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.References
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Additional Information
- Matthew Harrison-Trainor
- Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand
- MR Author ID: 977639
- Email: matthew.harrisontrainor@vuw.ac.nz
- Meng-Che “Turbo” Ho
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 1200055
- ORCID: setImmediate$0.7583630476368097$9
- Email: turboho@gmail.com
- Received by editor(s): September 19, 2018
- Published electronically: May 1, 2019
- Additional Notes: The first author was supported by an NSERC Banting Fellowship.
This work was conducted at the University of Waterloo during a visit of the second author, supported by NSERC and the Fields Institute. - Communicated by: Heike Mildenberger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3533-3545
- MSC (2010): Primary 03D80, 20K25, 20Kxx
- DOI: https://doi.org/10.1090/proc/14546
- MathSciNet review: 3981131