Group Homomorphism Preserves Inverses

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:
 * $\forall x \in G: \phi \left({x^{-1}}\right) = \left({\phi \left({x}\right)}\right)^{-1}$

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

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

Let $x \in G$.

Then:

So by definition of inverse, $\phi \left({x^{-1}}\right)$ is the inverse of $\phi \left({x}\right)$.

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 $x = e_G$ and $y = x$ we have: