Aschbacher, Michael Some multilinear forms with large isometry groups. (English) Zbl 0646.20033 Geom. Dedicata 25, 417-465 (1988). 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 Cited in 1 ReviewCited in 31 Documents 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 Keywords:groups of Lie type; maximal subgroups; maximal closed subgroups; isometry groups; multilinear forms; twisted groups Citations:Zbl 0537.20023; Zbl 0618.20030; Zbl 0629.20018 PDFBibTeX XMLCite \textit{M. Aschbacher}, Geom. Dedicata 25, 417--465 (1988; Zbl 0646.20033) Full Text: DOI