A Characterization Of Lie Type Group E₇(3)

To characterize finite groups by quantity properties is very significant in the theory of finite groups. Also it is always an important method and interesting subject to study finite groups by the normalizers'orders of their Sylow subgroups. As mentioned above, by studying the prime graph components of finite groups and putting N C theorem in use, we find the final theorem.There are three chapters in this paper,In chapter 1, we introduce the symbols and some of the basic concepts of finite group theory at first, and then we talk about the summaries of simple groups of Lie type and the group E₇ (q). At last we point out the main method in order to solve this problem.Chapter 2 is the center of this paper. We characterize the finite group of Lie type E₇(3) by the normalizers's orders of its Sylow subgroups, and get the following theorem,Theorem. Let G be a simple group and G = E₇(3). If |N_G(R₂)|=|N_E7(3)(R₂)|for every prime r , where R₁∈Syl_rG and R₂∈Syl_r( E₇(3)), then G≌E₇(3).In the last chapter, the final conclusion is given and the problems requiring further studies are discussed.