Characterizations of cancellable groups

Harrison-Trainor, Matthew; Ho, Meng-Che

doi:10.1090/proc/14546

Characterizations of cancellable groups
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by Matthew Harrison-Trainor and Meng-Che “Turbo” Ho PDF

Proc. Amer. Math. Soc. 147 (2019), 3533-3545 Request permission

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 $\Pi ^0_4$ $m$-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.

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