M_p- Embedded Subgroups On The Group Structure

The main task of group theory is to discern the structure and properties of various groups. The relationship between the properties of primary subgroups and the structure of finite groups has been widely investigated. In particular, the embedded properties of subgroups has became one of the active field in the research of finite groups.In this paper, we shall investigate the structure of finite groups by using Mp-embedded primary subgroups. Some new results about p-nilpotency, p-supersolvability and supersolvability of finite groups are obtained.A subgroup H of G is called Mp-embedded in G, if there exists a p-nilpotent subgroup B of G such that Hp∈Sylp(B) and B is Mp-supplemented in G.The paper divides into the following three chapters.Chapter1, we will introduce the background of group theory and some relevant results.Chapter2, we will introduce some preliminary knowledge of this paper.Chapter3, the main conclusion and the proof.The main results are as follows:Theorem3.3.1Let G be a finite group and P a Sylow p-subgroup of G where p is the smallest prime divisor of|G|. If every maximal subgroup of P is Mp-embedded in G then G is p-nilpotent.Theorem3.3.9Let G be a p-solvable group and p is a prime divisor of|G|. If every maximal subgroup of noncyclic Sylow p-subgroup of Fp(G) is Mp-embedded in G, then G is p-supersolvable.Theorem3.3.13Let G be a finite group, If F*(G) is solvable and every maximal subgroup of every noncyclic Sylow subgroup of F*(G) is Mp-embedded in G, then G is supersolvable.