Group Homomorphism Preserves Inverses/Proof 2

Theorem

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

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

Let:

$e_G$ be the identity of $G$

$e_H$ be the identity of $H$

Then:

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

Proof

A direct application of Homomorphism to Group Preserves Inverses.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms