On the torsion product and direct sums of primary cyclic groups

A classical question of R. Nunke asks when the torsion product of two primary Abelian groups is a direct sum of cyclic groups. For each ordinal α, we define a partially ordered set, Pα, on which there is a natural multiplication, and a function μα from the class of all primary Abelian groups to Pα. This construction is used to address Nunke's problem and a complete answer is given, at least for groups whose cardinality is less than the first weakly inaccessible cardinal.