Definition 8

Let $f : G \to G'$ be a homomorphism of groups with operators. The inverse image of the identity element of $G'$ is called the kernel of $f$.

The kernel of $f$ is often denoted by $\mathrm{Ker}(f)$ and the image $f(G)$ of $f$ is sometimes denoted by $\mathrm{Im}(f)$.