The Influences Of The Characteristic Properties Of Some Subgroups On The Structure Of Finite Groups

In this paper we will study how the characteristic properties of some subgroups influence the structure of finite groups and discuss those from the following five chapters:In the first chapter, we will recall the some recent definitions and results. It follows that we states our results and means in our paper.In the second chapter, we give the definition of CCAP-subgroups. Applying these definition, we obtained some sufficient conditions of p-supersolvable groups. There are the following some main results:Definition 1.2.1 A subgroup H of a finite group G is said to be a CCMP-subgroup of G if it has either the completely conditionally permutable or the cover-avoiding property in G, for short, it is either a CCP-subgroup or a CAP-subgroup of G.Theorem1.2.1 Let p be a prime, G be a group and H be a p-soluble normal subgroup of G such that G/H is p-supersoluble. If all maximal subgroups of the Sylow p-subgroups of H are CCAP-subgroups of G. Then G is p-supersoluble.Theorem1.2.2 Let p be a prime, G be a group and H be a p-soluble normal subgroup of G such that G/H is p-supersoluble. If all maximal subgroups of F_p(H) which contain O_p' (H) are CCAP-subgroups of G. Then G is p-supersoluble.And we generalize the results of L.M.Ezquerro and W.Guo.In the third chapter, we will give the definition of SC-subgroups and obtain some properties and results of supersovablities and nilpotences of finite groups who are based on the assumption that some subgroups are SC-groups. There are the following some main results:Definition1.2.2 A subgroup H of a finite group G is said to be a SC-subgroup of G if it has either the completely conditionally permutable or the semi cover-avoiding property in G, for short, it is either a CCP-subgroup or a SCAP-subgroup of G.Theorem1.2.3 Let p be an odd prime dividing the order of the group G and P a Sylow p-subgroup of G. If Ng(P) is p-nilpotent and every maximal subgroup of P is a SC-group of G, then G is p-nilpotent.Theorem1.2.4 Let p be the smallest prime number dividing the order of G and let P be a Sylow p-subgroup of G such that every cyclic subgroup of P of order 4 is a SC-group of G and every subgroup of P of order p is contained in Z_F(G), where F is the class of all p-nilpotent groups. P is quaternion-free. Then G is p-nilpotent.Theorem1.2.5 For a solvable group G, If all maximal subgroups of every Sylow subgroup of G are SC-subgroups of G. Then G is supersoluble.Theorem1.2.6 For a group G, If all minimal subgroups or cyclic subgroups of order 4 of every Sylow subgroup of G are SC-subgroups of G. Then G is supersoluble.In the fourth chapter, we consider the concept of c-permutable subgroups and obtain some results of the solvablity of finite groups by reducing the number of the maximal subgroup of group. There are the following some main results:Theorem1.2.7 Let G be a finite group. G is soluble if and only if there exists a soluble maximal subgroup M of G which is c-permutable in G.Theorem1.2.8 Let G be a finite group. G is soluble if and only if for every maximal subgroup of G which belongs to F^od(G), M is a GGP-subgroup of G.Theorem1.2.9 Let G be a finite group. G is soluble if and only if for every maximal subgroup of G which belongs to F²(G), M is a GGP-subgroup of G.Theorem1.2.10 Let G be a finite group. G is p-soluble if and only if for every maximal subgroup of G which belongs toκ(G), M is a c - permutable subgroup of G.Theorem1.2.11 Let G be a finite group. G is p-soluble if and only if for every maximal subgroup of G which belongs to L(G), M is a c - permutable subgroup of G.In the fifth chapter, we consider the concept of norm-subgroups and obtain some properties and results of the supersolvablity and nilpotence of finite groups.Theorem1.2.12 G is a finite group. If G/N(G) is nilpotent, then G is also nilpotent.Theorem1.2.13 G is a finite group. Suppose that G/N(G) is supersoluble, then G is also supersoluble.Theorem1.2.14 G has a norm series. If G/N_∞(G) is nilpotent, then G is nilpotent.Theorem1.2.15 G has a norm series. If G/N_∞(G) is supersoluble, then G is supersoluble.