DIAGRAMS OF AN ABELIAN GROUP

In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of Sigma-unique decomposition, in terms of a pair of point set topological properties of Eilenberg-Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have Sigma-unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.