Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?

Question

Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?

I think the statement is correct. But why do we have to write: $[G,G]N/N$ here instead of just $[G,G]/N$?

Thanks!

Answer 1

$[G,G]N/N$ is the image of $[G,G]$ under the canonical homomorphism from $G$ to $G/N$. This homomorphism respects taking commutators. So in particular $[G/N,G/N]=[G,G]N/N$.

Answer 2

(1) $N$ is normal subgroup of $G$; so it is normal subgroup of every subgroup between $N$ and $G$.

(2) When talking of (some subgroup)$/N$, this (some subgroup) should be taken containing $N$.

(3) We ask: given a normal subgroup $N$ of $G$, does it always contain $[G,G]$? You can easily see negative answer by taking simple examples of finite order groups $G$.

(4) Thus, when we want to factor $[G,G]$ by $N$, first look smallest subgroup above $N$(=containing $N$) which contains $[G,G]$; it is precisely $[G,G]N$.

I hope this will clarify your doubt.