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:
 * $e_G \in \map \ker \phi$

where $\map \ker \phi$ is the kernel of $\phi$.

Proof
From the definition of kernel:
 * $\map \ker \phi = \set {x \in G: \map \phi x = e_H}$

From Group Homomorphism Preserves Identity we have that:
 * $\map \phi {e_G} = e_H$

Hence the result.

Proof

 * : Chapter $2$: Group Homomorphism and Isomorphism: $\S 65$