The Groups of Isometries of Metric Spaces over Vector Groups

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,?} where rho is a reflection at the origin and T={t(a):A -> A, t(a)(x)=x+a,a is an element of A}. We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0 is not an element of S and -S=S, the maximum subgroup of IsomX(A,S) is RT; (2) D? RT if and only if D <= or D=RT' where T'={h(2 ): h is an element of T}; (3) for the vector groups over integers with finite generating set S={u is an element of Z(n): |u|=1}, IsomX(Zn,S)=O-n(Z)Z(n). The paper also includes a few intermediate technical results.