Conjugation Class Length And Finite Group Structure Of Special Elements

In the process of development of theory of finite groups,people find strong re-lationship between the structure of finite groups and some arithmetical conditions of the size of conjugacy classes.Group theoretists make a deep study about the size of conjugacy class and obtain many results.Especially,when the number of the size of conjugacy class is constant,group theoretists describe the structure of finite groups and classify the type of groups.Firstly,group theoretis consider all elements of finite groups to study the structure of finite groups.In order to reduce the number of elements,group theoretists improve the meth od from many aspects.For example,the elements of prime power order,the p-regular elements,vanishing elements and so on.In the paper,we also investigate the structure of group in the way of some special elements,such as the elements of prime power order,p-regular elements,vanishing elements.In particular,In the chapter 3,we describe the structure of finite groups by the p-regular elements of prim-power order of groups,and obtain some sufficient conditions that the finite groups are p-nilpotent group and supersolvable group.What’s more,we give some generations of some known results.In the chapter 4,we mainly study the structure of the finite p-solvable group with three the sizes of primary,biprimary orders of p-regular conjuagcy class.In the chapter 5,the author study the influence of vanishing elements of prime-power order on the structure of groups.Generalizations of some theorems will be given.