kernel (thing)

(Group Theory)

Given a homomorphism H between two groups G₁ and G₂, the homomorphism's "kernel" (symbolized Ker H) is the set of elements of G₁ that give rise to the identity element of G₂.

If we call the identity element of G₂ "i₂", we can say

g e Ker H -> H(g) = i₂

Ker H necessarily contains the identity element i₁ of G₁.

Since H is a homomorphism, for all p, q e G₁,

G₂(H(p), H(q)) = H (G₁(p, q)).

Let a = H (i₁).  Now, for any g e G₁,

G₂(H(i₁), H(g)) = H (G₁(i₁, g))

But this means that

G₂(a, H(g)) = H (g)

for all g e G₁,  Although Im H is not necessarily G₂, G₂ is still a group. Therefore, a = i₂, meaning i₁ e Ker H, which was to be proven.