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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. ==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'' = id''A''. Abelian groups ''A'' 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? 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Whitehead problem」の詳細全文を読む スポンサード リンク
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