Mahlo cardinals and the torsion product of primary Abelian groups. (Q1759644)

The author considers a fundamental question from the 60's that goes back to Nunke. Given two Abelian \(p\)-groups the problem asks when the torsion product of them is actually a direct sum of cyclic groups. In earlier works the author and others introduced a new approach to the problem using an invariant that worked for cardinals up to the first weakly Mahlo cardinal and was rather complicated. In this paper the author introduces a new invariant that helps to solve Nunke's problem completely. It is shown that the new invariant indeed tells when a group is \(\Sigma\)-cyclic and that it behaves well with respect to the torsion product. It is also pointed out why the limitations in earlier papers had been necessary and several applications and also independence results are given.