Let G be a torsion-free abelian group of rank n and X= {xl, *. , x,j a maximal set of rationally independent elements in G. It is well known that any g e G can be uniquely written g= oc1xl?+ +x, for some cci, . , ?C72, E Q, the rational numbers. This enables us to define, for any such (G, X), a collection of subgroups of Q and "natural" isomorphisms, denoted by S(G, X). It is known that if G is of rank two, then G may be recovered from S(G, X) is a natural way. The following result is obtained for groups of rank greater than two: THEOREM. Let G, G' be torsion free abelian groups offinite rank with S(G, X)=S(G', X') for suitable X, X'. Let F, F' be the free subgroups of G, G' generated by X, X'. Then G/F_?G'/F'. An additional condition isgiven for pairs (G, X), (G', X') such that S(G, X)=S(G', X') implies G_G'. 1. Schemes of groups. Let G be a group' of rank n and X= {xl, * , xn} a maximal set of rationally independent elements in G. It is well known that any g E G can be uniquely written g= Ocxl + * * + acxn, for some (1X, * , * Zn cE Q, the rational numbers. This enables us to define, for any such (G, X), a collection of subgroups of Q as follows: DEFINITION 1. Let li