Weak M - Complement Properties And Group Structure Of Given Subgroups

The central task of group theory is to study the structure of group and the supplementation of some primary subgroups plays a crucial role in the study of the structure of finite groups. Many scholars deeply studied the structure of solvable groups, supersolvable groups, nilpotent groups by supplementation and permutability of subgroups and obtained some new results.In this thesis, we mainly study the influence of some certain subgroups of G on the structure of finite groups and consider the relationship between the weakly M-supplemented subgroups of primary subgroups of G and the structure of finite groups.The thesis divides into the following three chapters.Chapter 1, we will introduce the background of group theory and some results related to our work.Chapter 2, we will give rise to the preliminaries including basic concepts and main lemmas which will be used in this thesis.Chapter 3, The main conclusions and the proofs.In this paper,we mainly consider the specific single prime number 5 and 7 and combine with the nice properties of weakly M-supplemented subgroup. It reveals the structure of finite groups from the point of composition and extends the results to order D. Then by considering the normal prime p 、 maximal subgroup and minimal subgroup,besides that, this paper reveals the structure of finite groups by chief factor and upgrade the results to the hypercenter of normal subgroups.