Kernel of group homomorphism

by Patrick Stevens Jun 14 2016 updated Jun 17 2016

The kernel of a Group homomorphism $~$f: G \to H$~$ is the collection of all elements $~$g$~$ in $~$G$~$ such that $~$f(g) = e_H$~$ the identity of $~$H$~$.

It is important to note that the kernel of any group homomorphism $~$G \to H$~$ is always a subgroup of $~$G$~$. 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.