On bireflectionality and trireflectionality of orthogonal groups
type: Item
instance of: scholarly article
title: On bireflectionality and trireflectionality of orthogonal groups
zbMATH Open document ID: 0623.20034
DOI: 10.1016/0024-3795(87)90090-5
author: Klaus Nielsen
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let V be a vector space endowed with a quadratic form Q. The \(f(x,y)=Q(x+y)-Q(x)-Q(y)\) is a bilinear form. The vector x is perpendicular to the vector y if \(f(x,y)=0\). The radical of V consists of all vectors that are perpendicular to all vectors in V. The weak orthogonal group \(O^*(V)\) consists of all form preserving bijections that keep every vector in the radical fixed. The author assumes that the field of scalars has characteristic distinct from 2. He shows (A) Every element in \(O^*(V)\) is a product of at most three involutions and (B) Every element in \(O^*(V)\) is a product of two involutions if and only if the index of V or the dimension of the radical is at most 1. This corrects an error in \textit{E. W. Ellers}, \textit{R. Frank}, and \textit{W. Nolte} [J. Algebra 88, 63-67 (1984; Zbl 0533.20020)].
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 51F25
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 4010777
zbMATH Keywords: quadratic form
zbMATH Keywords: bilinear form
zbMATH Keywords: radical
zbMATH Keywords: weak orthogonal group
zbMATH Keywords: product of two involutions
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90090-5
OpenAlex ID: W2049968546
cites work: Q3241211
cites work: Product of two involutions
cites work: Q5632776
cites work: Bireflectionality of orthogonal and symplectic groups
cites work: Bireflectionality of the weak orthogonal and the weak symplectic groups
cites work: Products of two involutions in classical groups of characteristic 2
cites work: Isometrien von Vektorräumen. I
cites work: Group-theoretic characterization of transvections using CDC
cites work: Q5553361
description: scientific article; zbMATH DE number 4010777