Some presentations for the special linear groups on finite fields.
type: Item
instance of: scholarly article
title: Some presentations for the special linear groups on finite fields.
zbMATH Open document ID: 1034.20029
author: G. Chiaselotti
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2004-02-01T00:00:00Z
review text: The so-called Steinberg presentation of the finite group \(\text{SL}(n,q)\), \(q=p^m\) a power of the odd prime \(p\) is a presentation of \(\text{SL}(n,q)\) as an abstract group with generators \(x_{ij}(\alpha)\), \(1\leq i,j\leq n\), \(i\neq j\), \(\alpha\in\text{GF}(q)\) together with the relations (E1) \(x_{ij}(\alpha)x_{ij}(\beta )=x_{ij}(\alpha+\beta)\), (E2) \([x_{ij}(\alpha),x_{k\ell}(\beta)]=1\) for \(1\leq i,j,k,\ell\leq n\), \(i\neq\ell\), \(j\neq k\), (E3) \([x_{ij}(\alpha),x_{jk}(\beta)]=x_{ik}(\alpha\beta)\), \(1\leq i,j,k\leq n\), \(i\neq j\neq k\neq i\), and \(\alpha,\beta\in\text{GF}(q)\). The author shows that these relations can be replaced by the same relations where \(\alpha\) and \(\beta\) range over the elements of a \(\text{GF}(p)\)-basis of \(\text{GF}(q)\) of the form \(\{1,\gamma,\gamma^2,\dots,\gamma^{m-1}\}\). Thus the number of relations is reduced considerably.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 2035111
zbMATH Keywords: Steinberg presentations
zbMATH Keywords: finite fields
zbMATH Keywords: generators
zbMATH Keywords: special linear groups
zbMATH Keywords: elementary matrices
zbMATH Keywords: relations
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
full work available at URL: s102310100016
OpenAlex ID: W2030599646
DOI: 10.1007/S102310100016
description: scientific article; zbMATH DE number 2035111