Finite Groups With Few Conjugate Classes Of Non-abelian Subgroups

In the study of group theory,it is always one of the hot topics to study the structure and properties of groups by means of the properties of subgroups.In this paper,we attempt to characterize the isomorphic classification of some finite groups by using the number of conjugate classes of non-abelian subgroups and second maximal subgroups.If G is a finite group,?(G)denotes the number of conjugacy classes of all non-abelian subgroups of G,?(G)denotes the set of the prime divisors of |G|.This paper is divided into three chapters.The first chapter lists the basic concepts and theorems of finite groups,the research results of some scholars on conjugate classes of non-abelian subgroups and second maximal subgroups in recent years are introduced.In chapter 2,we mainly discuss the finite groups with few conjugate classes of non-abelian subgroups.It is proved that the number of prime factors of a finite group satisfying?(G)=3 is no more than 3,and further determines the isomorphic classification of a finite group satisfying this condition.In chapter 3,we study the isomorphic classification of finite groups when the second maximal subgroups are all abelian.For the convenience of writing,SMA-group means that the second maximal subgroups are abelian groups.It is obvious that abelian groups and minimal non-abelian groups are all SMA-groups,It is proved that the number of prime factors of non-abelian SMA-groups is less than 3,and further determines the isomorphic classification of finite groups satisfying this condition.Finally,we give two examples of SMA-groups.