Characterizations of Cancellable Groups

2018/09/19 by Harrison-Trainor, Matthew, Ho, Meng-Che "Turbo"
#03D80 #20K25 #20Kxx #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.1809.07191

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 Π⁰₄ 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 Π¹₁ m-hard; we know of no upper bound, but we conjecture that it is Π¹₂ m-complete.

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