On Conjugacy Classes Craphs With No More Than Five Vertices And Related Questions

Finite group theory is the foundation of group theory,and it is the most widely used branch in group theory.The influence of conjugacy class on group structure is also studied extensively.In this paper,the group structure is studied by a graph of conjugacy class.The graph of conjugacy class?38??G?is the undirected graph that satisfies the following conditions:?1?The vertices are all the non-central conjugacy classes in the group G.?2?There is an edge between the two vertices if and only if the sizes of their conjugacy classes are not coprime.In this paper,the conjugacy class graphs with no more than 5 vertices are classified.When a conjugacy class graph has no vertice,if and only if the group is an abel group.There is no conjugacy class graph with one vertice.A conjugacy class graph has two vertices if and only if the group is isomorphic toS₃.There are 2 conjugacy class graphs with 3 vertices,and 4conjugacy class graphs with 4 vertices,and 4 conjugacy class graphs with 5 vertices.Using the method of group classification,the group is divided into nilpotent group,quasi-Frobenius group with abelian kernel and complement,other solvable groups and non solvable group.How the non-central conjugacy classes of a finite group influence its structure is investigated and the finite groups with at most 9 non-central conjugacy classes are classified.The conjugacy class graphs has at most two connected components.When the number of connected components is 2,the group is the quasi-Frobenius group with abelian kernel and complement.When the number of connected components is 1,the conjugacy class graphs contains at least one triangle.If a non solvable group G contains no more than 8 non-central conjugacy classes,the conjugacy class graphs of Gis complete.If a nilpotent group G contains no more than 9 non-central conjugacy classes,the conjugacy class graph of Gis complete.