A basic question, I seem to be struggling with the concept of quotient groups and cosets.
Suppose $G$ is a group, and $N$ is a normal subgroup.
I know that is $x$ and $y$ are in the same coset of $G/N$, then $xN = yN$.
I also know that cosets are either disjoint or equivalent, $G/N$ is the set of cosets. Also, $G$ can be written as a disjoint union of the cosets.
Is it correct that all cosets are isomorphic to $N$, since they are of the form $xN$ for some $x \in G$?
Or is it possible for more elements of $G$ to be in one coset than in another?