How do we know that this map is well-defined?
Independent from the choices we can choose for ?
Well, let us assume there are two ways to expess an element in
Then the function maps them into,
and because the conditon we had said that,
Thus, the map is well-defined.
The function is a group homomorphism,
The function is one-to-one,
are both in same coset of and,
are both in same coset of .
The function is onto,
That result is trivial.
Any, can be obtained from mapping the element,