Generalized Normalities Of Subgroups And Residuals Of Group Formations

The main content of this thesis is figuring out the relationship between some generalized normalities and the formation residuals.There are three aspects included in this paper,that is the relationship between the permutability of subgroups and formation residuals,the relationship between the supplement of subgroups and formation residuals,the relationship between the cover-avoiding subgroups and formation residuals.This thesis contains five chapters.In Chapter 1,we introduce some basic concepts and results used in this thesis.In Chapter 2,we discuss the the relationship between the permutability of subgroups and formation residuals by focusing on groups having factorization G=A（?）B.In the first part of this chapter,we prove that Gu=A^uB^u,when B permutes with every maximal subgroup of Sylow subgroups of A or every maximal subgroup of A or every nontrival normal subgroup of A.In the second part of this chapter,we prove that G^wu=A^wuB^wu,when B permutes with every maximal subgroup of Sylow subgroups of A and the generalized commutant G^A is nilpotent.In Chapter 3,we discuss the relationship between the weakly s-semipermutability and formation residuals.First,some new characterizations about p-supersolvability and p-nilpotence are obtained by studying the intersections of all normal subgroups of order d of Sylow p-subgroup P of G with formation residuals,which are weakly s-semipermutable in G.Second,a new criterion of p-supersolvability and p-nilpotence is given by studying the intersection of subgroups with order d of P with formation residuals.Some known results are generalized.In Chaptet 4,we discuss the relationship between Π-property and formation residuals.First,we obtain a new criterion about p-supersolvability by studying the intersections of all normal subgroups of order d of Sylow p-subgroup of G with formation residuals,which have Π-property in G.Second,we discuss the condition that the intersections of all noncyclic normal subgroups of order d of P with formation residuals,which having Π-property in G.Third,we obtain a new criterion that a normal subgroup is p-hypercyclic embedded.In Chapter 5,we discuss the relationship between some cover-avoiding subgroups and formation residuals.We study two kinds of cover-avoiding subgroups,that is partial CAP*-subgroups and CAP*-subgroups,and prove some new criteria of psupersolvability and p-nilpotence by considering formation residuals with these two kinds of cover-avoiding subgroups.