Epimorphism Preserves Inverses

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

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

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

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

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

That is:
 * $\map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$

Proof
Let $\struct {S, \circ}$ be an algebraic structure in which $\circ$ has an identity element $e_S$.

From Epimorphism Preserves Identity, it follows that $\struct {T, *}$ also has an identity element, which is $\map \phi {e_S}$.

Let $y$ be an inverse of $x$ in $\struct {S, \circ}$.

By definition of inverse element:
 * $x \circ y = e_S = y \circ x$

Then:

So $\map \phi y$ is an inverse of $\map \phi x$ in $\struct {T, *}$.

Also see

 * Group Homomorphism Preserves Inverses


 * Epimorphism Preserves Associativity
 * Epimorphism Preserves Commutativity
 * Epimorphism Preserves Identity


 * Epimorphism Preserves Semigroups
 * Epimorphism Preserves Groups


 * Epimorphism Preserves Distributivity