The Influence Of Local Properties Of Finite Groups On Group Construction

The main task of group theory is to discern the structure and properties of various groups. Now,relationship between properties of primary subgroups and structure of finite groups has been widely investigated. In particular, the local properties of subgroups has become one of the important means in the research of finite groups.In this paper, we shall investigate the structure of finite groups by using local properties of maximal subgroups of some primary subgroups. Some new results about p-nilpotency,p-supersolvability and supersolvability of finite groups are obtained.The paper divides into the following three chapters.Chapter1, we will introduce the background of group theory.Chapter2, we will introduce some preliminary knowledge of this paper and the main lemma.Chapter3, the main conclusion and the proof.The main results are as follows:Theorem3.1.1Let 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 weakly M-supplemented in NG(P), and P’is s-permutable in G, then G is p-nilpotent.Theorem3.2.1Let F be a saturated formation containing U, and let N be a solvable normal subgroup of G such that G/N∈F. If every maximal subgroup of every noncyclic Sylow subgroups of F(N) is a SCAP subgroup of G or M-permutable in G, then G∈F.Theorem3.2.4Let F be a saturated formation containing U, and let N be a normal subgroup of G such that G/N∈F. If every maximal subgroup of every noncyclic Sylow subgroups of F*(N) is a SCAP subgroup of G or M-permutable in G, then G∈F.