Conjugation Class Length Of Real Elements And Zero Elements And The Structure Of Finite Group

In the research of finite group theory,there are many relationships between the arithmetic information of groups and the structure of groups,for example,orders of groups,orders of elements in groups,conjugacy class sizes and the degree of characters,etc.We characterize the structure of finite groups by their conjugacy class sizes.In the early research of conjugacy class sizes,group theorists characterized the structure of groups by conjugacy class sizes of all elements.Moreover,they found that conjugacy class sizes of some key elements can control the structure of groups,such as real elements,vanishing elements,elements with prime power order and so on.In this thesis,we introduce some basic concepts and common propositions in the research of conjugacy class sizes,and review a number of classical results about conjugacy class sizes,real elements and vanishing elements.Based on the previous results,we study the structure of finite groups by the real elements of prime power orders and the vanishing elements of prime power orders,discuss properties of these two special elements,characterize solvability and supersolvability of finite groups,generalize a few existed results and raise some further problems about them.