Computing generators of groups preserving a bilinear form over residue class rings.
type: Item
instance of: scholarly article
title: Computing generators of groups preserving a bilinear form over residue class rings.
zbMATH Open document ID: 1264.20050
author: A. Hulpke
published in: Journal of Symbolic Computation
publication date: 2013-01-10T00:00:00Z
review text: Let \(R=\mathbb Z_q(=\mathbb Z/q\mathbb Z)\) be a residue class ring, and \(J\in M_n(\mathbb Z_q)\) a matrix for which \(J^T=\pm J\). Let \(F_n(R)\) be the group of matrices preserving the form given by \(J\), that is \[ F_n(R)=\{A\in M_n(R)\mid A^TJA=J\}. \] This group \(F_n(R)\) is called an orthogonal group if \(J\) is symmetric (\(J=J^T\)); it is called a symplectic group if \(J\) is alternating (\(J=-J^T\)). If \(q\) is prime, generators for symplectic groups have been given by Taylor (1987). Similarly generators for orthogonal groups are given by Rylands and Taylor (1998). This paper is to show how to extend this result to obtain generators for these groups over the ring \(R=\mathbb Z_q\) if \(q\) is a proper odd prime power. These generators have been implemented and are available in the computer algebra system GAP.
reviewed by: Nan Ji-Zhu
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 6124232
zbMATH Keywords: orthogonal groups
zbMATH Keywords: symplectic groups
zbMATH Keywords: generators
zbMATH Keywords: residue class rings
zbMATH Keywords: bilinear forms
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jsc.2012.08.002
OpenAlex ID: W2065714029
DOI: 10.1016/J.JSC.2012.08.002
description: scientific article; zbMATH DE number 6124232