Walker's cancellation theorem

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.