一类多重循环群的剩余有限性质

设A是秩为n(n≥2)的自由Abel群,A的自同构群Aut(A)= GL(n,Z).对整数m,取 α =(0 1 0…0 0 0(………)(…………)0 0 0…0 1 1 0…0 m)∈ Aut(A).记Γm(n)=A(×)〈α〉,则它是一个2元生成的多重循环群.本文给出了 Γm(n)的准确的剩余有限性质.

On the Residual Finiteness of a Class of Polycyclic Groups

Let $A$ be a free Abelian group of rank $n (n\geq2)$. For a rational integer $m$, take $$\begin{aligned}\begin{split}\alpha=\begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & \vdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & 0\\ 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & \cdots & 0 & m \\ \end{pmatrix}\in \mathrm{Aut}(A). \\\end{split}\\\end{aligned}$$Let $\Gamma_{m}(n)=A\rtimes\langle\alpha\rangle$, then $\Gamma_{m}(n)$ is a polycyclic group generated by two elements. In this paper, the residually finiteness of $\Gamma_{m}(n)$ is analyzed.\vskip 3mm