Since $H$ is a normal subgroup of $G$, we have that for any $g \in G$ and $h \in H$, $ghg^{-1} \in H$. Now, let $gN \in G/N$ and $hN \in H/N$. We want to show that $(gN)(hN)(gN)^{-1} \in H/N$.
Notice that $(gN)(hN)(gN)^{-1} = (ghg^{-1})N$. Since $ghg^{-1} \in
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