The nonabelian tensor square and Schur multiplicator of nilpotent groups of class 2
In search of van Kampen theorems for diagrams of spaces, \textit{R. Brown} and \textit{J. L. Loday} studied the notions of non-abelian tensor and exterior squares of a non-abelian group [Topology 26, 311-335 (1987; Zbl 0622.55009)]; the non-abelian tensor square was studied already by \textit{A. Lue} [J. Lond. Math. Soc., II. Ser. 14, 309-312 (1976; Zbl 0357.20030)]; and \textit{R. Dennis} [unpublished preprint (1978)]. In the paper, the non-abelian tensor and exterior squares, the Schur multiplicator, and the third homotopy group of the suspension \(\Sigma K(G,1)\), are determined for arbitrary nilpotent groups \(G\) of class 2. Functorial descriptions are given in terms of the quotient \(P_2(G) = I(G)/(I(G))^3\) of the augmentation ideal \(I(G)\) of the integral group ring of \(G\). Moreover, the tensor square being abelian for a nilpotent group of class 2, abelian presentations of these groups are given in terms of a presentation of \(G\) itself. For a finite \(p\)-group, the calculation of these groups is thereby reduced to matrix calculations. This is illustrated by explicit calculations for 2-generator groups.
scientific article; zbMATH DE number 843873