Main wiki: Quotient groups
This definition is the reason that must be normal to define a quotient group; it holds because the chain of equalities
holds, where utilizes the fact that for any (true iff is normal, by definition).
For example, consider the subgroup of (which is an additive group). The left cosets are
so . This can be more cleanly written as which is isomorphic to , or the cyclic group . Additional examples:
- The quotient group , where --the group of even integers--is a normal subgroup of , is isomorphic to as well.
- The quotient group , where --the group of integers--is a normal subgroup of the reals , is isomorphic to the circle group defined by the complex numbers with magnitude 1.
Recall that a homomorphism from to is a function such that
for all . The kernel of a homomorphism is the set of elements of that are sent to the identity in , and the kernel of any homomorphism is necessarily a normal subgroup of .
In fact, more is true: the image of under this homomorphism (the set of elements is sent to under ) is isomorphic to the quotient group , by the first isomorphism theorem. This provides a bijection between normal subgroups of and the set of images of under homomorphisms.
Thus normal subgroups can be classified in another manner:
A subgroup of is normal if and only if there exists a homomorphism on whose kernel is .