Quotients

Quotient

Let be a group and a normal subgroup. Let denote the set of cosets (left or right, they are the same)

Proposition

is a group with composition that for all . And the identity is , while the inverse is given by for any .

Proposition

for finite groups and .

Proof

Def Canonical(Natural) Map Let be a group and a normal subgroup, then the canonical map is defined to be:

Prop The canonical map is a surjective homomorphism whose kernel is .