OD - Characterization Of Finite Simple Groups L₁₅（2） And S₄（53）

Using some intuitive, simple properties(quantity, prime graph, etc.) of a group to recognize its abstract properties is helpful to learn its properties and structures. There are many kinds of recognition ways of finite simple groups, such as prime graph recognition, non-commuting graph recognition, spectrum recognition and OD -characterization. In this essay, we mainly study the OD - characterization of two finite non-abelian simple groups: the projective special linear group ( )15L2 and the projective symplectic simple group ( )4S53.This essay includes four chapters.At the beginning of the essay, we introduce the background, significance, some notations, concepts and definitions. Then we summarize the research situations of OD -characterization of finite simple groups and list the relevant research results in recent years. The motivation of this study comes from the study and analysis of these essays and the comparison of different research ways. At the end of this essay we list our main results and put forwards some questions.The main results and innovation points of this paper are listed as follows:1. Through analyzing the degree of vertices and the characteristics of the prime graph of15L(2), we characterize it by its order and degree pattern. Furthermore, in order to characterize it, we introduce some new lemmas. It is important to note that the prime graph (( ))15GL2 is connected. Meanwhile,15L(2) is also OG - characterizable and two-order- characterizable.2. Through analyzing the characteristics of the prime graph (( ))4GS53 and degree of vertices of 4S(53), we use the order and degree pattern to prove 4S(53), whose prime graph is unconnected, is OD - characterizable. Meanwhile,4S(53) is also OG -characterizable and two-order-characterizable.3. This paper uses unified tools(order, degree pattern) to characterize both15L(2)and4S(53).