The Arithmetical Conditions Of Conjugacy Classes And The Structures Of Finite Groups

It is an important subject in the theory of finite group to study how thearithmetical conditions of conjugacy classes of a finite group influence its structure.Many scholars have studied these respects and obtained many important results.This thesis continue to consider some problems of influences of conjugacy classes ofa finite group on its structure. Let G be a finite group and N a normal subgroup ofG, Shi wu-jie, Wang jin, M. Shahryari , M. A. Shahabi and U. Riese have studied thestructure of N when N is the union of 2, 3 and 4 conjugacy classes of G respectively,Qian guo-hua, Shi wu-jie and You xing-zhong consider the conversely extreme case,they have studied the structure of G when G have at most 3 G- conjugacy classesoutside N. On the basis of work of Qian guo-hua, Shi wu-jie and You xing-zhong,we continue to consider the influence of the number of conjugacy classes outside anormal subgroup of a finite group on its structure.In Chapter 1, we mainly introduce the works related to this thesis and contentand method to study in this thesis.In Chapter 2, we give some necessary preliminaries, including some basic con-ceptions ,lemmas and their proofs.In Chapter 3, let G be a finite solvable group and N a normal subgroup of G,we investigate the structure of G when there are exact four conjugacy classes of Goutside N. We study three cases When G is abelian, G is non-abelian but G/N isan abelian group of order great than 2, and G/N is non-abelian respectively.In Chapter 4, we study the influence of the number of conjugacy classes outsidethe center of a finite group on its structure. Let G be a finite solvable group, weknow that it is difficult to completely characterize the structure of G when there areexact 4 G- conjugacy classes outside a normal subgroup of G in Chapter 3, hence weconsider the case that the normal subgroup of G is the center Z(G) of G and deletethe condition of solvability of G, we investigate the structure of G when there are atmost five conjugacy classes of G outside Z(G) and give complete characterizationof the structure of G.