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First isomorphism theorem

Statement

Given a homomorphism $\phi: G_1 \to G_2$, there is an isomorphism $G_1/\ker(\phi) \to \phi(G_1)$ defined by $g\ker(\phi) \mapsto \phi(g)$.

The map $\phi$ can split into three stages:

1. The quotient map $G_1 \to G_1/\ker(\phi)$ mapping $g \to g \ker(\phi)$
2. The above isomorphism $G_1/\ker(\phi) \to \phi(G_1)$
3. The inclusion map $\phi(G_1) \to G_2$

Corollaries

If $\phi$ is injective, then $G_1 \cong \mathrm{im}\phi$.

If $\phi$ is surjective, then $G_1 / \ker\phi \cong G_2$.

all these pages adapted with probably insufficient credit from my university's lecture notes