Proof verification: If group $G$ is abelian, then quotient group $G/H$ is abelian.

Question

Could someone please verify my following proof?

If group $G$ is abelian and $H\triangleleft G$, then quotient group $G/H$ is abelian.

Proof: Let $G$ be abelian and let $G/H=\left \{ gH:g\in G \right \}$. Let $g_{1}H,g_{2}H\in G/H$. Then $(g_{1}H)(g_{2}H)=g_{1}g_{2}H=g_{2}g_{1}H$ ($G$ is abelian) $=(g_{2}H)(g_{1}H)$. Therefore, $G/H$ is abelian. $\square$

Answer 1

The proof is right. For a similar proof in slightly different language, you can observe that the natural group homomorphism $\varphi:G\to G/H$ is surjective, so that for every $h_1,h_2\in G/H$ we can find $g_1,g_2\in G$ so that $\varphi(g_1)=h_1,\varphi(g_2)=h_2$, from which it follows that $$h_1h_2=\varphi(g_1)\varphi(g_2)=\varphi(g_1g_2)=\varphi(g_2g_1)=\varphi(g_2)\varphi(g_1)=h_2h_1.$$