The nonabelian tensor square and Schur multiplicator of nilpotent groups of class 2
type: Item
instance of: scholarly article
title: The nonabelian tensor square and Schur multiplicator of nilpotent groups of class 2
zbMATH Open document ID: 0844.20024
author: Manfred Hartl
published in: Journal of Algebra
publication date: 1996-09-01T00:00:00Z
review text: 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 Ser. 14, 309-312 (1976; Zbl 0357.20030)]; and \textit{R. Dennis} [unpublished preprint (1978)].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.
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 843873
zbMATH Keywords: non-abelian tensor squares
zbMATH Keywords: non-abelian exterior squares
zbMATH Keywords: Schur multiplicators
zbMATH Keywords: van Kampen theorems
zbMATH Keywords: diagrams of spaces
zbMATH Keywords: suspension
zbMATH Keywords: nilpotent groups
zbMATH Keywords: augmentation ideal
zbMATH Keywords: integral group ring
zbMATH Keywords: 2-generator groups
reviewed by: J. Huebschmann
MaRDI profile type: Publication
full work available at URL: jabr.1996.0018
OpenAlex ID: W2034241560
DOI: 10.1006/JABR.1996.0018
description: scientific article; zbMATH DE number 843873