无限循环群的整群环上的两个矩阵群

Two Matrix Groups over the Integral Group Ring of an Infinite Cyclic Group

构造群例是群论研究的重要方面,本文研究了两个具体群例的剩余有限性.设p是任意素数,C=<c>是无限循环群,R=ZC是C上的整群环,UU(n,R)是R上的单位上三角矩阵群,其中n≥2,它是幂零类为n-1的无限秩的幂零群.本文首先证明了U(n,R)是剩余有限p-群.其次,记G=〈α〉×U(3,R),其中α=diag(c,1,c)是3阶对角矩阵.本文给出了G的结构,G是3元生成的导长为3的可解群,特别地,证明了G是剩余有限p-群.进一步地,本文构造了G的两个商群,它们均不是剩余有限的,这两个商群似乎比Hall发现的经典群例要初等具体.

Constructing examples of groups is an important aspect in the theory of groups.We will study the residual finiteness of two concrete matrix groups.Let p be a prime,let C=c be an infinite cyclic group,let R=ZC be the integral group ring over C,and let U(n,R)be the upper unitriangular group over R of order n,where n≥2,which is a nilpotent group of infinite rank of class n-1.Firstly,we prove that U(n,R)is a residually finite p-group.Secondly,let G=〈α〉×U(3,R),whereα=diag(c,1,c)is a diagonal matrix of order 3.We will study the structure of G and prove that G is a residually finite p-group,G is a 3-generated soluble group of derived length 3.Moreover,we will construct two quotient groups of G,neither of which is residually finite.These two quotient groups seem to be more elementary and concrete than the classical examples discovered by Hall.