Normality of the elementary subgroups of twisted Chevalley groups over commutative rings
type: Item
instance of: scholarly article
title: Normality of the elementary subgroups of twisted Chevalley groups over commutative rings
zbMATH Open document ID: 0840.20043
author: Kazuo Suzuki
published in: Journal of Algebra
publication date: 1995-08-29T00:00:00Z
review text: Let \(G\) be a twisted Chevalley group of type \({^2A_n}\) (\(n\geq 2\)), \({^2D_n}\) (\(n\geq 4\)) or \({^2E_6}\) over a commutative ring \(A\) with 1 and with an involutive automorphism \(\sigma\). Denote \(A_0=\{u\in A\mid u^\sigma=u\}\), \(U(A)=\{(a,b)\in A\times A\mid aa^\sigma=b+b^\sigma\}\) and \(U(A)^*=\{(a,b)\in U(A)\mid b\in A^*\}\), where \(A^*\) is the group of units of \(A\). Let \(E\) be the subgroup of \(G\) generated by root elements. It is proved that \(E\) is a normal subgroup of \(G\) if the ring \(A\) satisfies the following two hypotheses. (A1) For any maximal ideal \(m\) of \(A\), the natural map: \(A_0\to(A/(m\cap\sigma m))_0\) is surjective if \(G\) is not of type \({^2A_{2n}}\), and the natural map: \(U(A)\to U(A/(m\cap\sigma m))\) is surjective and \(U(A)^*\neq\emptyset\) if \(G\) is of type \({^2A_{2n}}\). (A2) For any maximal ideal \(m_0\) of \(A_0\), we have \(m_0=A_0\cap m_0A\). Note that if 2 is invertible in \(A\), then (A1) is satisfied.
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 783934
zbMATH Keywords: subgroup generated by root elements
zbMATH Keywords: twisted Chevalley groups
zbMATH Keywords: commutative rings
zbMATH Keywords: group of units
zbMATH Keywords: normal subgroups
zbMATH Keywords: maximal ideals
MaRDI profile type: Publication
full work available at URL: jabr.1995.1199
OpenAlex ID: W1984811046
DOI: 10.1006/JABR.1995.1199
description: scientific article; zbMATH DE number 783934