first isomorphism theorem

This theorem is extremely useful for identifying/ understanding groups by using a homomorphism from a known/ understood group.

Theorem

Let $f:G\to H$ be a group homomorphism.

1. $\text{im}f$ is a subgroup
2. $\ker f$ is a normal subgroup
3. $\text{im}f\simeq G/\ker f$

Special case (surjective)

Let $f:G\to H$ be a surjective group homomorphism. Then $G/\ker f\simeq H$.

Picture:

Proof

todo

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References