Characterizations of cancellable groups

An abelian group A is said to be cancellable if whenever A \oplus G is isomorphic to A \oplus H, G is isomorphic to H. We show that the index set of cancellable rank 1 torsion-free abelian groups is \Pi ^0_4 m-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is \Pi ^1_1 m-hard; we know of no upper bound, but we conjecture that it is \Pi ^1_2 m-complete.