N_p- Complemented Subgroups On The Group Structure

It is well-known that the supplementation of some primary subgroups plays a crucial role in the study of the structure of finite groups. Many scholars studied the structure of finite groups by supplementation and permutability of Sylow objects (primary subgroups, normalizers and centralizers of primary subgroups), and obtained some new results.In this thesis, we main study the influence of some certain subgroups of G on the structure of finite groups and consider the relationship between the Mp-supple-mented subgroup of p-nilpotent subgroup of G and the structure of finite groups.A subgroup H of G is called M p-supplemented in G if there exist a subgroup B of G such that G=HB and TB<G for every maximal subgroup T of H with|H:T|=pα.The thesis divides into the following three chapters.Chapter1, we will introduce the background of group theory and some relevant results. Moreover, we will bring our main results.Chapter2, we will give rise to the preliminaries including basic concepts and main lemmas which will be used in this thesis.Chapter3, as our main work, we will present the proof of our main results in detail. The main results are follows:Theorem3.1Let G be a finite group and H be a p-nilpotent subgroup containing a Sylow p-subgroup P of G, where p is the smallest prime divisor of|D|. Suppose H has a subgroup D with Dp≠1and|H:D|=pα. If every subgroup T of H with order|D|is Mp-supplemented in G, then G is p-nilpotent.Theorem3.3Let G be a p-solvable group and H be a p-nilpotent subgroup containing a Sylow p-subgroup P of G. Suppose H has a subgroup D with Dp≠1and|H:D|=pα. If every subgroup T of H with order|D|is Mp supplemented in G, then G is p-supersolvable.Theorem3.7Let G be a p-solvable group. Suppose D is a subgroup of Fp(G) containing Op’(G) and Dp≠1. If every subgroup T of Fp(G) with order|D|is Mp-supplemented in G, then G is P-supersolvable. Theorem3.8Let F be a saturated formation containing U. Suppose G has a normal subgroup N such that G/N∈F and F*(N) is solvable. If F*(N) has a subgroup D with Dp≠1for any p∈(F*(N)) and every subgroup E of F*(N) with|T|=|D|is Mp-supplemented in G, then G∈F.