The stabilizer of two-dimensional vector space of 27-dimensional module of type E6 over a field of characteristic two

Abstract

The purpose of this paper is to use the notion of [Formula: see text]-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beitrag Zur Algebra und Geometry 58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of [Formula: see text] to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type [Formula: see text] over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup [Formula: see text] of [Formula: see text] or a Borel subgroup. This construction is elementary on the account that we use not more than little naive linear algebra notions. For more information one can see [M. E. Aschbacher, The 27-dimensional module for [Formula: see text], 1, Invent. Math. 89 (1987) 159–195; M. E. Aschbacher, The 27-dimensional module for [Formula: see text], II, J. London Math. Soc. 37 (1988) 275–293; M. Bani-Ata, On Lie algebras of type [Formula: see text] and [Formula: see text] over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group [Formula: see text] which are generated by root-subgroups, J. Algebra 46 (1977) 355–388.].