Editing Whitehead problem (section)

==Refinement==
Assume that ''A'' is an abelian group such that every short [[exact sequence]]
:<math>0\rightarrow\mathbb{Z}\rightarrow B\rightarrow A\rightarrow 0</math>
must [[Split_exact_sequence|split]] if ''B'' is also abelian. The Whitehead problem then asks: must ''A'' be free? This splitting requirement is equivalent to the condition Ext<sup>1</sup>(''A'', '''Z''') = 0. Abelian groups ''A'' satisfying this condition are sometimes called '''Whitehead groups''', so Whitehead's problem asks: is every Whitehead group free? It should be mentioned that if this condition is strengthened by requiring that the exact sequence 
:<math>0\rightarrow C\rightarrow B\rightarrow A\rightarrow 0</math>
must split for any abelian group ''C'', then it is well known that this is equivalent to ''A'' being 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 Ext<sup>1</sup>(''A'', '''Z''') = 0. Whitehead's problem then asks: do Whitehead groups exist?