$\big(\frac{G}{N}\big)^{(n)}=\frac{G^{(n)}N}{N}$

Question

$G$ is a group and $N\unlhd G$ then, $\forall n\in \mathbb N, \big(\frac{G}{N}\big)^{(n)}=\frac{G^{(n)}N}{N}$

$(G/N)^{(n)}=\left \{ Nx :x\in G \right \}^{(n)}=\left \{ Nx^{n} :x\in G \right \}$ but from here I couldn't reach $G^{(n)}N/N$ part.

Any idea will be appreciated.

Answer 1

In general, if $\varphi: G\to H$ is a surjective homomorphism then $\varphi(G')=H'$. This is easy to check from the definitions, so I'll leave it to you. Then by induction it follows that $\varphi(G^{(n)})=H^{(n)}$.

So in particular if we take $\pi: G\to G/N$ to be the canonical homomorphism then:

$(G/N)^{(n)}=[\pi(G)]^{(n)}=\pi(G^{(n)})=G^{(n)}N/N$