Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?
$[G,G]N/N$ is the image of $[G,G]$ under the canonical homomorphism from $G$ to $G/N$. This homomorphism respects taking commutators. So in particular $[G/N,G/N]=[G,G]N/N$.
Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?
$[G,G]N/N$ is the image of $[G,G]$ under the canonical homomorphism from $G$ to $G/N$. This homomorphism respects taking commutators. So in particular $[G/N,G/N]=[G,G]N/N$.