First isomorphism theorem

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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.