Group Homomorphism Preserves Inverses

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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} \[email protected]+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.


Proof 1

Let $x \in G$.


Then:

\(\ds \map \phi x * \map \phi {x^{-1} }\) \(=\) \(\ds \map \phi {x \circ x^{-1} }\) Definition of Group Homomorphism
\(\ds \) \(=\) \(\ds \map \phi {e_G}\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds e_H\) Group Homomorphism Preserves Identity

So, by definition, $\map \phi {x^{-1} }$ is the right inverse of $\map \phi x$.

$\Box$


Similarly:

\(\ds \map \phi {x^{-1} } * \map \phi x\) \(=\) \(\ds \map \phi {x^{-1} \circ x}\) Definition of Group Homomorphism
\(\ds \) \(=\) \(\ds \map \phi {e_G}\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds e_H\) Group Homomorphism Preserves Identity

So, again by definition, $\map \phi {x^{-1} }$ is the left inverse of $\map \phi x$.

$\Box$


Finally, as $\map \phi {x^{-1} }$ is both:

a left inverse of $\map \phi x$

and:

a right inverse of $\map \phi x$

it is by definition an inverse.

From Inverse in Group is Unique, $\map \phi {x^{-1} }$ is the only such element.

Hence the result.

$\blacksquare$


Proof 2

A direct application of Homomorphism to Group Preserves Inverses.

$\blacksquare$


Proof 3

From Group Homomorphism of Product with Inverse, we have:

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

Putting $x = e_G$ and $y = x$ we have:

\(\ds \map \phi {x^{-1} }\) \(=\) \(\ds \map \phi {e_G \circ x^{-1} }\)
\(\ds \) \(=\) \(\ds \map \phi {e_G} * \paren {\map \phi x}^{-1}\)
\(\ds \) \(=\) \(\ds e_H * \paren {\map \phi x}^{-1}\) Group Homomorphism Preserves Identity
\(\ds \) \(=\) \(\ds \paren {\map \phi x}^{-1}\)

$\blacksquare$


Sources

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