Concept# Fundamental theorem on homomorphisms

Summary

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, or the first isomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and 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 N be a normal subgroup in G and φ the natural surjective homomorphism G → G/N (where G/N is the quotient group of G by N). If N is a subset of ker(f) then there exists a unique homomorphism h: G/N → H such that f = h∘φ.
In other words, the natural projection φ is universal among homomorphisms on G that map N to the identity element.
The situation is described by the following commutative diagram:
:
h is injective if and only if

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