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Nunke’s problem asks when the torsion product of two abelian p-groups is isomorphic to a direct sum of cyclic groups. A complete solution to the problem is given using a new invariant, denoted by L G, whose values are certain collections of finite sets of uncountable regular cardinals. This is a refinement of a previous approach to the problem that only worked up to the first cardinal that is weakly Mahlo. The multiplicative properties of L G are then related to the generalized continuum hypothesis.

Mahlo cardinals and the torsion product of primary abelian groups