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Let $G$ be group, $N$ is normal in $G$ and $\mathcal{L}:G \to G^\prime$ be a surjective group homomorphism, then prove that the image $\mathcal{L}(N)$ of $N$ is a normal subgroup of $G’$.

I let $g_{1},g_{2} \in G$ such that, $g_1,g_2 \in G/N$. Since $N$ is normal $g_1 N=N g_1$ and $g_2 N= N g_2$. $g_1 N g_2 N = g_1 g_2 N = N$, $N g_1 N g_2 = N g_1 g_2 = N$, $\mathcal{L}( g_1 g_2 N) = g_1^\prime g_2^\prime \mathcal{L}(N)= \mathcal{L}(N) g_1^\prime g_2^\prime$. Hence this is my proof, I need some corrections, and is it necessary to consider the fact $g_1,g_2 \in N$?

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I will assume that you assume $\mathcal L$ is surjective and that your definition of a normal subgroup $N\leq G$ is a subgroup such that $gN = Ng$ for all $g\in G$. In this case, you need to prove that g'\mathcal L(N) = \mathcal L(N)g' for all g'\in G' (I assume you already know that the image of a subgroup is a subgroup).

To do this, let g'\in G'. Using the fact that $\mathcal L$ is surjective, we have some $g\in G$ such that \mathcal L(g) = g', hence g'\mathcal L(N) = \mathcal L(g)\mathcal L(N) = \mathcal L(gN) = \mathcal L(Ng) = \mathcal L(N)\mathcal L(g) = \mathcal L(N)g', thus $\mathcal L(N)$ is normal.