The Influence Of Special Subgroups On The Structure Of Finite Groups

It is an important subject of group theory to study the properties and structures of finite groups by subgroups.We will consider two special subgroups:conjugate permutation subgroup and cyclic subgroup,and use these two subgroups to study the structure and properties of some finite groups.In the first part of this thesis,we study the influence of conjugate-permutable subgroups on the structure of a finite groups.A group G is called a NCCP group if all non-cyclic subgroups of G are conjugate-permutable in G.We obtain some properties of NCCP group and characterize the structure of two classes finite non-nilpotent NCCP groups.The first class:a finite group G with cyclic Sylow subgroups.By[7],we can write G(?)(a,b| am=1,bn=1,ab=ar,((r-1)n,m)=1,rn 三 1(mod m)>.We denote G by G(m,n).We obtain:Theorem 2.10 If a group G(m,n)is non-cyclic,then the following conditions are equivalent:(1)G(m,n)is a NCCP group;(2)m is a prime;(3)non-cyclic subgroups of G(m,n)are normal subgroups.The second class:a finite group G with an elementary abelian Sylow subgroups.We have:Theorem 2.11 If G is a NCCP group,with an elementary abelian Sylow r-subgroup R,then G=H(?)R,where H is cyclic.Let H=<a>,|a|=n and |R|=rγ.If H irreducibly acts on R,then we have:H(?)R(?)<c1,c2,…,cγ,a|cir=an=1,Cia=ci+1,(i=1,2,…,γ-1),cγa=c1t1c2t2…cγtγ>.Where f(x)=xγ-tγxγ-1-…-t2x-t1 is irreducible in the field Fr.For all m|n and m≠n,f(x)(?)xm-1,but f(x)|xn-1.For ni n,if fi(x)is a characteristic polynomial of Ani a representation matrix of ani then fi(x)is irreducible in the field Fr.In the second part of this paper,we study the relationship between the number of cyclic subgroups and the structure of groups,and classify all finite groups with exactly 9 cyclic subgroups.We have:Theorem 3.9 Let G be a finite group.G has exactly 9 cyclic subgroups if and only if G is isomorphic to one of the followings:(1)Cp(?)C7(p=2,3,5),(2)<a,b|a5=b4=1,ab=a-1>,(3)Cp8,Cp2 2,(4)<a,b|a8=b3=1,ba=b-1>.