Group Isomorphism Preserves Inverses/Proof 1

Theorem

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

Let:

$e_G$ be the identity of $\struct {G, \circ}$

$e_H$ be the identity of $\struct {H, *}$.

Then:

$\forall g \in G: \map \phi {g^{-1} } = \paren {\map \phi g}^{-1}$

Proof

An group isomorphism is by definition a group epimorphism.

The result follows from Epimorphism Preserves Inverses.

$\blacksquare$