U_c^*-normal Subgroups And Weakly Diagonal Subgroups Of Finite Groups

Let G be a finite group,H a subgroup of G.We say H is uc*-normal in G if there is a subnormal subgroup T containing HG of G such that G=HT and H∩T/HG is contained in the uΦ-hypercenter ZuΦ（G/HG）of G/HG.Let M be an abelian subgroup of G,and write W（M）={（m,m-1）|m∈M},we call W（M）a weakly diagonal subgroup of G×G.In this thesis,we investigate the influence of uc*-normality of subgroups and weakly diagonal subgroups on the structure of finite groups.The thesis consists of four chapters,the specific content is as follow:The first chapter mainly introduces some background,some notions that will be used in this thesis.The second chapter describes some lemmas that will be used in chapter 3 and chapter 4.In the third chapter,the influence of the uc*-normality of subgroups on the structure of finite groups is discussed.A necessary and sufficient condition for the supersolvability of finite groups is given by using the uc*-normality of a maximal subgroup of Sylow subgroups.A characterization of finite groups as subnilpotent is given by using the uc*-normality of Sylow subgroups.In this chapter,we also give a judgment of solvability of finite groups.The last chapter,the influence of the embedding property of weakly diagonal subgroups on the structure of finite groups is discussed.In this chapter,we use the normality and s-permutation of weakly diagonal subgroups W（M）to give the embedding properties of subgroup M in finite group G and the structure of G itself.