Subring subgroups in Chevalley groups with doubly laced root systems.
type: Item
instance of: scholarly article
title: Subring subgroups in Chevalley groups with doubly laced root systems.
zbMATH Open document ID: 1267.20073
author: Alexei Stepanov
published in: Journal of Algebra
publication date: 2013-02-21T00:00:00Z
review text: The author studies the lattice \(\mathcal L\) of subgroups of a Chevalley group \(G(\Phi,A)\) over a ring \(A\), containing its elementary subgroup \(E(\Phi,S)\) over a subring \(S\subseteq A\). The standard description asserts that for any \(H\in\mathcal L\) there exists a unique subring \(R\supseteq S\) of \(A\) such that \(H\) contains \(E(\Phi,R)\) as a normal subgroup. The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems. Now, suppose that \(\Phi\) is doubly laced, i.e. \(\Phi=B_l\), \(C_l\) or \(F_4\), and that 2 is invertible in \(S\). In the article it is proven that in these settings the standard description of \(\mathcal L\) holds for an arbitrary pair of rings \(S\subseteq A\).
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 6138243
zbMATH Keywords: Chevalley groups
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: elementary subgroups
zbMATH Keywords: normal subgroups
zbMATH Keywords: standard descriptions
zbMATH Keywords: small semisimple elements
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.04.007
OpenAlex ID: W2028774614
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DOI: 10.1016/J.JALGEBRA.2012.04.007
description: scientific article; zbMATH DE number 6138243