Special Conjugacy Classes Long Finite Group And Sub-group Of Generalized Permutation,

The topic of this dissertation is the investigation of the influence of the special conjugacy class sizes and generalized permutable subgroups on the structure of finite groups.In Chapter I, we introduce the research background and main results in this thesis.In Chapter II, we introduce some concepts, notations and known results which will be used frequently in the sequel.The Chapter III is devoted to investigate the effect of the special conjugacy class sizes on the structure of finite groups, especially simple groups. J. G. Thompson, noted for his work in the field of finite groups and awarded the Fields Medal in1970, the Wolf Prize in1992and the2008Abel Prize, conjectured that if G is a group with Z(G)=1, M a non-abelian simple group and cs(G)=cs(M), where cs(G) denotes the set of all conjugacy class sizes of G, then G-M. It has been proved the conjecture holds for PSLn(q), A11and all simple groups with non-connected prime graphs. In this conjecture, all the conjugacy class sizes of a group are involved. Here we want to weaken this condition and consider only some special conjugacy class sizes, such as the largest conjugacy class sizes and the smallest conjugacy class sizes larger than1. In this dissertation, we prove that the sporadic groups can be determined by their largest conjugacy class sizes or smallest conjuagcy class sizes larger than1. We weaken essentially the condition of Thompson's conjecture in the sense that the two groups G and M are of the same order under the hypothesis of this conjecture since Professor G. Y. Chen has proved that this conjecture holds for simple groups with non-connected prime graphs.The last chapter consists of three sections and includes the discussions of the generalized permutable subgroups. We first introduce a new kind of generalized permutable subgroups, that is the weakly S-embedded subgroups. We prove some properties of weakly S-embedded subgroups and present some relations between this kind of subgroups and the supersolubility and p-nilpotency of groups. In the second section of this chapter, we introduce the concept of the A-supplemented subgroups. The influence of the A-supplementary of some primary subgroups on the structure is investigated and some new characterizations of p-supersolubility of finite groups in terms of A-supplemented subgroups are obtained. At the end of this chapter, we study further the X-s-semipermutable subgroups and their relations with the structure of finite groups. In connection with X-s-semipermutable subgroups, some new characterizations about the finite groups are given. In this research, we mainly discuss how the global properties of groups can be obtained from the properties of some subgroups of Sylow subgroups, generalized Fitting subgroups and local subgroups. Moreover, we make a study of the structures of groups in the framework of formation theory, whose given subgroups possess some generalized permutability. The ideas and methods in the study of formation theory are applied extensively in our investigations.