An alternative proof of Hill's criterion of freeness for abelian groups

TL;DR

This paper offers a new proof of Hill's criterion for free abelian groups, emphasizing the construction of specific subgroup families to establish freeness.

Contribution

It introduces an alternative proof method based on constructing families of pure subgroups, differing from previous approaches.

Findings

Provides a different proof of Hill's criterion

Highlights the role of pure subgroup constructions

Enhances understanding of abelian group freeness

Abstract

In this note, we provide a different proof of Hill's criterion of freeness for abelian groups. Our proof hinges on the construction of suitable families of subgroups of the links in Hill's theorem and, ultimately, on the construction of such a family of pure subgroups of the group itself.