Finite Groups With The Number Of Conjugacy Classes Of Subgroups No More Than6

Suppose that G is a finite group.A subgroup H of G is called a co njugate-permutable subgroup of G,if HHf=HgH for all g∈G,and we name it of H<C-P G for short.Furthermore,we name the conjugate-permutable subgroup of CP-subgroup. On the contrary, we call non-conjugate-permutable subgroup of NCP-subgroup.In this paper,we use some properties of non-conjugate-permutable subgroups and non-cyclic subgroups to characterize the structure of a finite group G.The aim of this paper is to research the structures of finite groups with the numbers of conjugacy classes of non-conjugate-permutable subgroups and non-cyclie subgroups.The paper is divided into two chapters.In the first chapter,we introduce the research background, the basic concepts,the relative lemmas and Theorems.In the second chapter,we use the properties of the conjugacy classes of the non-conjugate-permutable subgroups and non-cyclic subgroups to reason the structures of finite gronps.Now the research results obtained as follows.Theorem2.1.1Let G be a finite non-nilpotent group,then all NCP-subgroups of G are conjugate if and only if G is an inner-nilpotent group andG=<a,b1,b2….,bβ|ap.=1=b1q=b2q=…=bβq,[bi,bj]=1,i,j=1,2,…,β,bia bi+1,i=1,2,...,β-1；bβ=b1dnb2d2....bβdβ>,where f(x)=xβ—dβxβ-1-…d2x—d1is an irreducible polynomial over the field Fd,which divides xp—1.Theorem2.1.2Let G be a finite non-nilpotent group,if G has two conjugacy classes of NCP-subgrbgroups,then G is soluble,and|G|has three prime factors at most. Theorem2.1.3Let G be a finite non-nilpotent group with three conjugacy classes of NCP-subgroups, then there is at least one NCP conjugacy class of Sylow subgroups and one conjugacy class of maximal NCP-subgroups, two conjugacy classes of maximal NCP-subgroups at most in|G|.(1)If G has only one conjugacy class of maximal NCP-subgroups, then G is solu-ble,|G|has four prime factors at most.(2)If G has two conjugacy classes of maximal non-subnormal subgroups and at least one normal maximal subgroup, then G is soluble and there are at most three prime factors in|G|.Theorem2.1.4Let G be a finite non-nilpotent group with three conjugacy classes of non-subnormal subgroups, then there is at least one non-subnormal conjugacy class of Sylow subgroups, one conjugacy class of maximal non-subnormal subgroups, two conjugacy classes of maximal non-subnormal subgroups at most in|G|.(1)If G has only one maximal conjugacy class of non-subnormal subgroups, then G is soluble and|G|has four prime factors at most.(2)If G has two conjugacy classes of maximal non-subnormal subgroups and at least one normal maximal subgroup, then G is soluble and there are at most three prime factors in|G|.Theorem2.2.1Let G be a finite nilpotent group with π(G)≥2, if δ(G)=3, then G is one of the following groups:(ⅰ)G≌Zp×Zp×Zq2；(ⅱ)G≌Q8×Zq2.Theorem2.2.2Let G be a finite nilpotent p-group, if δ(G)=5, then G is one of the following groups:(ⅰ)G≌Zp5×Zp；(ⅱ)G≌<a,b|ap5=bp=1,b-1ab=a1+p4>,p>2,n≥3;(ⅲ)G≌D16;(ⅳ)G≌<a,b|a32=b2=1,b-1ab=a17);(ⅴ)G≌<a,b|a16=1,a8=b2,b-1ab=a-1>, so G is Q32.Theorem2.2.3Let G be a finite nilpotent group with π(G)≥2, if δ(G)=5, then G is one of the following groups:(ⅰ)G≌Zp2×Zp×Zr2;(ⅱ)G≌Zp×Zp×Zq2×Zr;(ⅲ)G≌Zp×Zp×Zq4;(ⅳ)G≌<a,b|a8=b2=1,b-1ab=a-1>×Zq；(ⅴ)G≌Q8×Zq4.Theorem2.2.4Let G be a finite nilpotent group with π(G)≥2,if δ(G)=6, then G is one of the following groups:(i)G≌Zp3×Zp×Zr;(ⅱ)G≌Zp×Zp×Zq2×Zr;(ⅲ)G≌Zp×Zp×Zq3×Zr;(ⅳ)G≌Zp×Zp×Zq5;(ⅴ)G≌<a,b|ap3=bp=1,b-1ab=a1+p2>×Zq；(ⅵ)G≌<a,b|a8=b2=1,b-1ab=a-1>×Zq；(ⅶ)G≌Q8×Zq2×Zr；(ⅷ)G≌Q8×Zq5.