The Influence Of The Number Of Conjugacy Classes Of Some Subgroups On The Structure Of Finite Groups

Let G be a finite group and let μ(G)denote the number of conjugacy classes of all non-subnormal subgroups of G,μc(G)denote the number of conjugacy classes of all non-subnormal and non-circle subgroups of G.In this thesis,we show that all finite groups G with μ(G)≤ 2|π(G)| are solvable and classify finite group G with μ(G)≤ |π(G)| completely.we also classify finite simple group G with μc(G)≤9.In the study of finite groups,as we known,using the order of group or the properties of subgroup or element to protray the structure and discuss the properties of finite groups is a main direction and a common approach.Which through some special subgroups of the number of conjugations to explore the structure of the finite group has been very Important subject,also produced a wealth of results.In this thesis,we investigate the structure of finite groups by the number of conjugate classes of non-subnormal subgroups,and non-subnormal and non-circle subgroups.The main results are as follows:Theorem 2.1.1 Let G be a group with μ(G)≤ 2|π(G)|.Then G is solvable.Theorem 2.1.2 Let G be a group with μ(G)≤ 2|π(G)|-2.Then G has a normal Sylow p-subgroup P for some p ∈π(G).Theorem 2.1.3 Let G be a non-nilpotent group with μ(G)≤ |π(G)|.Then the following assertions hold:(1)If μ(G)=1.Then G is one of Lemma 1.2.3.(2)If μ(G)＝2.Then G is one of Lemma 1.2.4.(3)If μ(G)≥ 3.Then G is one of the following groups:(3a)G ＝<a,b,c|apo＝bq=1=ccr2,bn＝bd1.[a,c][b,c]= 1>,where x-d1∈Fq[x]divides xp-1,and q≡1(mod p),p,q,rr are distiinct primes.(3b)G=<a,c,b1,b2,…,bn|apo = crβ =1][a,c]>,where(a,b1,b2,…,bn>and(c,b1,b2,…bn)arc all q-basic groups,p,q,r are distinct primes.(3c)G =<a,c,b1,b2,...,bn| apα = cγ = 1 =[c,a]= 1 =[c,bi]>,where(a,b1,b2,...,bn>is q-basic group,γ = r or rs,p,q,r,s are distinct primes.(3d)G=<a,c,b1,b2,…,bn| apα=cr = 1 =[c,a]＝[c,bi]v=1,2,...,n,where(a,b1,b2,…,bn)is q-basic group,p,q,r are distinct primes.Theorem 2.2.1 Let G be a finite simple group.Then the following assertions hold:(1)G = PSL(2,q),q = 4,5,7,8,11,13,27,the number of corresponding conjugacy classes of non-subnormal and non-circle subgroups is 4,4,9,5,9,8,8.For group G = SL(2,5)or G = SL(2,7),the number of corresponding conjugacy classes of non-subnormal and non-circle subgroups is 4 or 9.(2)For group G = S5,A7,PSL(3,3),PSL(3,4),U3(3),U4(2),then μc(G)>10.Theorem 2.2.2 let G = PSL(2,q),whichq= pf is a prime power,9≥4,then the following assertions hold:(1)If q = 2f then q = 8:q =5 or μc(G)>10;(2)If q=3f,then μc(G)≥10;(3)If q does not belong to the collection {5,7,8},,then μc(G)>10.Theorem 2.2.5 let G is finite simple group with μc(G)≤ 9,then G≌ PSL(2,q),which q ∈{4,5,7,8,11,13,27}.