Isomorphism Preserves Inverses/Proof 2

Theorem

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

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

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


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


That is, if and only if:

$\map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$


Proof

We have that an isomorphism is a fortiori an epimorphism.

The result follows from Epimorphism Preserves Inverses.

$\blacksquare$

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