Groups generated by \(k\)-root subgroups
type: Item
instance of: scholarly article
title: Groups generated by \(k\)-root subgroups
zbMATH Open document ID: 0794.20057
DOI: 10.1007/BF01243925
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143954
review text: The paper is a successor of the author's paper [Invent. Math. 100, No. 1, 167-206 (1990; Zbl 0697.20018)]. The author studies groups \(G\) generated by so called \(k\)-root subgroups (\(k\) a commutative field). This is a set \(\Sigma\) of abelian groups with \(G= \langle \Sigma\rangle\) and \(\Sigma^ g=\Sigma\) for \(g\in G\) such that for each pair \(A,B\in\Sigma\) one of the following holds: (1) \([A,B]=1\), (2) \(\langle A,B\rangle\cong\text{(P)SL}(2,k)\) and \(A\) and \(B\) are full unipotent subgroups, (3) \(\langle A,B\rangle'\leq Z(\langle A,B\rangle)\) and \([a,B]= [b,A]= [A,B]\) is in \(\Sigma\) for all \(a\in A^ \#\) and \(b\in B^ \#\). Furthermore he assumes the maximality condition for unipotent \(\Sigma\)-subgroups. In the paper mentioned above the author treated the case that (3) never occurs. The conclusion is too lengthy to be stated here but roughly it says that there is a nilpotent normal subgroup \(R(G)\) such that \(G/R(G)\) is a quasisimple group, which then is a central extension of a well defined class of algebraic groups defined over certain extension fields of \(k\), which are described in a paper of \textit{J. Tits} [Proc. Symp. Pure Math. 9, 33-62 (1966; Zbl 0238.20052)] (misquoted as [25] in the paper under review). The proof of the theorem is geometric in nature. The author investigates three graphs on the set \(\Sigma\). This is the commuting graph, the graph whose edges are the pairs \((A,B)\) with \(\langle A,B\rangle \cong\text{(P)SL}(2,k)\), and a graph whose edges are pairs \((A,B)\) such that \(\Sigma\cap AB\) gives a partition of \(AB\). The latter is very important as the proof is done by induction on the number of generators from \(\Sigma\) needed to generate the group given by the neighbours of \(A\) in the last-mentioned graph. Investigation of these graphs together with a very detailed analysis of the groups generated by three members from \(\Sigma\) gives at the end that the group \(G\) has to act on some building, and the…
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 59788
zbMATH Keywords: root-group
zbMATH Keywords: root subgroups
zbMATH Keywords: nilpotent normal subgroup
zbMATH Keywords: quasisimple group
zbMATH Keywords: algebraic groups
zbMATH Keywords: graphs
zbMATH Keywords: building
author: F. G. Timmesfeld
reviewed by: Gernot Stroth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59788