The Collection Process Of Nilpotent Groups Of Class 3 And 4

In this thesis,we mainly focus on the basic commutator calculus formula,for the nilpotent groups of class 3 and class 4.By calculating the collection formula of some elements respectively,we get the collection formula of finite elements by induction.This thesis is divided into four parts:The first part is the introduction,we introduce the research background of this thesis and the previous work in this field,and give the conclusion of this thesis.The second part is prelimineries,we make a list of some basic theory and theorem in this article.In the third part,we give the derivation process of the nilpotent group of class 3.Theorem 3.1 Let G be a group over the nilpotent group of class 3,x,y,z ?G,we have(xyz)n=xnynzn[y,x]Cn2[z,y]Cn2[z,x]Cn2[y,x,x]Cn3[y,x,y]Cn2+2C3[y,x,z]2Cn2+2Cn3[z,x,x]Cn3[z,y,y]Cn3[z,x,z]Cn2+2Cn3[z,y,z]Cn2+2Cn3[z,x,y]Cn2+2Cn3Theorem 3.2 Let G be a group over the nilpotent group of class 3.x,y,z,k?G,we have(xyzk)n=xnynznkn[y,x]Cn2[z,y]Cnx[z,x]Cn2[k,z]Cn2[k,y]Cn2[k,x]Cn2[y,x,x]Cn3[k,y,y]C3[z,y,y]Cn3[k,z,z]Cn3[z,x,x]Cn3[k,x,x]Cn3[k,x,y]Cn2+2Cn3[y,x,k]2Cn2+2Cn3[z,x,y]Cn2+2Cn3[y,x,y]Cn2+2Cn3[k,x,z]Cn2+2Cn3[k,y,z]Cn2+2Cn3[z,x,z]Cn2+2Cn3[z,y,z]C2n2+2Cn3[k,z,k]Cn2+2Cn3[k,y,k]Cn2+2Cn3[k,x,k]Cn2+2Cn3[y,x,z]2Cn2+2Cn3[z,y,k]2Cn2+2Cn3[z,x,k]2Cn2+2Cn3Theorem 3.3 Let G be a group over the nilpotent group of class 3,x1,x2,…,xm?G,we have(x1x2·xm)n=x1nx2n·xmn(?)[xj,xi]Cn2(?)[xj,xi,xi]Cn3(?)[xk,xj,xi]Cn2+2Cn3(?)[xk,xj,xi]2Cn2+2Cn3 In the fourth part,we give the derivation process of the nilpotent group of class 4.Theorem 4.1 Let G be a group over the nilpotent group of class 4,x,y?G,we have(xy)n=xnyn[y,x]Cn2[y,x,y]Cn2+2Cn3[y,x,x]Cn3[y,x,x,x]Cn4[y,x,x,y]2Cn3+3Cn4[y,x,y,y]2Cn3+3Cn4Theorem 4.2 Let G be a group over the nilpotent group of class 4,x,y,z ? G,we have(xyz)n=xnynzn[y,x]Cn2[z,y]Cn2[z,x]Cn2[y,x,z]2Cn2+2Cn3[y,x,y]Cn2+2Cn3[z,x,z]Cn2+2Cn3[z,y,z]Cn2+2Cn3[z,x,y]Cn2+2Cn3[z,x,x]Cn3[z,y,y]Cn3[y,x,x]Cn3[z,x,y,y]2Cn3+3Cn4[y,x,z,z]Cn2+4Cn3+3Cn4[z,x,y,z]Cn2+6Cn3+6Cn4[y,x,x,z]3Cn3+3Cn4[y,x,y,y]2Cn3+3Cn4[z,x,x,z]2Cn3+3Cn4[z,x,x,y]2Cn3+3Cn4[y,x,x,y]2Cn3+3Cn4[y,x,y,z]2Cn2+8Cn3+6C4n4[z,y,z,z]2Cn3+3Cn4[z,y,y,z]2Cn3+3Cn4[z,x,z,z]2Cn3+3Cn4[z,y,y,y]Cn4[y,x,x,x]Cn4[z,x,x,x]C4n4[[z,x],[y,x]]2Cn3+3Cn4[[z,y],[y,x]]2Cn3+3Cn4[[z,x],[z,y]]2Cn3+3Cn4Theorem 4.3 Let G be a group over the nilpotent group of class 4,x1,x2,…,xm ? G,we have(x1x2…xm)n=x1nxn2…xnn(?)[xi,xj]Cn2(?)[xi,xj,xj]Cn3(?)[xi,xj,xk]Cn2+2Cn3(?)[xj,xj,xk]2Cn2+2Cn3(?)[xi,xj,xj,xj]Cn4(?)[xi,xj,xk,xl]2Cn3+3Cn4(?)[xi,xj,xj,xk]3Cn3+3Cn4(?)[xi,xj,xk,xl]2Cn2+8Cn3+6Cn4(?)[[xi,xj],[xk,xl]]2Cn3+3Cn4(?)[xi,xj,xk,xl]Cn2+6Cn3+6Cn4(?)[xi,xj,xk,xk]Cn2+4Cn3+3Cn4(?)[xi,xj,xk,xk]4Cn2+10Cn3+6Cn4...