Characterizations of Cancellable Groups

Abstract

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 ${\Pi }_4^0$ $m$-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 ${\Pi }_2^1$ $m$-complete.

14 pages