What is different between $G/N$ and $GN/N$?
($G$ denotes a group, and N does a normal subgroup of G)
If you look at the element of each group above,
I think $G/N$ contains element of form $gN$, where $g$ is an element of G but not in $N$.
and $GN/N$ contains element of form $gnN$ which is equivalent to $gN$,
because $n$ is an element of $N$.
Then it looks like they indicate same group.
Is there anything wrong with my reasoning?
If so, could you point out and fix it?
The notation $GN/N$ may be useful in derived group.
For example:
$N$ is the normal subgroup of $G$, the $(G/N)^{\prime}=\phi(G)^{\prime}=\phi(G^{\prime})=G^{\prime}N/N=\{gH: g \in G^{\prime}\}$
$N$ may not be the normal subgroup of $G^{\prime}$, so the notation above is a right way to explain all the cosets.
By the way, $G^{\prime}N/N \cong G^{\prime}/(N \cap G^{\prime})$ is really useful.