Metric Dimensions of Metric Spaces over Vector Groups

Let (X,rho) be a metric space. A subset A of X resolves X if every point x is an element of X is uniquely identified by the distances rho(x,a) for all a is an element of A. The metric dimension of (X,rho) is the minimum integer k for which a set A of cardinality k resolves X. We consider the metric spaces of Cayley graphs of vector groups over Z. It was shown that for any generating set S of Z, the metric dimension of the metric space X=X(Z,S) is, at most, 2maxS. Thus, X=X(Z,S) can be resolved by a finite set. Let n is an element of N with n >= 2. We show that for any finite generating set S of Zn, the metric space X=X(Zn,S) cannot be resolved by a finite set.