The Groups of Isometries of Metric Spaces over Vector Groups

Abstract

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.

Keywords

• abelian group
• automorphism
• isometry
• vector group