Constructing faithful representations of finitely-generated torsion-free nilpotent groups
type: Item
instance of: scholarly article
title: Constructing faithful representations of finitely-generated torsion-free nilpotent groups
zbMATH Open document ID: 1021.20006
author: Werner Nickel
author: Willem A. De Graaf
published in: Journal of Symbolic Computation
publication date: 2002-06-11T00:00:00Z
full work available at URL: https://dspace.library.uu.nl/handle/1874/2330
review text: A finitely generated torsionfree nilpotent group \(G\) has a finite series, whose factors are central infinite cyclic groups. Every such group can be embedded into a group of upper unitriangular matrices with integer coefficients. Using the series the authors show how one obtains a consistent polycyclic presentation for \(G\) and then formulate an efficient algorithm for calculating a representation by unitriangular matrices. The inductive construction consists of an algorithm which extends a representation of this kind for a normal subgroup \(N\) of \(G\) to one for \(G\), where \(G/N\) is infinite cyclic. The central step is to describe the action of \(G\) on the dual space of the group ring \(\mathbb{Z} N\) by means of a suitable finite-dimensional faithful submodule. A table of examples is included which indicates the significant reduction in the embedding dimension this method achieves.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1753636
zbMATH Keywords: finitely generated torsionfree nilpotent groups
zbMATH Keywords: polycyclic presentations
zbMATH Keywords: efficient algorithms
zbMATH Keywords: representations by unitriangular matrices
zbMATH Keywords: group rings
reviewed by: Martin L. Newell
describes a project that uses: GAP
MaRDI profile type: Publication
OpenAlex ID: W2078306804
cites work: Q5614712
cites work: Q4942163
cites work: Q3253828
cites work: Collection from the left and other strategies
cites work: Q4867140
cites work: Q4285784
cites work: Collection from the left
DOI: 10.1006/JSCO.2001.0497
description: scientific article; zbMATH DE number 1753636