I'm quite confused about the concept of quotient groups. From what I gather from my course so far:

Let $\displaystyle \psi :G \rightarrow H$ and $\displaystyle K = \text{kernel} \, \, \psi$. Then

$\displaystyle G/K = $ the collection of fibres $\displaystyle \{ \psi ^{-1} (a) | a \in H \}$

Let $\displaystyle N \unlhd G$, then

$\displaystyle G/N$ = the collection of left cosets (or right cosets).

But how is that a group? You get a collection of sets, and only one of those sets contain the identity so only one of those sets are a group?

Looking at the Wikipedia example:

Quotient group - Wikipedia, the free encyclopedia

they get G/N = { {0, 3}, {1, 4}, {2, 5} } which is a collection of sets... so how are they a group?