Rigid cyclic subgroups in Chevalley groups. I
type: Item
instance of: scholarly article
title: Rigid cyclic subgroups in Chevalley groups. I
zbMATH Open document ID: 0727.20035
DOI: 10.1016/0021-8693(90)90203-Z
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The author studies overgroups of rigid r-subgroups (where r is a prime) of a Chevalley group G defined over a finite field \(F_ q\), where \(q=p^ k\) for some prime p. Rigid r-subgroups R of G are those for which \(C_ G(R)\) is abelian and has a cyclic r-subgroup. The paper gives a characterization of the overgroups of such R in terms of arithmetic and algebraic-group-theoretic properties, and thereby obtains necessary conditions for a quasisimple group to be such an overgroup. Subsequent papers will use the classification of finite simple groups to describe the lattice of overgroups of rigid r-subgroups. Overgroups of maximal tori have been studied by \textit{G. M. Seitz} [Pac. J. Math. 106, 153-244 (1983; Zbl 0522.20031)] by means of root groups. The author obtains more complete results for fields of small characteristic. The nontrivial elements of rigid r-subgroups are regular semisimple elements of the corresponding connected reductive almost simple algebraic groups \(\bar G\) for which \(G=\bar G'_{\sigma}\), where \(\bar G_{\sigma}\) is the finite kernel of the surjective endomorphism \(\sigma\) of G. (Types \(A_ 1(2\) or 3), \({}^ 2A_ 2(4)\), \({}^ 2C_ 2(2)\), \({}^ 2F_ 4(2)\), and \({}^ 2G_ 2(3)\) are excluded.) Rigid r-subgroups are contained in certain maximal regular tori called strongly regular. Work of \textit{T. A. Springer} [Invent. Math. 25, 159-198 (1974; Zbl 0287.20043)] is the point of departure for the study of the structure of root lattices, which gives the basis for several key results that tie together the arithmetic, algebraic-geometric, and finite-group- theoretic properties of rigid r-subgroups R. A representative example of this is Theorem 3.4, which characterizes rigid r-elements (that is, nontrivial elements of such R) as follows. Let \(r\neq p\) and let t be an r-element of G such that \(C_ G(t)\) (misprinted on p. 698 of the paper) has a cyclic \(S_ r\)-subgroup. Then t is a rigid r-element if and only if it is a regular semisimple ele…
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 4200488
zbMATH Keywords: overgroups of rigid r-subgroups
zbMATH Keywords: Chevalley group
zbMATH Keywords: cyclic r-subgroup
zbMATH Keywords: quasisimple group
zbMATH Keywords: finite simple groups
zbMATH Keywords: lattice of overgroups
zbMATH Keywords: maximal tori
zbMATH Keywords: root groups
zbMATH Keywords: regular semisimple elements
zbMATH Keywords: connected reductive almost simple algebraic groups
zbMATH Keywords: root lattices
reviewed by: James F. Hurley
MaRDI profile type: Publication
cites work: Q5668957
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cites work: The root subgroups for maximal tori in finite groups of Lie type
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author: John H. Walter
description: scientific article; zbMATH DE number 4200488