M - Almost Complemented Subgroup And Its Properties

The structures and the properties of finite groups is an important research field of group theory, researches considering some kinds of generalizations of the normality of groups and the structure of groups have been studied extensively.The aim of this paper is to study the structure of finite groups in which primary subgroups are nearly M-supplemented, and we get some new results about the group is p-nilpotence,p-supersolvability and supersolvability.In chapter1, we introduce some research background and some main results about this paper.In chapter2, we introduce some definitions, notations and lemmas.In chapter3, the main results of this paper and the proof.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|.Then G is p-nilpotent if and only if every maximal subgroup of P having no p-nilpotent supplement is nearly M-supplemented in G.Theorem3.1.4Let G be a finite group where p is an odd prime divisor of|G|. Then G is p-nilpotent if and only if NG(P) is p-nilpotent and every maximal subgroup of P is nearly M-supplemented in G.Theorem3.2.2Let 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 nearly M-supplemented in G, then G is p-supersolvable.Theorem3.3.2Let F be a saturated formation containing U, suppose G has a solvable normal subgroup N with G/N∈F. If every maximal subgroup of noncyclic Sylow subgroup of F(N) having no supersolvable supplement is nearly M-supplemented in G, then G∈FTheorem3.3.6Let F be a saturated formation containingU, suppose G has a normal subgroup H with G/H∈F. If every maximal subgroup of noncyclic Sylow subgroup of F*(H) having no supersolvable supplement is nearly M-supplemented in G, then G∈F...