$X$-torsion and universal groups

TL;DR

This paper introduces the concept of $X$-torsion in groups, proves the existence of a universal finitely presented $X$-torsion-free group for recursively enumerable $X$, and analyzes the complexity of their presentations.

Contribution

It establishes the existence of a universal finitely presented $X$-torsion-free group using two different proofs and characterizes the complexity of their finite presentations.

Findings

Existence of a universal finitely presented $X$-torsion-free group.

Two independent proofs: group-theoretic and model-theoretic.

The set of finite presentations of $X$-torsion-free groups is $ ext{Pi}_2^0$-complete.

Abstract

For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is $\Pi_{2}^{0}$-complete in Kleene's arithmetic hierarchy.