The Graph Of Conjugacy Class Sizes With Five Vertices

The size of conjugacy class is very important to the studying of finite groups in group theory. In recent years, many scholars are researching the subject and have a lot of outstanding achievements. In this paper, we study the structures of the finite groups according to the graph of conjugacy sizes. First, the graph of conjugacy class sizes has vertex set Cl(G) and an edge between two vertices if they are not coprime.By routine combination calculation, we know there are 34 graphs with five vertices.According to some characteristic properties of the graphs of conjugacy class sizes, 19 graphs can be ruled out. In other words, there are not groups to correspond to this 19 graphs. Then using GAP to find large amounts of data about groups, seven graphs have the corresponding groups, and I make a summing up of the smallest order groups of this 7 graphs in the small groups library(those of order at most 2000). In addition, only the complete graph has examples of nonsolvable groups.A theorem of conjugacy class number and size is given and proved: When the graph of conjugacy class sizes has only one vertex, the number of conjugacy classes is five at least; if the graph of conjugacy class sizes has only two vertices, the number of conjugacy classes is three at least; the number of conjugacy classes is five at lowest when the graph of conjugacy class sizes has only three vertices; the minimal number of conjugacy classes is six if the graph of conjugacy class size has only four vertices; when the graph of conjugacy class sizes has only five vertices, the number of conjugacy classes is at least seven. Then the proof of a special graph of conjugacy class size with five vertices is provided: For any prime number p,if the nonsolvable group G has four conjugacy classes at most and the size of conjugacy classes is p times, G is isomorphis to5 A or)7(2L, and the diameter of the graph with 5vertices is 3.