The Influence Of The Normality And The Condition Of Normalizer Of Subgroups On The Structure Of Finite Groups

It has been one of the important topics to study the structure of the finite groups using the normality of subgroups for a long time. In particular, after finite simple groups were classified. Not only many new concepts have been introduced but also fruitful results have been ob-tained, which pushed forward the study and the development of finite groups theory. The concept of normal subgroup is most fundamental and the normalities of subgroups are very important in the research of finite groups. In the first few parts of this thesis, we generalize the normalities of subgroups and then get some weaker normalities of sub-groups, give the dual concept of conjugate permutable, self-conjugate permutable and the dual of normality, the normalizer condition. And we study the structure of finite groups by using these weaker normali-ties of subgroups, embedded subgroups and the normalizer condition. It consists of five chapters.Chapter 1. We introduce some notations and list some known re-sults needed in the paper.Chapter 2. We introduce a new embedded subgroups concept, self-conjugate permutable subgroup and we investigate new characterations of solvabel T-groups, minimal non-T-groups and give the structure of groups whose maximal subgroups of even order are PSC-groups by con-sidering self-conjugate permutable subgroup. In particular, we give new characteration of some special simple groups and non-solvable groups by considering self-conjugate permutable subgroup and by Wujie Shi's results about the structure of subgroups of special liner groups. A part content of this chapter will be published in Communication in Algebra, Joural of Algebra and its application and Acta Mathematica Sinica, En-glish Series. Chapter 3. We define the SNS-permutable subgroups, SS-quasinormally embedded subgroups and weakly SS-quasinormally embedded subgroups by permutablity and complement. Furthemore, we investigate the struc-ture of the finite groups whose the maximal subgroups and the elments of the set of subgroups with the same order satisfy the SNS-permutability, SS-quasinormally embedded subgroups and extend related results to for-mation. A part content of this chapter will be published in Publpcation. Mathematics. Debrecen.Chapter 4. We investigate the influence of the quasinormally embed-ded subgroups,H-subgroups, SS-quasinormality of maximal subgroups of Sylow subgroup P with rank d (i.e., the smallest generator number) on the structure of finite groups and obtain some new criterions for p-nilpotency and supersolubility of finite groups. The obtained results improve, unify and generalize some recent related results. In particular, we solve a open problem of Li Shirong and improve Duan Xuefu, Chen Zhongmu, Zhang Jiping and Zhang Laiwu giving the idea of reducing the maximal set of Sylow subgroup. In 1984, they gave the idea of reducing the maximal set of Sylow subgroup. They also considered the smellest generator number d and gave DCZZ(P) is a set of maximal subgroups of Sylow subgroup P such that|DCZZ(P)|= f(d)= (pd-2-1)/(p-1) (In fact, the number of all maximal subgroups of Sylow subgroup P is (pd-1)/(p-1)). Moreover, we have|f(d)|》d when d→∞.At the same time, we give a example such that the smallest generator number d is the best possible. A part content of this chapter will be published in Journal of Group Theory, Fronter Mathematic in China and Algebra Collq.For the dual of normality. In 1934, Baer define the concept of Norm. Let G be a group and the Norm of G by N(G)= (?)H≤GNG(H).Chapter 5. We generalize the concept of Norm and give the in- tersection of normalizer of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups, i.e., the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpo-tent residual of subgroups. In particular, we generalize the p-nilpotent criterion of Ito and Burnside and give the p-lentgh of p-solvable. At last, we study the Norm of non-cyclic subgroups, the dirived subgroups of subgroups and the nilpotent residual of subgroups and obtain some basic results of infinite groups. We first investigate the structure of the infinite groups by Norm. A part content of this chapter will be published in Journal of Algebra and Journal of Keroa Mathematic Soc.