| In group theroy, subgroups and quotient groups are very important approachs to describe structures of a group. Also, it is a very interesting topic to discuss the stuctrue of finite group through investigating the properties of subnormal groups. In this aspect, many important conclusions have been obtained. Here the authors discussed its dual issue, i.e, the influence of non-subnormal subgroups on the structrue of finite groups.Franciosi Silvana, etc, characterised the groups with finite conjugacy classes of non-subnormal subgroups. Aifang Feng gave complete classification of finite groups whose non-subnormal subgroups are all conjugate in [1], and abtained the complete classification of finite groups having exactly 2 conjugacy classes of non-subnormal subgroups in [3]. The author continues the research on this topic.This paper mainly consists of two parts:In the first part of this paper, the authors use mainly the complete classifications of finite groups having exactly 1, or 2 conjugacy classes of non-subnormal subgroups to give the complete classification of finite groups having exactly 3 conjugacy classes of non-subnormal subgroups.Theorem 3 Seeing the third section.In the second part of this paper, the authors employ mainly the complete clas-sifications of finite groups having exactly 1, or 2 conjugacy classes of non-subnormal subgroups and the relation between the number of conjugacy classes of non-subnormal subgroups and solvability to give the complete classifications of finite groups whose orders are paqbτcsd and paqbrc and have exactly 4 conjugacy classes of non-subnormal subgroups.Theorem 4(4.1)If G is a finite group with paqbτcsd order,p,q,τ,s are different prime numbers andμ(G)=4,then G is isomorphie to one of the following groups:(1)G=PQRs,where P(?)Q,P(?)R,P(?)S are non-nilpotent inner-abelian groups,[Q,R]=[Q,S]=[R,S]=1.(2)G=PQRS=,and f(x)=xm-lmxm-1-…-l2x-l1 is an irreducible polynomid over the field Fs,which divides xp-1 and sm≡1(mod p).(4.2)If G is a finite group with paqbτc order,p,q,τare different prime numbers andμ(G)=4,then G is isomorphic to one of the fowllowing groups:(1)G=,and f(x)=x-d is an irreducible polynomid over the field Fq,which divides xp-1 and q≡1(mod p)；9(x)=xn-dnxn-1-…-d2x-d1 is an irreducible polynomid over the field Fτ, which divides xp-1 and rn≡1(mod p);h(x)=xn-lnxn-1-…-l2x-l1 is an irreducible polynomid over the field Fr,which divides xq-1 andτn≡1(mod q).(2)G=, and f(x)=xn-dnxn-1-…-d2x-d1 is an irreducible polynomid over the field Fτ,which divides xp-1 andτn≡1(mod p);g(x)=xn-lnxn-1-…-l2x-l1 is an irreducible polynomid over the field Fτ,which divides xq-1 andτn≡1(mod q).(3)G=,and f(x)=xn-dnxn-1-…-d2x-d1 is an irreducible polynomid over the field Fτ,which divides xp-1 andτn≡1(mod p).(4)G=,and f(x)=xn-dnxn-1…-d2x-d1 is an irreducible polynomid over the field Fτ,which divides xp-1 andτn≡1(mod p)(5)(a)G=PQR,where P=,Q=,apm=bq=1,[P,Q]:[Q,R]=1.(b)μ(P(?)R)=2,where P(?)R is isomrphic to (4)or(5) in[3]. |
