- Fundamental theorem on homomorphisms
In

abstract algebra , the, also known as thefundamental theorem on homomorphisms**fundamental homomorphism theorem**, relates the structure of two objects between which ahomomorphism is given, and of the kernel and image of the homomorphism.The homomorphism theorem is used to prove the

isomorphism theorem s.**Group theoretic version**Given two groups "G" and "H" and a

group homomorphism "f" : "G"→"H", let "K" be anormal subgroup in "G" and φ the naturalsurjective 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" = ker("f") we immediately get the

first isomorphism theorem .**Other versions**Similar theorems are valid for

monoid s,vector space s, modules, and rings.**ee also***

Quotient category **External links*** [

*http://planetmath.org/encyclopedia/FundamentalHomomorphismTheorem.html A proof at planetmath*]

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