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

Let and . Then

the collection of fibres

Let , then

= 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?