Second Isomorphism Theorem

theoremgroup-theory

Intuition

Formal Statement

For a group $(G,\cdot )$, with subgroup $H\le G$, and normal subgroup $N⊴G$, we have

$${}^{HN}{/}_N\simeq {}^H{/}_{H\cap N}.$$

Note that $HN=\{hn\mid h\in H,n\in N\}\le G$, $N⊴HN$, and $H\cap N⊴H$.

Furthermore, there exists an isomorphism $F:{}^H{/}_{H\cap N}\to {}^{HN}{/}_N$, defined by $F(h\cdot (H\cap N))=hN$.

Corollaries