Study On The Classification Of The Prime Graph Of Finite Group With Five Vertices

The study of the property and structure of finite group is an important part of the finite group theory.In recent years,many scholars realized finite group using arithmetical conditions and obtained some important results.This article uses prime graph which is weaker than group order and element order to study the properties of finite group.Prime graph of a finite group is defined as follows:（1）the vertex set is prime divisor of group order;（2）two different vertices p and q are adjacent if and only if the group contains element whose order is pq.Firstly,the thesis obtained 34 kinds of non-isomorphism and undirected graph of five vertices and be divided into 11 categories,according to the number of edges.Then analyzing them,it obtained the non-isomorphism graph with only five vertices is not prime graph and the others are prime graphs.At last,it analyzed the properties of finite group of prime graph with special structure and had the following properties.Prime graphs with five vertices contain at least one edge.Finite groups which prime graphs are tree or chain are unsolvable.Finite groups which prime graphs are loop are solvable.Finite groups which prime graphs contain at least three connected components are unsolvable.The prime graphs of solvable groups which the number of connected components is two must be cliques.Studying by 33 kinds of prime graph with five vertices,it reached the following conclusions.If prime graph contains chain,the length of chain is at most five.The prime graphs of alternating groups and sporadic groups are not trees.Finally,the paper realized prime graphs with only three or four vertices and obtained that simple groups whose prime graphs with only three vertices are isomorphism to A₅,A₆,L₂（8）,L₂（17）,L₃（2）,simple groups whose prime graphs with only four vertices are isomorphism to A₂（4）²,B₂（8）²,B₂（32）and simple groups whose prime graphs with three vertices and one edge are isomorphism to M₂₂ and some of the Suzuki groups.