes the centre of \(K\). Among many other results considered, the author defines a group \(G_{{\mathcal C},A}\) from a given small category \({\mathcal C}\) and an associative ring \(A\) with identity. The Steinberg groups \(ST_ n(A)\) and the McLain groups \(M(\Lambda,F)\) then arise as special cases of this group. Sufficient conditions on \({\mathcal C}\) so that \(G_{{\mathcal C},A}\) becomes a superperfect group are given (Theorem 4.1).
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 3935304
zbMATH Keywords: Ext
zbMATH Keywords: equivalence classes of extensions
zbMATH Keywords: exact sequence of pointed sets
zbMATH Keywords: hypercentre
zbMATH Keywords: small category
zbMATH Keywords: Steinberg groups
zbMATH Keywords: McLain groups
zbMATH Keywords: superperfect group
reviewed by: Lekh Raj Vermani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3935304