Group extensions and their trivialisation
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).
scientific article; zbMATH DE number 3935304