Identity is in Kernel of Group Homomorphism

Theorem
Let $G$ and $H$ be groups.

Let $e_G$ and $e_H$ be the identity elements of $G$ and $H$ respectively.

Let $\phi: G \to H$ be a (group) homomorphism from $G$ to $H$.

Then:
 * $e_G \in \ker \left({\phi}\right)$

where $\ker \left({\phi}\right)$ is the kernel of $\phi$.

Proof
From the definition of kernel:
 * $\ker \left({\phi}\right) = \left\{{x \in G: \phi \left({x}\right) = e_H}\right\}$

From Group Homomorphism Preserves Identity we have that:
 * $\phi \left({e_G}\right) = e_H$

Hence the result.