On The Self-Conjugate-Permutable Subgroups And SS-Quasinormal Subgroups To The Influnce Of The Structure Of Finite Groups

In this paper,we will give new concepts of self-conjugate-permutable subgroups and SS-quasinormal subgroups and consider the self-conjugate-permutable subgroups and SS-quasinormal subgroups to the influence of the structure of finite groups,there are three sections in our results.For detail,as the following:In Chapter One,we introduce the backgrounds and the present investigations of this paper.In Chapter Two,at first,we give new concepts of self-conjugate-permutable subgroups and PSC-groups.Definition 1.Let G be a group.A subgroup H of G is said to be self-conjugate-permutable if HH~x=H~xH implies H~x=H.Definition 2.Let G be a group.G is said to be a PSC-group if every cyclic subgroup of G of prime order or order 4 is self-conjugate-permutable.Then,we classify the PSC-groups,non-PSC-groups,finite groups whose every maximal subgroups of are PSC-groups,finite simple groups whose all of second maximal subgroups are PSC-groups and finite groups whose all of second maximal subgroups are PSC-groups.We also give new characterization of finite solvable T-groups in terms of the requirement that certain subgroups are self-conjugate-permutable and obtain new sufficient and necessary conditions for supersolvability and nilpotency. A part content of this chapter will be published in JOURNAL OF MATHEMAT-ICAL RESEARCH AND EXPOSITION(to appear)by the title"Finite groups all of whose second maximal subgroups are PSC~*-groups" and submitted to J.Group Theory by the title"Finite groups with self-conjugate-permutable subgroups".In Chapter Three,at first,we give new concept of SS-quasinormal subgroups. Definition 3.Let G be a finite group.A subgroup H of G is said to be SS-quasinormal subgroup(Supplement-Sylow-quasinormal subgroup)of G if there is a supplement B of H to G such that H permute with every Sylow subgroup of B.Then,we will consider that the SS-quasinormality of minimal subgroups,maximal subgroups,second minimal subgroups and second maximal subgroups of Sylow subgroups of groups,Fitting subgroups and general Fitting subgroups to the influence of p-nilpotent, nilpotent and supersolvable of finite groups and give new characterizations of p-nilpotent, nilpotent and supersolvable of finite groups.A part content of this chapter will be published in Comm.in Algebra(2008.9)by the title"On SS-quainormal Subgroups of Finite Groups" and in J.Algebra 319(2008) 4275-4287 by the title"The influence of SS-quasinormality of some subgroups on the structure of finite groups".