On The Residually Finiteness Of Several Infinite Soluble Groups

The residually finiteness is a very basic property of the infinite group.Usually,this topic mainly deals with two kinds of problem:one is to determine the residually finiteness of some groups,so as to understand many“locally”properties of these groups;the other is to“approximate”the residually finite group with its finite quotient groups.This dissertation gives out the precise residually finiteness of two kinds of polycyclic groups,investigates the isomorphism problem,supersoluability and nilpotency of a class of polycyclic groups,which are constructed by using the companion matrices,studies the residually finiteness of a class of infinite soluble groups,which are constructed by using the completely decomposable homogeneous torsion-free Abel groups of finite rank.The main contents of this dissertation are summarized as follows:In the first part,using a free Abel group of finite rank and its automorphism,we construct two kinds of polycyclic groups and give out the precise residually finiteness of them.Let A=＜a₁＞⊕＜a₂＞⊕···⊕＜a_n＞be a free Abel group of rank n.The automorphism group of A is Aut（A）=GL（n,Z）.For integer m,takeα,β∈Aut（A）such thatLetΓ_m（n）=A（?）＜α＞,Λ_m（n）=A（?）＜β＞.Obviously,both of them are polycyclic groups generated by two elements.In this part the residually finiteness of these polycyclic groups have been given out.In the second part,a general kind of polycyclic groups is constructed by using the companion matrices.Let A as before.TakeTake G=A（?）＜α＞,H=A（?）＜β＞.We present an effective method to determine the isomorphism of such kind of polycyclic groups:if±1 are not the roots of|λI-β|,and|λI-β|or has a non-unit root or has a multiple unit root,then G（?）H if and only ifαequalsβorαis similar toβ^?.Then we give some necessary and sufficient conditions for this kind of polycyclic groups to be supersoluabe or nilpotent:（1）G is supersoluable if and only if|λI-α|=（λ-1）ⁱ（λ+1）^j,i,j∈N,i+j=n;（2）G is nilpotent if and only if|λI-α|=（λ-1）ⁿ.In the third part,we construct a class of infinite soluble groups,which is the semi-direct product of a completely decomposable homogeneous torsion-free Abel group of finite rank by its isomorphism.Let A=X a₁⊕Xa₂⊕···⊕Xa_n,where 0 X Q is a torsion-free Abel group of rank 1.And letπ={prime integer p|p X=X}.The automorphism group of A is Aut（A）=GL（n,Q_π）.For integer m,takeα∈Aut（A）such thatTakeΘ_m（n）=A（?）＜α＞.We show the residually finiteness of infinite soluble groups of this kind.