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Some multilinear forms with large isometry groups. (English) Zbl 0646.20033

This paper is a basic part of a program for the study of the subgroup structure of the groups of Lie type over arbitrary fields, all maximal subgroups if the field is finite and all maximal closed subgroups if the field is algebraically closed. The groups of Lie type may be best described as the isometry groups of some multilinear forms on modules of minimal dimensions. It is to be desired that the subgroup structure is determined from the geometry associated to the forms. In case of the finite classical groups the work has been done [in Invent. Math. 76, 469- 514 (1984; Zbl 0537.20023)]. In this paper 3 or 4-forms for groups of type \(G_ 2\), \(F_ 4\), \(E_ 6\), and \(E_ 7\), and the twisted groups \({}^ 3D_ 4\) and \({}^ 2E_ 6\) are described. Some of the presentations of these forms are well-known and go back to Dickson and Cartan. The identification of the isometry group with a group of Lie type is left to other papers. Some of them have already been published [J. Algebra 109, 193-259 (1987; Zbl 0618.20030), Invent. Math. 89, 159-195 (1987; Zbl 0629.20018)].
Reviewer: H.Yamada

MSC:

20F65 Geometric group theory
20D06 Simple groups: alternating groups and groups of Lie type
20G15 Linear algebraic groups over arbitrary fields
20D30 Series and lattices of subgroups
20E28 Maximal subgroups
20F29 Representations of groups as automorphism groups of algebraic systems
20G40 Linear algebraic groups over finite fields
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