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EspañolFUNDAMENTAL THEOREM ON HOMOMORPHISMS
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.
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''. If ''K'' ⊂ ker(''f'') then there exists a unique homomorphism ''h'':''G''/''K''→''H'' such that ''f'' = ''h'' φ.
The situation is described by the following commutative diagram:
By setting ''K'' = kernel(''f'') we immediately get the first isomorphism theorem.
Similar theorems are valid for monoids, vector spaces, modules, and rings.
★ Quotient category
★ A proof at planetmath
The homomorphism theorem is used to prove the isomorphism theorems.
| Contents |
| Group theoretic version |
| Other versions |
| See also |
| External links |
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''. If ''K'' ⊂ ker(''f'') then there exists a unique homomorphism ''h'':''G''/''K''→''H'' such that ''f'' = ''h'' φ.
The situation is described by the following commutative diagram:
By setting ''K'' = kernel(''f'') we immediately get the first isomorphism theorem.
Other versions
Similar theorems are valid for monoids, vector spaces, modules, and rings.
See also
★ Quotient category
External links
★ A proof at planetmath
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