>[!info] Definition (Whitehead group). > Let $G$ be a group. We say that $G$ is a *Whitehead group* (or that $G$ is *Whitehead*) if for every group $G^{\prime}$, every surjective homomorphism $\rho: G^{\prime} \rightarrow G$ such that $\operatorname{ker} \rho \cong \mathbb{Z}$ *splits*, i.e. there is $\tau:G \to G'$ such that $\rho \circ \tau = \mathrm{id}_{G}$. **Fact.** Whitehead groups are torsion free. **Fact.** A group is *free* iff every surjective homomorphism onto it splits. Hence all free groups are Whitehead. >[!question] Whitehead problem >Is every Whitehead group free? **Theorem (Stein).** Every countable Whitehead group is free. **Theorem (Shelah).** Whitehead's problem is independent of ZFC. In particular: 1. [[Axiom of constructibility]] implies that every Whitehead group is free. 2. [[Martin's Axiom]] (with $\neg \mathrm{CH}$) implies there are non-free Whitehead groups of cardinality $\aleph_{1}$.