On torsion-free minimal abelian groups

An abelian group is said to he minimal if it is isomorphic to all its subgroups of finite index. In this article we show that torsion-free groups which are complete in their Z-adic topology or are of p-rank not greater than 1, for all primes p, are minimal. A criterion is found for the minimality of all finite rank and for large classes of infinite rank completely decomposable groups. Separable minimal groups are also considered.