natural projection
Proposition. If is a normal subgroup
of a group , then the mapping
is a surjective homomorphism
whose kernel is .
Proof. Because every coset appears as image, the mapping is surjective. It is also homomorphic, since for all elements of , one has
The identity element of the factor group is the coset , whence
The mapping in the proposition is called natural projection or canonical homomorphism.