The Influence Of Generalized Permutable And Generalized Supplemented Subgroups On The Structure Of Finite Groups

Throughout this thesis, all the groups are considered to be finite groups. The main purpose of this thesis is to study the structure of finite groups by using some generalized permutable subgroups and generalized supplemented subgroups.The whole thesis contains five chapters.In Chap I, we introduce the background of our research and list some main results of the dissertation briefly.In Chap â…¡, we introduce some notations, basic concepts and lemmas, which will be used all through this paper.In Chap â…¢, we study the influence of partially s-permutable subgroups on the structure of finite groups. First of all, we give a sufficient condition by using this new concept, under which, every G-chief factor of a normal subgruop of G or the gener-alized Fitting subgroup of this normal subgroup is cyclic. Further, we use this result to give some new theorems, which make some groups belong to some saturated for-mation (Theorem3.2.6,3.3.1). Secondly, we use partially s-permutability to describe the p-nilpotency and p-supersolubility of finite groups (Theorem3.2.7,3.2.9,3.2.11,3.2.14). Finally, we research the relationship between the partially s-permutablity and solubility of finite gropus. According to, for instance, maximal subgroups,2-maximal subgorups of Sylow subgroups, we prove some new criteria about solubility (Theorem3.1.1, Theorem3.2.15, Theorem3.2.16). At the end of this chapter, we discuss about the application of the results achieved above.In Chap IV, we apply the concept of(?)s-quasinormal subgroups to research the structure of finite groups. At the beginning, we use maximal subgrups,2-maximal subgroups to describe the solubility of groups. Later on, we try to use the maximal sub-groups of Sylow subgroups and the generalized Fitting subgroup to get some new results such that a group is supersoluble or belongs to some saturated formation under some assumptions (Theorem4.2.5, Theorem4.2.6). Finally, we talk about the p-nilpotency of a group by the (?)s-quasinormality of n-maximal subgroups, Fitting subgroups, minimal subgroups. A series of new results are obtained.In Chap V, we research the structure of finite groups according to the(?)-supplemented subgroups. In fact, we focus on the criteria of finite p-nilpotent groups. By assuming that the maximal subgroups,2-maximal subgroups,3-maximal subgroups of the Sylow subgroups of a group G and by assuming that the maximal subgroups,2-maximal subgroups,3-maximal subgroups of the Sylow subgroups of a normal sub- group of a group G are(?)-supplemented in G, we get some new criteria of p-nilpotent groups.The method of the theory of finite groups and the theory of class of groups are used throughout the work of this thesis.