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"&rarr;"H", let "K" be a normal subgroup in "G" and &phi; the natural surjective homomorphism "G"&rarr;"G"/"K". If "K" &sub; ker("f") then there exists a unique homomorphism "h":"G"/"K"&rarr;"H" such that "f" = "h" &phi;.

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

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

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