Refinement
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?
Read more about this topic: Whitehead Problem
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