He Guo LIU, Ji Ping ZHANG, Xing Zhong XU, Jun LIAO
Let $A$ be a free abelian group of rank $n$. It is well known that the automorphism group $\operatorname{Aut}(A)$ of $A$ is $\operatorname{GL}(n,\mathbb{Z})$. Let $f(\lambda)=\lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}$ be an irreducible polynomial in $\mathbb{Z}[\lambda]$, where $a_{0}=\pm1$. Let $T=\langle\alpha\rangle$ be an infinite cyclic group. Let $\alpha$ act on $A$ via the automorphism of $A$ induced by the Frobenius companion matrix of the monic polynomial $f(\lambda)$. Assume that $G=A\rtimes T$. Let $p$ be a prime. We prove that $G$ is a residually-finite $p$-group if and only if $p$ divides $f(1)$.