The 27-dimensional module for \(E_ 6\). IV
type: Item
instance of: scholarly article
title: The 27-dimensional module for \(E_ 6\). IV
zbMATH Open document ID: 0698.20031
DOI: 10.1016/0021-8693(90)90164-J
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The author continues the study of the geometry of the 27-dimensional vector space V over a field F with a certain symmetric trilinear form f and the subgroup structure of the universal Chevalley group G of type \(E_ 6\), which is the group of isometries of the form f [for Part III see the preceding review Zbl 0698.20030]. Let \(\Gamma\) be the group of semilinear maps on V preserving f up to a scalar and a twist by a field automorphism. Then \(\Gamma\) is the subgroup of Aut(G) of index two trivial on the Dynkin diagram of G. In the present paper it is shown that if the field F is finite or algebraically closed and M is a subgroup of \(\Gamma\) such that \(M\cap G\) is a closed subgroup of G then either (1) M stabilizes a certain structure on V, or (2) \(F^*(M\cap G)Z(G)/Z(G)\) is a nonabelian simple group. In case (1) the group M is determined by the preceding papers. A version of the Baer-Suzuki Theorem for linear groups is also proved.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 4144257
zbMATH Keywords: geometry
zbMATH Keywords: 27-dimensional vector space
zbMATH Keywords: symmetric trilinear form
zbMATH Keywords: universal Chevalley group
zbMATH Keywords: isometries
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: closed subgroup
zbMATH Keywords: nonabelian simple group
zbMATH Keywords: Baer-Suzuki Theorem
author: Michael Aschbacher
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
cites work: The 27-Dimensional Module for E6 , II
cites work: Chevalley groups of type \(G_ 2\) as the group of a trilinear form
cites work: Involutions in Chevalley groups over fields of even order
cites work: Some groups generated by transvections
cites work: Some subgroups of \(SI_ n(F_2)\)
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description: scientific article; zbMATH DE number 4144257