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? 


A coset $xN$ of $N\unlhd G$ for nontrivial $x$ cannot be isomorphic to $N$ if $xN\neq N$, since $xN$ has no identity element (why?) and is thus not a group.
There is, however, a bijection $\varphi:N\to xN$ given by $n\mapsto xn$. So the sets are of the same cardinality.
