The Influence Of Several Generalized Normal Properties Of Subgroups On The Structure Of Finite Groups

Many properties of finite groups can be characterized by their sub-group's generalized normal properties. The relationships between the structures of a finite group and it's generalized normal subgroups, is one of the most important topics in the studying of finite groups. In the process of research, many new generalized notions of normality were introduced successively, this indeed pushed forward the development of group theory. The Sylow subgroup's maximal subgroups,2-maximal subgroups, minimal subgroups and 2-minimal subgroups are several kind of very important subgroups, which playing important roles in investi-gating the structure of a finite group. From their generalized normal properties, the scholars have got a lot of classical characterizations about the structure of a finite group. In this paper, we introduce several new generalized notions of normality, and characterize the p-nilpotency and supersolvability of a finite group by it's generalized normal subgroups with prime power order. We also get some results about the saturated formations. This paper is organized as the following five chapters:In Chapter 1, we introduce some symbols, basic concepts, important lemmas and conclusions which will be used in the thesis.In Chapter 2, we mainly study the relationships between the struc-tures of a finite group and it's semi cover-avoiding subgroups. We give out some characterizations about the p-nilpotency and supersolvability of a group G, some new results are obtained. In Section 1, by con-sidering the semi cover-avoiding properties of some minimal subgroups or 2-minimal subgroups, we obtain several sufficient and necessary con-ditions about a group belonging to a saturated formation. In Section 2, we mainly discuss the case that some maximal sugroups of a Sylow subgroup of G having the semi cover-avoiding property, but the others having the s-permutably embedded property. Our results improved and covered many nice ones.In Chapter 3, we define a new kind of generalized normal subgroup: weakly s-supplemently embedded subgroup. By assuming some maximal subgroups,2-maximal subgroups, minimal subgroups or 2-minimal sub-groups of some Sylow subgroups of G have the weakly s-supplemently embedded property, we obtained some new characterizations about the (ρ-)nilpotency and super solvability of G. Our main results are extended to saturated formations. In addition, we also discuss the case that some maximal subgroups of a Sylow subgroup of G having the weakly s-supplemently embedded property in it's normalizer. Our results unified and generalized a lot of meaningful works.In Chapter 4, we mainly investigate the influence of H-subgroups with the same order on the structrues of an even order group. Our results completed a remained problem which was proposed by professor Guo Xiuyun etc. on a paper published on the Journal of Group Theory.In Chapter 5, we introduce the concept of S(?)-supplemented sub-group. By assuming some primary subgroups of a group G with the same order have the S(?)-supplemented properties, we obtain some new characterizations about theρ-nilpotency and supersolvability of G. We also get some neat results about formations.