Abstract
An abelian group A A is said to be cancellable if whenever A ⊕ G A \oplus G is isomorphic to A ⊕ H A \oplus H , G G is isomorphic to H H . We show that the index set of cancellable rank 1 torsion-free abelian groups is Π 4 0 \Pi ^0_4 m 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 Π 1 1 \Pi ^1_1 m m -hard; we know of no upper bound, but we conjecture that it is Π 2 1 \Pi ^1_2 m m -complete.
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