Conjugate Class Lengths And Structures Of Finite Groups Of Two Special Elements

In the process of development of finite group theory,it is found that some arithmetical conditions of the conjugacy class sizes of elements are closely related to the structure of groups.Many group theorists have studied some arithmetical conditions of the conjugacy class sizes of elements in finite groups and obtained lots of research results.At first,group theorists characterize the structure of finite groups by the conjugacy class sizes of all elements.Later,group theorists characterize the structure of finite groups by reducing the number of elements,and only consider the conjugacy class sizes of some elements.For example,real elements,the elements of prime power order,vanishing elements and so on.In the paper,on the basis of previous results,we continue to study the structure of finite groups by the conjugacy class sizes of some special elements,such as the real elements of prime power order,the vanishing elements of prime power order.In particular,in chapter 3,we characterize the structure of finite groups by studying the conjugacy class sizes of real elements of prime power order.In chapter 4,we characterize the structure of finite groups by studying the conjugacy class sizes of vanishing elements of prime power order.In chapter 5,we present the application of the main theorems and the problem that the structure of finite groups can be further studied by using the conjugacy class sizes of real elements and vanishing elements.