Normal Subgroup is Kernel of Group Homomorphism

Theorem

Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Then there exists a group homomorphism of which $N$ is the kernel.

Proof

Let $G / N$ be the quotient group of $G$ by $N$.

Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:

$\forall x \in G: \map {q_N} x = x N$

Then from Quotient Group Epimorphism is Epimorphism, $N$ is the kernel of $q_n$

Thus $q_N$ is that group homomorphism of which $N$ is the kernel.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $6$