Walker groups

Abstract

A reformulation of Walker’s theorem on the cancellation of $\mathbf {Z}$ says that any two homomorphisms from an abelian group W onto $\mathbf {Z}$ have isomorphic kernels. It does not have a constructive proof, even for W a subgroup of $\mathbf {Z}^{3}.$ In this paper we give a constructive proof of Walker’s theorem for W a direct sum, over any discrete index set, of groups of the following two kinds: Butler groups with weakly computable heights, and finite-rank torsion-free groups B with computable relative heights (that is, all quotients of B by finite-rank pure subgroups have computable heights). Throughout, “group” means abelian group. The infinite cyclic group, and the ring of integers, is denoted by $\mathbf {Z}.$ The nonnegative integers are denoted by $\mathbf {N},$ the positive integers by $\mathbf {Z}^{+},$ and the rational numbers by $\mathbf {Q}.$ We say that a set is discrete if any two elements are either equal or different. A subset A of a set B is detachable (from B) if for each $b\in B,$ either $b\in A$ or $b\notin A.$ A group is discrete if and only if its subset $\{0\}$ is detachable. Walker, in his dissertation [7], and Cohn in [2], showed that $\mathbf {Z}$ is cancellable in the sense that if $\mathbf {Z}\oplus B\cong \mathbf {Z}\oplus B^{\prime },$ then $B\cong B^{\prime }.$ It is somewhat of an oddity that $\mathbf {Z}$ is cancellable. A rank-one torsion-free group A is cancellable if and only if $A\cong \mathbf {Z}$ or the endomorphism ring of A has stable range one [3], [1, Theorem 8.12]. (A ring R has stable range one if whenever $aR+bR=R,$ then $a+bR$ contains a unit of R.) In fact, any object in an abelian category whose endomorphism ring has stable range one is cancellable. The endomorphism ring of $\mathbf {Z}$ does not have stable range one, so $\mathbf {Z}$ is the unique rank-one torsion-free group that is cancellable for some reason other than its endomorphism ring. Walker’s theorem can be reformulated to say that any two maps from an abelian group W onto $\mathbf {Z}$ have isomorphic kernels. Accordingly, we define a Walker group to be such a group W. Of course, Walker’s theorem says that every abelian group is a Walker group. However, a counterexample in the (abelian) category of diagrams $\cdot \rightarrow \cdot \rightarrow \cdot$ of abelian groups provides a Kripke model which shows that there is no constructive proof that even every subgroup of $\mathbf {Z}^{3}$ is a Walker group [4]. Thus, from a constructive point of view, it is of interest to explore the class of Walker groups. We will say that a group is a cZ-group if every homomorphism from it into $\mathbf {Z}$ has a cyclic image. An easy classical argument shows that every group is a cZ-group. It is an immediate (constructive) consequence of [4, Theorem 1] that every cZ-group is a Walker group. This is not a complete triviality because it provides a classical proof of Walker’s theorem! The question remains as to how extensive the class of cZ-groups is. That question motivated the current paper. Our main results along these lines are Theorem 1.3, which says that Butler groups with weakly computable heights are cZ-groups, and Theorem 1.6, which says that a finite-rank torsion-free group with computable relative heights is a cZ-group (Butler groups with computable heights have computable relative heights). The relevance of the ability to compute heights to the study of Walker groups was suggested by the fact that this was not possible for the group corresponding to the counterexample. The notions of weakly computable heights and computable heights already appeared in [5, 6], papers that are over 20 years old. The notion of computable relative heights is stronger than these and originates in the current paper, just after Theorem 1.3. Note that B is a cZ-group if and only if $\mathbf {Z}\oplus B$ is a cZ-group. Finitely generated groups are clearly cZ-groups. Finite direct sums of cZ-groups, and quotients of cZ-groups, are cZ-groups. As any map into $\mathbf {Z}$ kills all torsion elements, we will focus on torsion-free groups B. However, not even nonzero subgroups of $\mathbf {Z}$ need be cZ-groups: for example, $\{x\in \mathbf {Z}:\,x\,\mathrm{is\,even,\, or}\,P\}.$ In Sect. 2 we show that a direct sum of cZ-groups over a discrete index set is a Walker group (Corollary 2.3). This gives essentially the largest class of Walker groups that we currently know (Corollary 2.4), although in [4, Theorem 5] it was shown that if B is a torsion-free group such that every nonzero map from B into $\mathbf {Z}$ is one-to-one, then $\mathbf {Z}\oplus B$ is a Walker group. Rank-one torsion-free groups B have that property, as do subgroups of $\mathbf {Z},$ and any group with no nontrivial maps into $\mathbf {Z}.$ The question regarding a group B that is a finite direct sum of such groups was left open, and is still open as far as I know. Section 3 deals with the idea of the height of a subgroup. This idea arose in an effort to formulate a strong height condition that would imply that a group was a cZ-group. That approach failed and was replaced by the notion of computable relative heights. However, I still feel that the idea is interesting and may prove fruitful for some other purpose.