If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely.

This can be used to prove the following proposition:

A chain map between chain complexes and induces homomorphisms between the homology groups of the two complexes.

Proof:

The relation implies that takes cycles to cycles since implies . Also takes boundaries to boundaries since . Hence induces a homomorphism , by universal property of quotient groups.

For , we have . Therefore .