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In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. The homomorphism theorem is used to prove the isomorphism theorems. == Group theoretic version == Given two groups ''G'' and ''H'' and a group homomorphism ''f'' : ''G''→''H'', let ''K'' be a normal subgroup in ''G'' and φ the natural surjective homomorphism ''G''→''G''/''K'' (where ''G''/''K'' is a quotient group). If ''K'' is a subset of ker(''f'') then there exists a unique homomorphism ''h'':''G''/''K''→''H'' such that ''f'' = ''h'' φ. In other words, the natural projection φ is universal among homomorphisms on ''G'' that map ''K'' to the identity element. The situation is described by the following commutative diagram: File:FundHomDiag.png By setting ''K'' = ker(''f'') we immediately get the first isomorphism theorem. 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「fundamental theorem on homomorphisms」の詳細全文を読む スポンサード リンク
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