Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Characterizations of cancellable groups


Authors: Matthew Harrison-Trainor and Meng-Che “Turbo” Ho
Journal: Proc. Amer. Math. Soc. 147 (2019), 3533-3545
MSC (2010): Primary 03D80, 20K25, 20Kxx
DOI: https://doi.org/10.1090/proc/14546
Published electronically: May 1, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03D80, 20K25, 20Kxx

Retrieve articles in all journals with MSC (2010): 03D80, 20K25, 20Kxx


Additional Information

Matthew Harrison-Trainor
Affiliation: School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand
Email: matthew.harrisontrainor@vuw.ac.nz

Meng-Che “Turbo” Ho
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: turboho@gmail.com

DOI: https://doi.org/10.1090/proc/14546
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
Article copyright: © Copyright 2019 American Mathematical Society