Nunke’s problem for abelian $p$-groups of uncountable length

Abstract

Nunke's problem asks when the torsion product of two abelian $p$-groups is a direct sum of countable reduced groups. In previous work the author gave a complete answer to this question when the groups involved have countable length. In this paper a complete answer is given in the case of groups of uncountable length, at least in any set-theoretic universe in which $2^{{\aleph }_1}={\aleph }_2$.