Subgroups of type \(A_1\) containing a fixed unipotent element in an algebraic group
type: Item
instance of: scholarly article
title: Subgroups of type \(A_1\) containing a fixed unipotent element in an algebraic group
zbMATH Open document ID: 0981.20035
author: Richard Proud
author: Jan Saxl
author: Donna M. Testerman
published in: Journal of Algebra
publication date: 2001-12-05T00:00:00Z
review text: The main result proved by the authors is the following theorem: Let \(G\) be a simple algebraic group defined over an algebraically closed field of characteristic \(p>0\). Let \(\sigma\) be either a Frobenius morphism of \(G\), or \(\sigma=1\). Let \(u\in G\) be a \(\sigma\)-invariant element of order \(p\). Then \(u\) is contained in a closed \(\sigma\)-invariant subgroup of \(G\) of type \(A_1\), except in the following cases: (i) \(G=G_2\), \(p=3\), and \(u\) is an element of order \(3\) of type \(A_1^{(3)}\), (ii) \(G=G_2\), \(p=3\), and \(\sigma\) is a morphism involving the graph morphism of \(G\), (iii) \(G=B_2\) or \(G=F_4\), \(p=2\), and \(\sigma\) is a morphism involving the graph morphism of \(G\).
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1525266
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: unipotent elements
zbMATH Keywords: subgroups of type \(A_1\)
zbMATH Keywords: Frobenius morphisms
zbMATH Keywords: graph morphisms
zbMATH Keywords: finite groups of Lie type
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
full work available at URL: jabr.2000.8353
OpenAlex ID: W2062341042
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DOI: 10.1006/JABR.2000.8353
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description: scientific article; zbMATH DE number 1525266