We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.
In this paper, we consider the groups of isometries of metric spaces arising from finitely generated additive abelian groups. Let A be a finitely generated additive abelian group. Let R={1,ϱ} where ϱ is a reflection at the origin and T={ta:A→A,ta(x)=x+a,a∈A}. We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0∉S and −S=S, the maximum subgroup of IsomX(A,S) is RT; (2) D⊴RT if and only if D≤T or D=RT′ where T′={h2:h∈T}; (3) for the vector groups over integers with finite generating set S={u∈Zn:|u|=1}, IsomX(Zn,S)=On(Z)Zn. The paper also includes a few intermediate technical results.
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