Walker's cancellation theorem
Logic
2015-10-09 v1 Group Theory
Abstract
Walker's cancellation theorem says that if B+Z is isomorphic to C+Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.
Cite
@article{arxiv.1510.02137,
title = {Walker's cancellation theorem},
author = {Robert Lubarsky and Fred Richman},
journal= {arXiv preprint arXiv:1510.02137},
year = {2015}
}