Chevalley groups of type \(G_ 2\) as the group of a trilinear form
type: Item
instance of: scholarly article
title: Chevalley groups of type \(G_ 2\) as the group of a trilinear form
zbMATH Open document ID: 0618.20030
DOI: 10.1016/0021-8693(87)90173-6
author: Michael Aschbacher
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: This paper is a part of the author's investigation on the subgroup structure of groups of Lie type. The classical groups are conveniently studied with the help of the natural module and the corresponding bilinear form. The author tries to transfer this method to exceptional groups of Lie type too. It turns out, that the module of minimal dimension for exceptional groups of Lie type carries a 3- or 4-linear form, such that this group can be studied as an isometry group of such a form. Here the particular case of the exceptional Lie group \(G=G_ 2(F)\) and the 7-dimensional F-module V with a trilinear form for G is studied in detail. The field F is arbitrary for more general results, while F is chosen to be finite or algebraically closed for more special results. The paper treats geometric and group theoretic aspects. The long list of theorems can not be reproduced here but we indicate the nature of a few results: The characterization of various isometry types of subspaces and the action of the isometry group on these spaces (like analogues of Witt's theorem). The description of stabilizers of various geometric structures. The determination of the maximal subgroups of G if F is finite or the maximal closed subgroups if F is algebraically closed. This paper belongs to the program of studying maximal subgroups of finite simple groups initiated by the author [Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)]. It gives also new access to former results on \(G=G_ 2(F)\) as for instance [\textit{A. M. Cohen}, \textit{D. B. Wales}, Commun. Algebra 11, 441-459 (1983; Zbl 0517.20020)] and [\textit{B. N. Cooperstein}, J. Algebra 70, 23-36 (1981; Zbl 0459.20007)].
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20E28
zbMATH DE Number: 4002378
zbMATH Keywords: subgroup structure of groups of Lie type
zbMATH Keywords: exceptional groups of Lie type
zbMATH Keywords: module of minimal dimension
zbMATH Keywords: isometry group
zbMATH Keywords: trilinear form
zbMATH Keywords: maximal subgroups
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90173-6
OpenAlex ID: W2094183439
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description: scientific article; zbMATH DE number 4002378