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

A classical research field in the theory of finite groups is the analysis of the interplay between the structure of a group and the conjugacy class sizes.As a key tool in this study,many authors considered the prime graph of conjugacy class sizes.In this paper,we denote it: the vertices are prime factors of the conjugacy class sizes,and two vertices p,q are adjacent if and only if there is an element in conjugacy class size that is divisible of pq.In this paper,we mainly study the classification problem of the prime graph of conjugacy class sizes with five vertices.Firstly,we determine the non-isomorphic graphs with five vertices are 34.By the properties of the connected component number at most 2 and the diameter not greater than 3 of the prime graph of conjugacy class sizes,20 graphs can be ruled out.And then using GAP to analyze the group within the order of 10000,there are 10 graphs corresponding groups.There are 2 graphs of the prime graph of conjugacy class sizes with diameter of 3,then the corresponding groups are semidirect product of two abelian groups of coprime order,and has some additional properties;There are 6 graphs of the prime graph of conjugacy class sizes with diameter is 2,since their complete vertices is not greater than 2,can be concluded that their corresponding groups are solvable groups;In addition,when the number of connected components of the graph is 2,there is only one graph corresponding to group is quasi-Frobenius with abelian kernel and complements,also proves that the non-central conjugacy classes set is?2,n?,and1 23 4n=p q r s? ?? ?,(1,,4)i?i(28)is positive integer,p,q,r,s distinct odd prime numbers.Then,we prove a sufficient condition for the group of the prime graph of conjugacy class sizes to be solvable group,that is,the number of vertices is at most 5,only the group corresponding to the prime graph of conjugacy class sizes is a complete graph,there is a non-solvable group,other 9 graphs corresponding groups are solvable group.In the end,we proved that the Thompson's conjecture is valid for the non-abelian simple group3C(3).