Second isomorphism theorem for groups ($H/(H\cap N)\cong HN/N$)

For a subgroup $H\le G$ and a normal subgroup $N⊴G$, the map $h\mapsto hN$ from $H$ onto $HN/N$ is a group homomorphism with kernel $H\cap N$. The first isomorphism theorem gives the displayed group isomorphism. The product $HN$ is a subgroup because $N$ is normal. In particular, if $HN=G$ and $H\cap N=1$, then $H\cong G/N$.