The purpose of this paper is to use the notion of M-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type E6(K) for fields of characteristic two, Beitrag Zur Algebra und Geometry 58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of E6(K) to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type E6 over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup P2 of E6 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 E6, 1, Invent. Math. 89 (1987) 159–195; M. E. Aschbacher, The 27-dimensional module for E6, II, J. London Math. Soc. 37 (1988) 275–293; M. Bani-Ata, On Lie algebras of type F4 and D4 over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group E6(q) which are generated by root-subgroups, J. Algebra 46 (1977) 355–388.].

中文翻译：

---

E6型27维模的二维向量空间在特征二场上的稳定器

本文的目的是使用米-sets (cocliques) 由第二作者在 [S. Aldhafeeri 和 M. Bani-Ata，关于类型李代数的构造乙6(ķ)对于特征二的领域，Beitrag Zur 代数和几何 58(2017) 529–534.] 并使用 Levi 组件和单能自由基子群乙6(ķ)给出类型为 27 维模的二维向量空间稳定器的基本且自包含的构造乙6在特征二的领域。这个稳定器实际上是最大抛物线子群磷2的乙6或 Borel 子群。这种构造是基本的，因为我们使用的只是很少的朴素线性代数概念。有关更多信息，请参阅 [ME Aschbacher, The 27-dimensional module for乙6, 1,发明。数学。 89(1987) 159–195；ME Aschbacher，27 维模块乙6, 二,J.伦敦数学。社会党。 37(1988) 275–293；M. Bani-Ata，关于类型的李代数F4和D4在特征二的有限域上，预印本；B. Cooperstein，小组的子组乙6(q)由根子组生成，J.代数 46(1977) 355–388.]。