On The Study Of The Sizes Of Conjugacy Classes And The Structure Of Finite Groups

In the study of representations of finite group, lots of evidence suggested that there is a great relationship between the structure of conjugacy classes and the character degrees of finite group. In this paper, we consider some opposing problem about character degree of finite groups. It consists of the following three chapters.In the first chapter, we introduce the problems which we will discuss in this paper and the related results in this paper.In the second chapter, we study the relationship between the fitting height and conjugacy class graphΓ^*(G) of a solvable group. HereΓ*(G) is a graph whose vertices are the prime divisiors of conjugacy classes size, and two vertices are connected if they both divide a size of some conjugacy classes. We say that the conjugacy class graphΓ^* has bounded Fitting height if there is a bound on the Fitting heights of the solvable groups whose conjugacy class graph isΓ^*. In this chapter, we showed thatΓ^* has bounded Fitting height if and only ifΓ^* has at most one vertex of degree n-1. We also found the bounds of some conjugacy class graph which have bounded Fitting height.In the third chapter, we study the structures of 11- and 12-decomposable non-perfect finite groups. A finite group G is called n-decomposable if every proper non-trivial normal subgroup of G is a union of n distinct conjugacy classes of G. We showed that if G is a non-perfect 11-decomposable finite group, then G is isomorphic to an Abel group of order 121, or a non-Abel group of order pq, here p, q are primes which satisfy p-1-=10q or p=11, q=(?),r is a prime, too. If G is a 12-decomposable finite solvable group, then G is ismorphic to D₄₆.In the forth chapter, we study some arithmetical conditions on the size of conjugacy classes of a finite group. And showed that |cs(G)|≤2k^*(G)+1 in many cases, where k^*(G):=(?){|cs_p(G)|} and cs_p(G) is a set of all elements in |cs(G)|that can be divisible by a prime p.