Whitehead problem

Independent of the axiomsFoundations of mathematicsAlgebra

Statement

An abelian group $A$ is called a Whitehead group if $\operatorname{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}) = 0$ , meaning that every short exact sequence $0 \to \mathbb{Z} \to B \to A \to 0$ of abelian groups splits. Must every Whitehead group $A$ be a free abelian group?

Saharon Shelah proved in 1974 that Whitehead's problem is undecidable in ZFC: assuming $V = L$ , every Whitehead group is free, whereas assuming Martin's Axiom and $2^{\aleph_0} > \aleph_1$ , there exists a non-free Whitehead group of cardinality $\aleph_1$ .

Shelah's resolution launched set-theoretic algebra, extending independence results to Whitehead modules over general rings, the structure of $\operatorname{Ext}^1(A, G)$ for torsion-free groups, and Kaplansky's test problems on abelian groups.

References

Saharon Shelah (1974). Infinite Abelian groups, Whitehead problem and some constructions · DOI:10.1007/BF02757281
Saharon Shelah (1977). Whitehead groups may be not free, even assuming CH. I · DOI:10.1007/BF02760647
Paul C. Eklof, Alan H. Mekler (2002). Almost Free Modules: Set-Theoretic Methods

History of the problem

1950J. H. C. Whitehead poses the question of whether $\operatorname{Ext}^1_{\mathbb{Z}}(A, \mathbb{Z}) = 0$ forces an abelian group $A$ to be free, motivated by the universal coefficient theorem in algebraic topology.
1951Karl Stein proves that every countable Whitehead group is free, and shows that every countable subgroup of any Whitehead group is free (so every Whitehead group is $\aleph_1$ -free).
1974Saharon Shelah proves that the Whitehead problem for groups of cardinality $\aleph_1$ is independent of ZFC, and introduces his singular compactness theorem to extend the $V = L$ freeness proof to arbitrary cardinalities.
1977Saharon Shelah proves that even assuming the Generalized Continuum Hypothesis (GCH), the existence of non-free Whitehead groups remains independent of ZFC.