The Effect Of The Number Of Conjugacy Classes Of Finite Groups On The Structure Of Groups

In the study of finite groups,the order of groups and the number of conjugacy classes of elements of groups are two very important quantities of groups.These two quantities have a great influence on the structure and properties of groups.Many finite groups can be determined by these two quantities.For finite groups with order n,n belongs to 2p,4p,23p,pq,2pq,p3,2p2,3p2(p,q are odd prime numbers),according to the classifica-tion theorem,there must be more or less mutual isomorphism.For these n-order groups,their conjugacy classes are determined by using their generators and generative relations,number theory and group theory knowledge,and the number of their conjugacy classes is obtained.Further comparing the number of conjugacy classes of these n-order groups,the minimum number of conjugacy classes of n-order groups is obtained.The minimum number of conjugacy classes of order groups and the order of groups are compared by using the classification of n-order groups.The concrete structure of n-order groups with the minimum number of conjugacy classes is determined.