The Quantitative Property Of Non-Abelian Subgroups Of Finite Groups

Recently,exploring the relationship between some quantites of subgroups and the struc-ture of finite groups has become a hot topic in group theory.More and more scholars find that the quantitative property of some subgroups can directly determine the structure of finite groups.Let G be a finite group.?(G)the number of conjugacy classes of non-abelian subgroups of G,?0(G)the number of same order classes of all non-abelian subgroup of G and ?(G)the set of the prime divisors of |G|.In this thesis we discuss the influence of the number of conjugate classes and same order classes of non-abelian subgroups on the structure of finite groups.In chapter 3,we consider the influence of the number of conjugate classes of non-abelian subgroups on the structure of finite groups.We investigate the solvable groups with?(G)=|?(G)|+1 and prove that the number of prime factors of these kinds of groups is no more than 3,and we give complete classification of these kinds of groups.In chapter 4,we extend the number of conjugate classes of non-abelian subgroups to same order classes.We say that subgroups H1 and H2 of G belong to the same order class if H1 and H2 have the same order.Firstly,we give complete classification of the finite groups with ?0(G)=2.Secondly,we give the lower bounds on ?0(G)by the |?(G)| and show that this lower bounds is appropriate.Finally,we give the isomorphic classifition of groups with ?0(G)=|?(G)|+1.