Characterizations of Cancellable Groups

TL;DR

This paper investigates the complexity of classifying cancellable abelian groups, showing that the problem is highly complex within descriptive set theory and computability theory.

Contribution

It establishes the exact complexity of the classification problem for cancellable abelian groups, revealing it is $ ext{Pi}^0_4$ $m$-complete for rank 1 torsion-free groups and $ ext{Pi}^1_1$ $m$-hard for general groups.

Findings

Classification of rank 1 torsion-free cancellable groups is $ ext{Pi}^0_4$ $m$-complete.

Cancellability for arbitrary non-finitely generated groups is $ ext{Pi}^1_1$ $m$-hard.

Conjecture that the general problem is $ ext{Pi}^1_2$ $m$-complete.

Abstract

An abelian group $A$ is said to be cancellable if whenever $A \oplus G$ is isomorphic to $A \oplus H$, $G$ is isomorphic to $H$. We show that the index set of cancellable rank 1 torsion-free abelian groups is $\Pi^0_4$ $m$-complete, showing that the classification by Fuchs and Loonstra cannot be simplified. For arbitrary non-finitely generated groups, we show that the cancellation property is $\Pi^1_1$ $m$-hard; we know of no upper bound, but we conjecture that it is $\Pi^1_2$ $m$-complete.