Acharacterization Of Lie Type Simple Groups E₈(3)and E₈(5)

The quantitative characterization of finite groups has a very important position in the research of the finite groups. This is because some fundamental quantitative characterization such as the orders of groups, its elements and its subgroups has a certain effect on the characters and structures of the groups.But so far, there are still some Lie type simple groups which have not been characterized. The purpose of this article is to do some further research on the characterization of some special Lie type simple groups. We use the orders of normalizers of the Sylow - p subgroups of the finite simple groups to characterize E₈(3)and E₈(5) which belong to Lie type simpe groups.The paper is divided into three chapters. The main content can be listed as follows.In the first chapter, we first introduce some usual symbols, especially the basic concepts and some basic theorems such as Sylow theorem, group of isomorphism theorem, Frattini theorem and so on. Then, we introduce the research background of the finite simple groups and some research results. Finally we put forward the question which is researched in this paper.In the second chapter, we use some of the elementary number theory and the knowledge of orders of normalizers of finite simple groups Sylow - p subgroups.We give a new method to characterize E₈(3)and E₈(5).Thus we reach the following conclusions:In the third chapter, we mainly illustrate some conclusions and put forward series of problems unsolved and discuss the further work direction.