Overview of the group isomorphism theorems

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Statements

$\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N \ar@{.>}[ur]_{\bar{\varphi} } & }$

Group factorisation theorem

$\xymatrix{G \ar[r]^\varphi \ar[d]^\pi & H \\ G/\text{Ker}(\varphi) \ar@{.>}[r]_-\theta & \text{Im}(\varphi) \ar@{_{(}->}[u]^i }$ For $\theta$ an isomorphism
First isomorphism theorem

First group isomorphism theorem (AKA: Group homomorphism theorem)

$\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N & }$

Second group isomorphism theorem

$\xymatrix{G \ar[r]^\varphi \ar[d]_\pi & H \\ G/N & }$