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:
\(\ds \map \phi x * \map \phi y\) | \(=\) | \(\ds \map \phi {x \circ y}\) | Definition of Morphism Property | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {e_S}\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi {y \circ x}\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi y * \map \phi x\) | Definition of Morphism Property |
So $\map \phi y$ is an inverse of $\map \phi x$ in $\struct {T, *}$.
$\blacksquare$
Warning
Note that this result is applied to epimorphisms.
For a general homomorphism which is not surjective, nothing definite can be said about the behaviour of the elements of its codomain which are not part of its image.
Also see
- Epimorphism Preserves Associativity
- Epimorphism Preserves Commutativity
- Epimorphism Preserves Identity
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.2$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.1$