A countable universal torsion abelian group for purity

TL;DR

The paper constructs a countable universal abelian p-group for purity, solving a key problem by introducing and characterizing $eth_0$-strongly homogeneous p-groups, advancing the understanding of pure embeddings in abelian groups.

Contribution

It introduces the concept of $eth_0$-strongly homogeneous p-groups and constructs a universal abelian p-group for purity, completing a significant step in the classification of such groups.

Findings

Existence of a countable universal abelian p-group for purity.

Complete characterization of countable $eth_0$-strongly homogeneous p-groups.

Resolution of Problem 5.1 of [Fuc15] below $eth_ ext{omega}$.

Abstract

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below $\aleph_\omega$. We introduce $\aleph_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.