Normal Subgroup

up:: Subgroup (Mathematics)

Given a Group and a Subgroup , is a normal subgroup (denoted ) if

Equivalent definition

Normal subgroups imply left cosets equal to right cosets.
This is equivalent to requiring

that is, the Coset of , , are equal to its right cosets .

Equivalence to a homomorphism’s kernel

Given a Group Homomorphism , then A homomorphism’s kernel is a normal subgroup.

The reciprocal is also true: a normal subgroup induces a homomorphism

whose kernel is .