Finite Groups In Which Sylow Objects Have Given Local Properties

The theory of groups is one of the oldest branch of modern algebra. Theory of finite groups is an important research field in Group Theory. The main task of group theory is to discern the structure and properties of various groups. In the last sixties and seventies, in parallel to the prominent effort to classify the finite simple groups, a large number of papers created a beautiful and comprehensive view of finite soluble groups. In1980, when the classification was almost completed, Helmut Wielandt proposed giving priority after the classification to the extension of these brilliant results of the theory of finite solvable groups to the more ambitious universe of all finite groups.The so-called Sylow Theorem published130years ago is one of the central directions within the development of finite groups. Sylow objects, including primary subgroups, normalizers and centralizers of primary subgroups and so on, have been studied extensively in amounts of theorems such as Frobenius Theorem, Burnside Theorem, Glauberman Theorem and so on. A primary subgroup is a subgroup of prime power order.According to previous studies, a subgroup H is said to be supplemented in G if there exists a subgroup K of G such that G=HK and K is called a supplement of H in G. Furthermore, if H∩K=1, H is called complemented in G. Actually, various restricting conditions on supplements have been investigated in the past.It is well known that the supplementation of subgroups plays an important role in the study of Group Theory. For example, Hall(1937) proved that a group G is soluble if and only if every Sylow subgroup of G is complemented in G. Kegel(1965,1961) has shown that a group G is soluble if every maximal subgroup of G has a cyclic supplement in G or if some nilpotent subgroup of has a nilpo-tent supplement in G. Later on, Arad and Ward (1982) proved that a group G is solvable if and only if every Sylow2-subgroup and every Sylow3-subgroup of G is complemented in G. By considering some special supplement s(c-supplernent) of some primary subgroups. Wang (1996) obtained that a group G is soluble if and only if every maximal subgroup of G is c-normal in G. Ballester-Bolinches and X.Guo (1999) proved that a group G is supersoluble if either every maximal sub-group of any its Sylow subgroup is complemented in G or all minimal subgroups of G are complemented. Recently. W.Guo (2008) proposed the new concept of F-supplemented subgroup in combination with the theory of classes of groups and formation and obtained some new criteria of solubility and supersolubility of finite groups.Our main objective in this paper is to present several new local properties of subgroups and to continue and develop the work of scholars mentioned above. Meanwhile, extend the methods of the normality and soluble case to classes of finite non-necessarily soluble groups. Furthermore, we observe deeply and systemically the structure of finite groups, in which primary subgroups have been given local property. In addition, the detailed texture of some saturated formations will be explored.This paper is divided into three parts and considers the influence of the local properties of subgroups on the structure of finite groups.Firstly, we present the definition of the Fs-supplemented subgroups and investi-gate the general properties of the Fs-supplemented subgroups. We study the struc-ture of p-nilpotency and p-supersolubility of groups by using Fs-supplementation of the maximal subgroups and normalizers of Sylow subgroups, and obtain a series of new results about p-nilpotency and p-supersolubility of finite groups.Let p be an odd prime divisor of|G|and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if NG(P) is p-nilpotent and every maximal subgroup of P is Fs-supplemented in G, where T is the class of all p-nilpotent groups.Let G be a p-soluble group and p a prime divisor of|G|. Then G is p-supersoluble if and only if every maximal subgroup of noncyclic Sylow p-subgroup of Fp(G) is Fs-supplemented in G, where F is the class of all p-supersoluble groups.Meanwhile, we also consider the construction of the saturated formation con-taining all supersoluble groups by considering some subgroups of Fitting subgroups and generalized Fitting subgroups. We obtain some sufficient conditions which a group belong to given saturated formations: Let F be a saturated formation containing U and H he a normal subgroup of G such that G/H∈F. Suppose that every maximal subgroup of noncyclic Sylow subgroup of F*(H) has a supersoluble supplement in G or is Us-supplemented in G, then G∈F. Which generalizes the result of Prof.Wei:Let F be a saturated formation containing U and H be a normal subgroup of G such that G/H∈F. Suppose that every maximal subgroup of Sylow subgroup of F*(H) is c-normal in G, then G∈F.Secondly, we discuss weak c-normality of subgroups and get some conditions for the solubility of a finite group through some weakly c-normal primary subgroups. Moreover, by using the weak c-normality of maximal subgroups and2-maximal subgroups, we obtain some sufficient conditions for p-nilpotency and p-solubility and p-supersolubility of finite groups. The main result are as follows:Let H be a Hall π-subgroup of G and2∈π. If NG(H) is supersoluble and there exists a maximal subgroup of NG(H) which is weakly c-normal in G, then G is soluble.Let G be a finite group. If every maximal subgroup of noncyclic Sylow sub-group of F(H) is weakly c-normal in G, then G is supersoluble.Let G be a finite group. Then G is soluble if and only if M is weakly c-normal in G for every non-nilpotent maximal subgroup M in Fsc.Finally, we introduce the conception of the F-s-supplemented subgroups, and discuss the construction of class all supersoluble groups by considering of some subgroups of Fitting subgroups and generalized Fitting subgroups. We obtain some new criterions for supersolubility:Let G be a group with a soluble normal subgroup H such that G/H∈U. Then G is supersoluble if and only if all non-normal minimal subgroups and all cyclic subgroups of order4of F(H) are supersoluble s-supplemented in G.Let G be a group with a normal subgroup H such that G/H∈U. Then G is supersoluble if and only if all non-normal minimal subgroups and cyclic subgroups of order4of F*(H) are supersoluble s-supplemented in G.