The Generalized Normality Of Subgroups And The Structure Of Finite Groups

In this paper, we study the relationship between the generalized normality of somesubgroups and the structure of finite groups. This paper is organized as the followingfive chapters.In Chapter 1, we introduce some symbols and results that we usually use in thepaper.In Chapter 2, we study the relationship between the semi CAP -or c-supplementedsubgroups and the structure of finite groups. We obtain some su?cient conditionson p-nilpotency and supersolvability of groups by using the semi cover and avoid-ance property or c-supplementation of the maximal subgroups of the Sylow subgroups.Meanwhile, we get some results about certain formation.In Chapter 3, we investaget the relationship between the join of a pair of conjugatesubgroups and the structure of finite groups from four aspects.(1) We investigate the properties of G from the properties of the subgroups whichare generated by two conjugate elements. We get some interesting results and alsoimprove some results of Baer and Thompson.(2) We discuss the in?uence of the index of the subgroup H in H, Hg on thestructure of G and get some interesting results.(3) We study the structure of finite groups by the maximality of a subgroup H inH, Hg . We get some interesting results.(4) We consider the in?uence of the generalized normality of the subgroup H inH, Hg on the structure of G. We introduce a new generalized normality of subgroups(CSQ-normal), and obtain a criterion for nilpotency and supersolvablity of groups byusing the CSQ-normality of subgroups. Our results unify and generalize some earlierresults and we proved that an even QCLT -group is also supersolvable if the maximalsubgroups of its Sylow 2-subgroups are CSQ-normal subgroups of G.In Chapter 4, we study the relationship between theθ-pairs of proper subgroupsand the structure of finite groups. We investigate directly the conditions of normal θ-pairs of subgroups with the same order of Sylow subgroups of a normal subgroup Nof G such that G becomes supersolvable and nilpotent. We generalize some results.In Chapter 5, we continue the work of master’s thesis, study the relationshipbetween the weakly s-semipermutable subgroups and the structure of finite groups.We obtain some su?cient conditions on p-nilpotency of groups by using the weaklys-semipermutability of the maximal subgroups of the Sylow subgroups. Many knownresults on this topic are generalized.