Universal Property of the Quotient Group
The universal property of the quotient group is a fact about the quotient group and group homomorphisms that is useful particularly in proving the first isomorphism theorem.
Theorem
Let and be groups, and be a normal subgroup of .
There is a bijection
given explicitly by
where is the quotient epimorphism (natural projection onto the quotient group).
Furthermore, .
While the map here is chosen in the direction where it is natural to define, in practice we wish to take a homomorphism and use this to define a homomorphism .
Lemma
Let and be groups, and be a normal subgroup of .
For any homomorphism satisfying there is a well defined homomorphism given by such that .

Intuitively, if a map vanishes on the normal subgroup , then there is no information lost when attempting to define an equivalent map from the quotient group , as elements of are treated as identical.

We will first prove the lemma above.
Proof
We will prove this map is well defined by showing if and are in the same coset, then .
First note that by assumption for any . Let , then for some and hence
Now, is a homomorphism because .
Furthermore, for all and thus .
Now we prove the main result. In particular, note that the lemma above will give surjectivity of our map .
Proof
Suppose that . Then, we have that . Clearly as a composition of homomorphisms this is a homomorphism , and if then and hence this composition vanishes on .
Surjectivity of follows from the lemma above.
Injectivity of comes about because if then , and for any element , this means . This implies that .