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 ?(G)denote the number of conjugacy classes of all non-subnormal subgroups of G,?c(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 ?(G)? 2|?(G)| are solvable and classify finite group G with ?(G)? |?(G)| completely.we also classify finite simple group G with ?c(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 ?(G)? 2|?(G)|.Then G is solvable.Theorem 2.1.2 Let G be a group with ?(G)? 2|?(G)|-2.Then G has a normal Sylow p-subgroup P for some p ??(G).Theorem 2.1.3 Let G be a non-nilpotent group with ?(G)? |?(G)|.Then the following assertions hold:(1)If ?(G)=1.Then G is one of Lemma 1.2.3.(2)If ?(G)?2.Then G is one of Lemma 1.2.4.(3)If ?(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 = cr? =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| ap? = c? = 1 =[c,a]= 1 =[c,bi]>,where(a,b1,b2,...,bn>is q-basic group,? = r or rs,p,q,r,s are distinct primes.(3d)G=<a,c,b1,b2,…,bn| ap?=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 ?c(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 ?c(G)>10;(2)If q=3f,then ?c(G)?10;(3)If q does not belong to the collection {5,7,8},,then ?c(G)>10.Theorem 2.2.5 let G is finite simple group with ?c(G)? 9,then G? PSL(2,q),which q ?{4,5,7,8,11,13,27}.