Inverse of Group Isomorphism is Isomorphism/Proof 1

Theorem

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a mapping.

Then $\phi$ is an isomorphism if and only if $\phi^{-1}: \struct {H, *} \to \struct {G, \circ}$ is also an isomorphism.

Proof

A specific instance of Inverse of Algebraic Structure Isomorphism is Isomorphism.

$\blacksquare$