Chevalley groups of type \(G_ 2\) as the group of a trilinear form
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)].
subgroup structure of groups of Lie type
scientific article; zbMATH DE number 4002378
scientific article