Fundamental theorem on homomorphisms
- Fundamental 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.
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" = ker("f") we immediately get the first isomorphism theorem.
Other versions
Similar theorems are valid for monoids, vector spaces, modules, and rings.
ee also
* Quotient category
External links
* [http://planetmath.org/encyclopedia/FundamentalHomomorphismTheorem.html A proof at planetmath]
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