Kernel of group homomorphism

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:

• if \(f(g_1) = e_H\) and \(f(g_2) = e_H\) then \(e_H = f(g_1) f(g_2) = f(g_1 g_2)\), so the kernel is closed under \(G\)’s operation;

• if \(f(x) = e_H\) then \(e_H = f(e_G) = f(x^{-1} x) = f(x^{-1}) f(x) = f(x^{-1})\) (where we have used that the image of the identity is the identity), so inverses are contained in the putative subgroup;

• \(f(e_G) = e_H\) because the image of the identity is the identity, so the identity is contained in the putative subgroup.

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.