Abstract
Assume $\mathsf{GCH}$ and that $\kappa$ is the first uncountable cardinal such that there is a non-free $\kappa$-free Abelian Whitehead group of cardinality $\kappa$. We prove that if all $\kappa$-free Abelian groups of cardinality $\kappa$ are Whitehead then $\kappa$ is necessarily an inaccessible cardinal.
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