Group Homomorphism Preserves Identity

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

Let:
 * $e_G$ be the identity of $G$;
 * $e_H$ be the identity of $H$.

Then $\phi \left({e_G}\right) = e_H$.

Proof 1
The result follows directly from the morphism property of $\circ$ under $\phi$:

That is:

Proof 2
A direct application of Homomorphism to Group Preserves Identity and Inverses.

Proof 3
From Homomorphism of Product with Inverse, we have:
 * $\forall x, y \in G: \phi \left({x \circ y^{-1}}\right) = \phi \left({x}\right) * \left({\phi \left({y}\right)}\right)^{-1}$

Putting $x = y$ we have: