High Quality Content by WIKIPEDIA articles! In group theory, a branch of abstract algebra, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? Shelah (1974) proved that Whitehead's problem is undecidable within standard ZFC set theory. The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B -> A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : A -> B with fg = idA. Abelian groups satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?

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