The Grothendieck ring of invertible modules over nilpotent groups
type: Item
instance of: scholarly article
title: The Grothendieck ring of invertible modules over nilpotent groups
zbMATH Open document ID: 0798.20001
author: Esther Beneish
published in: Journal of Algebra
publication date: 1994-11-03T00:00:00Z
review text: For a finite group \(G\), a permutation \(ZG\)-module \(P\) is defined as a \(ZG\)-lattice with a \(Z\)-basis permuted by \(G\), and the direct summands of such \(P\) are called invertible \(ZG\)-modules. Let \({\mathcal L}(QG)\) denote the Grothendieck ring of \(QG\)-modules \(V\) containing a full invertible \(ZG\)-lattice. The author considers \({\mathcal L}(QG)\) in the case that \(G\) is nilpotent of odd order. Then \({\mathcal L}(QG)\) is the sum of the images of \(i_ *: {\mathcal L}(QH)\to{\mathcal L}(QG)\) for all \(p\)-elementary subgroups \(i:H\hookrightarrow G\). As a main result, the author determines the \(QH\)- modules containing a full invertible \(ZH\)-lattice, which yields a particular set of generators of \({\mathcal L}(QG)\).
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 19A31
zbMATH DE Number: 571353
zbMATH Keywords: finite group
zbMATH Keywords: permutation \(ZG\)-module
zbMATH Keywords: direct summands
zbMATH Keywords: invertible \(ZG\)- modules
zbMATH Keywords: Grothendieck ring
zbMATH Keywords: nilpotent of odd order
zbMATH Keywords: \(p\)-elementary subgroups
zbMATH Keywords: generators
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: jabr.1993.1164
OpenAlex ID: W2035552602
DOI: 10.1006/JABR.1993.1164
description: scientific article; zbMATH DE number 571353