The Classification Of Some Finite Simple Groups With Complete Graphs

It is well known that finite simple groups,is the basis of finite groups,and it’s group quantitative and structural characteristics has become the core of modern mathematics.A finite group G is called a group with complete prime graph,if and only if all the connected components of its prime graph are cliques.That is to say if r,s∈πi(i=1,2,…),then r～s.Now we have many ways to describe finite simple group,this paper focuses on a plain description,to satisfy a vertex(5,7)degree has full pigment figure 1 classifies the groups.This paper first briefly introduced the background and significance of the study of finite groups,summed up the meeting the importance of certain characteristics of finite simple groups,then introduced to the basic concepts and definitions of symbols.By reading lots of documentation is needed to prove that conclusion ideas and methods,so as to expand their research and ideas.Main achievement and innovation of this paper as follows:1.By using an adjacency criterion for a finite simple group,grain classification of connected components,as well as for several kinds of common solution of the Diophantine equation,we classify the complete prime graph G satisfying 5∈π(G)and dG(5)= 1 and given the evidence.2.We classify the complete prime graph G satisfying 7∈π(G)and dG(7)= 1,and given the evidence.3.Innovation of this paper is to vertex properties of currently focuses on vertex degree is 0(outlier),this paper has to classify the complete prime graph G with degree of a vertex is 1.