On the Residual Finiteness of a Class of Infinite Soluble Groups

Let $A$ be a completely decomposable homogeneous torsion-free abelian group of rank $n$ ($n\ge 2$). Let ${\mathrm{\Theta }}_m(n)=A⋊⟨\alpha ⟩$ be the split extension of $A$ by an automorphism $\alpha$ which is a cyclic permutation of the direct components twisted by a rational integer $m$. Then ${\mathrm{\Theta }}_m(n)$ is an infinite soluble group. In this paper, the residual finiteness of ${\mathrm{\Theta }}_m(n)$ is investigated.