A Characterization Of Lie Type Simple GroupsE₆(2) And E₆(3)

In recent years, using subgroups and quotients to describe the nature of finite groups has become a hot topic. Many scholars have made a lot of work, and got a large number of results. Finite simple group is the cornerstone of the structure of finite groups, with the number of properties of finite simple groups to characterize a finite simple group, help us to more effectively study the finite simple group. This paper is intended to study the following properties of finite simple groups: the order of their Sylow normalizers.The author of the paper divides the paper into four chapters. The main content can be listed as follows.In chapter 1, it is described that the development of group theory and the development of classification of finite simple groups, and the problem is proposed at the end of this paper.In chapter 2, preparatory work will be used on some symbols and concepts and common theorem, also other preparatory work.In chapter 3, some simple groups are proved to be isomorphic to E 6(2)and E 6(3) by the order of their Sylow subgroups'normalizers. Some of the results partially or completely extend the predecessors'results in content and methods. And we get the following results:Theorem 3.1 Suppose G is a finite group with |N_G(R₁)|=|N_E6(2)_R2|for every primer , where R1∈Syl rG and R2∈Sylr( E6(2)), then G≌E6(2).Theorem 3.2 we shall characterize E 6(3) under the same conditions as theTheorem 3.1. Also we give a completely different method to characterize E 6(3). The following result is proved:Theorem 3.2 Suppose G is a finite group with |N_G(R₁)|=|N_E6(3)_R2|for every primer , where R1∈Syl rG and R 2∈Sylr( E6(3)), then G≌E6(3).Normalizers are found by N C theorem in this paper. From that point, finite simple groups are characterized. Also, in the end of the paper we make two problems to be settled.In chapter 4, prospect is given.