Group Homomorphism Preserves Inverses/Proof 1

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:

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

Proof

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.

$\blacksquare$

Sources

1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.2$ Homomorphisms and their elementary properties
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms