Characterizations of cancellable groups

Abstract

An abelian group A is said to be cancellable if whenever A ⊕ G is isomorphic to A ⊕ H, G is isomorphic to H. We show that the index set of cancellable rank 1 torsion-free abelian groups is Π 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 Π 1 1 m-hard; we know of no upper bound, but we conjecture that it is Π 1 2 m-complete. * Supported by an NSERC Banting Fellowship.