Are Cosets Isomorphic to One Another 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? 


You are correct arguing that cosets are equicardinal (the same size), since for fixed $x$ the map $n \to xn$ is a bijection.
Your mistake (pointed out in comments and the other answer) is saying that makes them isomorphic. They can't be, since only $N$ itself is a group. When you are studying group theory, "isomorphic" means "isomorphic as groups".
In general, isomorphisms preserve the structure of the objects you are studying. When you are studying sets, an isomorphism is just a bijection, since sets have no other structure. So the map $n \to xn$ is indeed a set isomorphism.
