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
.