Group Homomorphism Preserves Inverses

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

Hence the notation $\map \phi x^{-1}$ can be used unambiguously.

This can be illustrated using the following commutative diagram:


 * $\begin{xy} \xymatrix@L+2mu@+1em{

G \ar[r]^*{\phi} \ar[d]_*{\iota_G} & H \ar[d]^*{\iota_H} \\ G \ar[r]_*{\phi} & H }\end{xy}$

where $\iota_G$ and $\iota_H$ are the inversion mappings on $G$ and $H$ respectively.