First isomorphism theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Statement

Let $G$ be a group and $\varphi:G \to H$ be a homomorphism of groups. The first isomorphism theorem states that the kernel of $\varphi$ is a normal subgroup, say $N$, and there is a natural isomorphism:

$G/N \cong \varphi(G)$

where $\varphi(G)$ denotes the image in $H$ of $G$ under $\varphi$.