Nunke's problem for abelian \(p\)-groups of uncountable length
All groups in this paper are abelian $p$-groups for some fixed prime $p$. The author considers a fundamental question from the 60's that goes back to \textit{R. J. Nunke} [Pac. J. Math. 22, 453--464 (1967; Zbl 0158.02805)]. Nunke's problem asks when the torsion product of two abelian $p$-groups is a direct sum of countable reduced groups. In [Pac. J. Math. 259, No. 1, 117--139 (2012; Zbl 1261.20058)], the author introduced set-theoretic invariants and gave a complete answer when the groups have countable length. In this paper, the author gives a complete answer for groups of uncountable length. \par The main result is formulated in set-theoretic invariants and holds in any set-theoretic universe in which $2^{\aleph_1}=\aleph_2$. However, the author notes ``it would perhaps be more accurate to say that the result shifts the problem from a question regarding the torsion product, to a question of computing invariants whose values, even in easy cases, depend upon undecidable statements from set theory. Still, it gives a way to analyse the problem by considering each group as a separate entity without any reference to their torsion product''.
scientific article; zbMATH DE number 6898049