Indices Of Subgroups And Some Kinds Of Generalized Normalities And The Structure Of Finite Groups

The studies of the relationships between the structure of a finite group and its subgroups having certain properties is one of the most important topics in the theory of finitc groups.Among thcsc studics,rescarches by considering the quantity property of groups,or some kinds of generalizations of the normality of groups and the structure of groups are full of interest to us.Our main purpose in this thesis is try to combine the quantity property of groups and some kinds of generalizations of the normality of groups so that they can bring some new topics for us.In Chapter 2 of this thesis,we first concentrate on the relationships between the structure of a finite group and the indices of the normalizes of its cyclic subgroups. By considering normalizers of cyclic subgroups with indices of prime power or square-free,we obtain some sufficient conditions of solvability,supersolvability and p-nilpotency of a finite group G.It is noted that all those cyclic subgroups in chapter 2 that be involved are very close to normal subgroups,so to a certain extent,they are generalized normal subgroups.Secondly,in Chapter 3,we introduce a kind of new subgroup,i.e.,quasi-c-normal subgroup.Quasi- c-normal subgroup is a kind of subgroup which not only possesses the quantity property of groups,but also is a kind of generalized normal subgroup.By means of the quasi-c-normality of maximal subgroups, minimal subgroups and Sylow subgroups of a group G,we obtain solvability, supersolvability,p-nilpotency and nilpotency of a group.Some of results in this part are extended to formations.Traditionally,researches of a given group G,which are based on the maximal and minimal subgroups of the Sylow subgroups of G,are devoted on the supersolvability and p-nilpotency.In Chapter 4,we obtain some sufficient conditions for a finite group G to be p-nilpotent,and supersolvable by considering all subgroups H of Sylow subgroups P of a group G,where those subgroups H having s-semipermutability in G and satisfying |H| = |D| and 2|D|(if P is a non-abelian 2-group and |P:D|＞2),where 1＜|D|＜|P|.