Walker's Cancellation Theorem

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. In fact, in our example B and C are subgroups of Z<sup>2</sup>. 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.