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

It is an important subject in the theory of finite groups to characterize the structures of finite groups from some arithmetic properties of their conjugacy classes, which is also an important means to study finite groups. The sizes and number of congjugacy classes of a finite group are two important arithmetic properties, this thesis mainly consider how the number of conjugacy classes outside a normal subgroup of a finite group influence its structures.Let G be a finite group, N a normal subgroup of G and Z(G) the center of G. Denote by ka(N) the number of conjugacy classes of G contained in N and by ko(G-N) the number of conjugacy classes of G outside N. Shi wujie, Wang jing, M. Shahryari, M. A. Shahabi and U. Riese have studied the structure of N when ko(N)=2.3,4, respectively. Qian Guohua, Shi Wujie and You Xingzhong considered the opposite extreme situation, they studied the structure of the finite group G when ka(G-N)≤3, Chen Weimin discussed the structures of the finite group G in the cases of kG(G-N)=4and ka(G-Z(G))=4or5, respectively. On the basis of the researches of Qian Guohua, Shi WuJie, You Xingzhong and Chen Weimin. we continue to investigate the structures of the finite group G for the cases of kG(G-N)=5and kG(G-Z(G))=6respectively.This thesis consists of four Chapters. In Chapter1, we mainly introduce the works related to this thesis and problems that will be solved in this thesis. In Chapter2, we list some basic concepts,lemmas and proofs of the lemmas related to this thesis. In Chapter3, we study the structure of finite solvable group G when there are five conjugate classes of G outside a normal subgroup N of G. In Chapter4, we investigate the structure of the finite group in which there are exactly six non-central conjugacy classes and give a complete classify of such groups.