A classification of subgroups of \(\Lambda\)-quadratic groups normalized by relative elementary groups
type: Item
instance of: scholarly article
title: A classification of subgroups of \(\Lambda\)-quadratic groups normalized by relative elementary groups
zbMATH Open document ID: 0821.19002
author: Günter Habdank
published in: Advances in Mathematics
publication date: 1995-09-25T00:00:00Z
review text: In ``Subgroups of the general linear group normalized by relative elementary groups'' [Lect. Notes Math. 967, 1-22 (1982; Zbl 0536.20030)], \textit{A. Bak} proved the following sandwich theorem: Let \(A\) be a commutative ring satisfying Bass' stable range condition \(SR(M,A)\) and let \(m \geq \max (3,M)\). If \(H\) is a subgroup of \(GL_ m (A)\) normalized by some relative elementary subgroup, then \(H\) lies between a relative elementary subgroup and a slightly enlarged congruence subgroup. In the paper under consideration the author extends this result to subgroups of other classical groups. This is done in the framework of form-rings \((A, \Lambda)\) introduced by Bak [cf. \textit{A. Bak}, ``\(K\)- theory of forms'', Ann. Math. Stud. 98 (1981; Zbl 0465.10013)] under a suitable \(\Lambda\)-quadratic stable range condition.
Mathematics Subject Classification ID: 19G38
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 20H05
zbMATH DE Number: 745281
zbMATH Keywords: \(\Lambda\)-quadratic groups
zbMATH Keywords: \(\Lambda\)-quadratic stable range condition
zbMATH Keywords: sandwich theorem
zbMATH Keywords: Bass stable range condition
zbMATH Keywords: congruence subgroup
zbMATH Keywords: classical groups
zbMATH Keywords: form-rings
reviewed by: Manfred Kolster
MaRDI profile type: Publication
full work available at URL: aima.1995.1008
OpenAlex ID: W2051181235
DOI: 10.1006/AIMA.1995.1008
description: scientific article; zbMATH DE number 745281