The Image of normal subgroup is also normal subgroup?

Question

Let $G$ be a group , $N\lhd G$ ,

$ \varphi:G\rightarrow G'$ is homomorphism onto $G'$,

prove that $\varphi(N)=\{\varphi(n):n\in N\}$ is normal subgroup of $G'$

Answer 1

Let $\phi(n)\in\phi(N)$. Let $a\in G'$. Then $\exists$ $g\in G$ such that $\phi(g)=a$. Then $a\phi(n)a^{-1}=\phi(g)\phi(n)\phi(g)^{-1}=\phi(gng^{-1})\in\phi(N)$ because $gng^{-1}\in N$ because $N$ is normal. QED.