The Study Of Arithmetical Properties Of Conjugacy Classes 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 class-es, which is also an important means to study finite groups. In finte groups, the sizes of congjugacy classes, the number of congjugacy classes and the orders of the representatives of conjugacy classes are some important arithmetic quantities, In this paper, we mainly consider how the number of noncentral conjugacy classes and co-primeness among the orders of the representatives of noncentral conjugacy class-es of a finite group influence its structures respectively. We divided this paper into four chapters. The main of the paper is the following:In Chapter 1, we mainly introduce the works related to this thesis and problems that will be solved in this thesis.In Chapter 2, we list some basic concepts,lemmas and proofs of the lemmas related to this thesis.In Chapter 3, we investigate how the number of noncenteral conjugacy classes of a finite group influences its structure. Let G be a finite group and Z?G? the enter of G, let g be an element of G, denote by gG the conjugacy class of G in which the representative is g. For a nonempty subset A of G, let k_G?A? be the least positive integer k such that A is a subset of the union of k conjugacy classes of G. It is clear that k_G?A?=|{x^G|x^G ? S?0,x?G}|. Let N be a. normal subgroup of G, Shi wujie, Wang jing, M. Shahryari, M. A.Shahabi and U. Riesee studied the structures of the group N in case k_G?N?= 2,3,4 respectively, Qian guohua, Shi wujie and You xingzhong consider the converse extreme cases, they investigated the structure of G in case k_G{G-N)?3. On the basis of the work of Qian guohua, Shi wujie and You xingzhong, Chen weimin and Zhu weihua studied the structure of G in case k_G{G-N)=4,5 and classify the finite groups satisfying k_G?G-Z?G??=5 or 6, respectively. In this chapter, we classify the finite groups satisfying k_G?G-Z?G??=7.In Chapter 4, we investigate how the co-primeness of the orders of the repre-sentatives of noncenteral conjugacy classes of a finite group influences its struc-ture. Let G be a finite group and g an element of G, denote by o?g? the or-der of g. Let g^G,g₁^G,g₂^G,…g_n^G be n distinct noncentral conjugacy classes of G, if ?o?g₁?,o?g₂?,…,o?g_n??=1, that is, the orders of g₁,g₂,…,g_n are setwise co-prime, then we say G has the property P_n. You xingzhong studied the structures of finite groups satisfying property P₃ and P-4, respectively, on the basis of his work, we investigate the structures of finite groups with property P-5.