The 27-dimensional module for \(E_ 6\). IV
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.
scientific article; zbMATH DE number 4144257
scientific article