The kernel of a group homomorphism is the collection of all elements in such that the identity of .
It is important to note that the kernel of any group homomorphism is always a subgroup of . Indeed:
It turns out that the notion of "normal subgroup" coincides exactly with the notion of "kernel of homomorphism". (Proof.) The "kernel of homomorphism" viewpoint of normal subgroups is much more strongly motivated from the point of view of category theory; Timothy Gowers considers this to be the correct way to introduce the teaching of normal subgroups in the first place.