\(K_{1}\) of Chevalley groups are nilpotent
type: Item
instance of: scholarly article
title: \(K_{1}\) of Chevalley groups are nilpotent
zbMATH Open document ID: 1012.19001
DOI: 10.1016/S0022-4049(02)00292-X
author: R. Hazrat
author: Nikolai Vavilov
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-09T00:00:00Z
review text: This paper extends the main result of \textit{A. Bak} [\(K\)-Theory 4, No. 4, 363-397 (1991; Zbl 0741.19001)] to the case of Chevalley groups. More precisely, let \(\Phi\) be a reduced irreducible root system of rank at least two and let \(R\) be a commutative ring of finite Bass-Serre dimension or finite Krull dimension. Let \(G(\Phi,R)\) be a Chevalley group of type \(\Phi\) over \(R\) and let \(E(\Phi,R)\) be its elementary subgroup. It is shown that \(G(\Phi,R)/E(\Phi,R)\) is nilpotent-by-abelian. If one takes \(G(\Phi,R)\) simply connected, then \(G(\Phi,R)/E(\Phi,R)\) is called \(K_1 (\Phi,R)\) and they show it to be nilpotent. The authors feel the method deserves better publicity and therefore make an effort to explain it well.
Mathematics Subject Classification ID: 19B14
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 1895983
zbMATH Keywords: localisation-completion method of Bak
zbMATH Keywords: Bass-Serre dimension
zbMATH Keywords: Chevalley groups
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
cites work: Chevalley groups over local rings
cites work: On normal subgroups of Chevalley groups over commutative rings
cites work: Nonabelian \(K\)-theory: The nilpotent class of \(K_ 1\) and general stability
cites work: Normality for elementary subgroup functors
cites work: Structure of Hyperbolic Unitary Groups I: Elementary Subgroups
cites work: Q5562621
cites work: Q3993275
cites work: A module structure on certain orbit sets of unimodular rows
cites work: THE STABILIZATION OF SYMPLECTIC GROUPS OVER A POLYNOMIAL RING
cites work: An algorithmic proof of Suslin's stability theorem for polynomial rings
cites work: On the stability of the \(K_1\)-functor for Chevalley groups of type \(E_7\)
cites work: On the word length of commutators in \(\text{GL}_n(R)\)
cites work: Q4187969
cites work: Decomposition of transvections: A theme with variations
cites work: Quadratic modules and the orthogonal group over polynomial rings
cites work: Normality of the elementary subgroups of twisted Chevalley groups over commutative rings
cites work: Q3737654
cites work: Q4161799
cites work: Schur multiplier of a group of elementary matrices of finite order
cites work: Q4010232
cites work: Q4271238
cites work: Chevalley groups over commutative rings. I: Elementary calculations
Recommended article: Non-Zero Ext1 for Chevalley Groups (via Algebraic Groups)
Recommended article: Q4022822
Recommended article: On absolutely-nilpotent of class \(k\) groups
Recommended article: Groups in which every proper subgroup is Chernikov-by-nilpotent or nilpotent-by-Chernikov
Recommended article: Q4015762
Recommended article: A Krengel-type theorem for finitely-generated nilpotent groups
Recommended article: Q5868852
Recommended article: Q5583897
Recommended article: EVERY (k + 1)-AFFINE COMPLETE NILPOTENT GROUP OF CLASS k IS AFFINE COMPLETE
Recommended article: Q4488003
description: scientific article; zbMATH DE number 1895983