# Homomorphism to Group Preserves Inverses

## Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {T, *}$ be a group.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Let $x^{-1}$ be an inverse of $x$ for $\circ$.

Then $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.

## Proof

By hypothesis, $\struct {T, *}$ is a group.

By Group Axiom $\text G 2$: Existence of Identity Element, $\struct {T, *}$ has an identity.

Thus Homomorphism with Identity Preserves Inverses can be applied.

$\blacksquare$