Group theory/Isomorphism theorem for residue class groups/Fact

Isomorphism theorem for residue class groups

Let ${}G$ be a group, and let ${}N\subseteq G$ be a normal subgroup with residue class group ${}Q=G/N$. Let ${}H\subseteq G$ be another normal subgroup in ${}G$ with ${}N\subseteq H$.

Then the

image ${}{\overline {H}}$ of ${}H$ in ${}Q$ is a normal subgroup, and we have a canonical isomorphism

${}G/H\cong Q/{\overline {H}}\,.$

Proof, Write another proof