Normal Subgroup is Kernel of Group Homomorphism
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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 $6$