Basic Subgroups and Freeness, A Counterexample

TL;DR

This paper constructs a specific torsion-free abelian group with unique properties, providing a counterexample that clarifies the limitations of existing theorems in the structure theory of such groups.

Contribution

It presents a counterexample of a non-free, torsion-free abelian group with a pure free subgroup, demonstrating the boundaries of previous theorems.

Findings

Constructed a non-free, aleph_1-separable torsion-free abelian group

Showed all subgroups disjoint from a certain free subgroup are free

Demonstrated G/B is divisible, countering previous assumptions

Abstract

We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a theorem of Blass and Irwin cannot be strengthened so as to give an exact analog for torsion-free groups of a result proved for p-groups by Benabdallah and Irwin.