The Influence Of Some New Generalized Normal Subgroups On The Structure Of Finite Groups

It is one of a basic topic in the study of finite groups that characterize the structure of groups whose some subgroups enjoy the given properties.In this paper,the structure of finite groups with some generalized normal subgroups are studied.The main contents studied in this paper are as follows:1.We generalize the concept of c-normal subgroup to cp-normal subgroup and generalize the concept of CN-group to CNp-group,respectively,by using the idea of localization:a subgroup H is called cp-normal in G,if there exists a normal subgroup T of G containing HG such that G=HT and H∩T/HG is a p’-group,where HG is the core of H in G,p is a prime dividing the order of G;a group G is called a CNp-group if every p-subgroup of G is cp-normal in G.In this paper,the structure of groups with cp-normal n-maximal subgroups are characterized,where n=1,2.Moreover,the structure of CNp-group are characterized,further,we obtain some necessary and sufficient conditions for a direct product of two CNp-groups to be a CNp-group.2.We generalize the concept of c-supplemented subgroup to cp-supplemented subgroup and generalize the concept of c-supplemented group to CSpgroup,respectively,by using the idea of localization:a subgroup H is called cp-supplemented in G,if there exists a subgroup T of G containing HG such that G=HT and H∩T/HG is a p’-group,where p is a prime dividing the order of G;a group G is called a CSp-group,if every p-subgroup of G is cp-supplemented in G.In this paper,the structure of CSp-groups are characterized and some sufficient conditions for a direct product of two CSp-groups to be a CSp-group are given.3.We further generalize the concept of cp-normal subgroup to cp*normal subgroup and introduce a concept of c*-normal subgroup:a subgroup H is called cp*-normal in G,if there exists a normal subgroup T of G containing HG such that G=HT and |H∩T/HG|p≤p.where p is a prime dividing the order of G;a subgroup H is called c*-normal in G,if there exists a normal subgroup T of G containing HG such that G=HT and H∩T/HG is a group of square-free order.In this paper,we characterize the structure of groups with cp*-normal meet-irreducible subgroups and groups with cp*-normal 2-maximal subgroups.Meanwhile,we characterize the structure of groups whose some p-subgroups are cp*-normal.4.Inspired by the weakly s-permutable subgroup,we further generalize the concept of c*-normal subgroup to weakly s*-permutable subgroup:a subgroup H is called weakly s*-permutable in G,if there exists a subnormal subgroup T of G containing HsG such that G=HT and |H∩T:HsG|is square-free,where HsG is the largest s-permutable subgroup of G contained in H.In this paper,we characterize the structure of groups in which some n-maximal subgroups are weakly s*-permutable.Meanwhile,we classify the non-abelian simple groups whose every 3-maximal subgroup is weakly s*-permutable.Further,we obtain some sufficient conditions for groups whose every 3-maximal subgroup is weakly s*-permutable to be solvable.5.We discuss the influence of normal index of general subgroups on the structure of groups,and obtain some characterizations of groups to be p-supersolvable by using the idea of localization.6.We characterize the detailed structure of non-supersolvable groups which can be factorized as a product of two subnormal supersolvable subgroups.Some conclusions in this paper generalize and unify a lot of previous results.