The Influence Of Weak M - Complementarity On Group Structure In Given Subgroups

It is well known that discerning structures and properties of various groups is a main task of group theory, while the property of the subgroup plays an important role in indicating the structure of the group. Particularly, the property of the primary subgroup has been characterized in detail, and some new wonderful results are obtained.In this thesis, we shall investigate the structure of finite groups by using the weak M-supplementation of given subgroups. Some new results about p-nilpotency, p-supersolvability and FΦ-hypercenter of finite groups are obtained.A nonidentity subgroup H of a group G is called weakly M-supplemented in G if there exists a subgroup B of G provided that (1) G=HB, and (2) if H1/HG is a maximal subgroup of H/HG, then H1B=BH1<G, where HG is the largest normal subgroup of G contained in H.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:Theorem3.1Let G be a p-solvable group and H be a p-nilpotent subgroup containing a Sylow p-subgroup of G. If every maximal subgroup H1of H with|H:H1|=p is weakly M-supplemented in G, then G is p-supersolvabe.Theorem3.3Let G be a p-solvable group and p be a prime divisor of|G|. If every maximal subgroup H of Fp(G) containing Op,(G) with Hp≠1is weakly M-supplemented in G, then G is p-supersolvable.Theorem3.5Let 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 (?)∈π(F*(N)), F*(N)has a subgrup D with|F*(N):D|=p for any p∈π(F*(N))and every subgroup E of F*(N) with|E|=|D|is weakly M-supplemented in G, then G∈F.Theorem3.6Let E be a normal subgroup of G, where p is the smallest prime divispr of|E|.suppose E has a Sylow p-subgroup P, such that every subgroup of P is weakly M-supplemented in NG(P) and P’ is s-quasinormal in E,where P’ is the commutator subgroup of P,then E/Op’(E) is uΦ-hypercentered in G/Op’(E).