I've looked at this a few different ways. I don't really get it.
I don't see how
or for all .
If we may conclude from the partitioning of cosets that . Thus, we may define a group to be the set with . Clearly is the identity element, it's associativity is associated from (why)? It is easy to prove that etc. Then, one can prove that is a well define homomorphism and . So?