Effectiveness of Walker's cancellation theorem

• Published in: Mathematical logic quarterly Vol. 70; no. 3; pp. 347 - 355
• Main Authors: Al‐Hellawi, Layth, Alvir, Rachael, Csima, Barbara F., Xie, Xinyue
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.08.2024
• Subjects: Complexity; Group theory; Isomorphism; Theorems
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202400030

Walker's cancellation theorem for abelian groups tells us that if A$A$ is finitely generated and G$G$ and H$H$ are such that A⊕G≅A⊕H$A \oplus G \cong A \oplus H$, then G≅H$G \cong H$. Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between G$G$ and H$H$, given indices for A$A$, G$G$, H$H$, the isomorphism between A⊕G$A \oplus G$ and A⊕H$A \oplus H$, and the rank of A$A$, is 0′$\mathbf {0^{\prime }}$. Moreover, we find that the complexity remains 0′$\mathbf {0^{\prime }}$ even if the generators in the copies of A$A$ are specified.