Walker's Cancellation Theorem

Abstract

Walker's cancellation theorem says that if B + Z isisomorphic to C + Z in the category of abeliangroups, then B is isomorphic to C. We construct an example ina diagram category of abelian groups where the theorem fails. As aconsequence, the original theorem does not have a constructiveproof. In fact, in our example B and C are subgroups ofZ<sup>2</sup>. Both of these results contrast with a group whoseendomorphism ring has stable range one, which allows aconstructive proof of cancellation and also a proof in any diagramcategory.