Definition:Kernel of Group Homomorphism

Definition
Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups.

Let $\phi: \left({G, \circ}\right) \to \left({H, *}\right)$ be a group homomorphism.

The kernel of $\phi$ is the subset of the domain of $\phi$ defined as:
 * $\ker \left({\phi}\right) = \left\{{x \in G: \phi \left({x}\right) = e_H}\right\}$

where $e_H$ is the identity of $H$.

That is, $\ker \left({\phi}\right)$ is the subset of $G$ that maps to the identity of $H$.

Also see

 * Identity in Kernel of Group Homomorphism where it is shown that $e_G \in \ker \left({\phi}\right)$ where $e_G$ is the identity of $G$.