- Fundamental theorem on homomorphisms
abstract algebra, the fundamental theoremon homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphismis given, and of the kernel and image of the homomorphism.
The homomorphism theorem is used to prove the
Group theoretic version
Given two groups "G" and "H" and a
group homomorphism"f" : "G"→"H", let "K" be a normal subgroupin "G" and φ the natural surjectivehomomorphism "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
By setting "K" = ker("f") we immediately get the
first isomorphism theorem.
* [http://planetmath.org/encyclopedia/FundamentalHomomorphismTheorem.html A proof at planetmath]
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