Degree 1 and 2 representations of nilpotent groups and applications to units of group rings
type: Item
instance of: scholarly article
title: Degree 1 and 2 representations of nilpotent groups and applications to units of group rings
zbMATH Open document ID: 0834.16034
DOI: 10.1007/BF02568007
author: Guilherme Leal
author: Eric Jespers
published in: Manuscripta Mathematica
publication date: 1996-04-08T00:00:00Z
full work available at URL: https://eudml.org/doc/156067
review text: This is a rather technical paper concerned with the unit group \(U(ZG)\) of the integral group ring \(ZG\) of the finite group \(G\). In studying generators for these unit groups, certain components \(eQG\) of the rational group ring \(QG\) cause difficulties and are considered exceptional. These troublesome components are isomorphic to: (1) noncommutative division rings which are not totally definite quaternion algebras, (2) \(M_2(Q)\), (3) \(M_2(F)\) where \(F\) is an imaginary quadratic extension of the rationals, or (4) \(M_2(D)\) where \(D\) is a noncommutative division ring. This paper starts by classifying those finite nilpotent groups \(G\) having an exceptional component and then uses this information to describe generators for a subgroup of finite index of \(U(ZG)\).
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 784054
zbMATH Keywords: integral group rings
zbMATH Keywords: finite groups
zbMATH Keywords: generators
zbMATH Keywords: unit groups
zbMATH Keywords: rational group rings
zbMATH Keywords: division rings
zbMATH Keywords: quaternion algebras
zbMATH Keywords: finite nilpotent groups
zbMATH Keywords: subgroups of finite index
reviewed by: D. S. Passman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784054