Group Isomorphism Preserves Inverses/Proof 2

Theorem

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism.

Let:

$e_G$ be the identity of $\struct {G, \circ}$

$e_H$ be the identity of $\struct {H, *}$.

Then:

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

Let $g \in G$.

$\ds \map \phi g * \map \phi {g^{-1} }$

$=$

$\ds \map \phi {g \circ g^{-1} }$

Definition of Group Isomorphism

$\ds $

$=$

$\ds \map \phi {e_G}$

Definition of Inverse Element

$\ds $

$=$

$\ds e_H$

It follows from Inverse in Group is Unique that $\map \phi {g^{-1} }$ is the unique inverse element of $\map \phi g$ in $\struct {H, *}$.

That is:

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

$\blacksquare$

1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.5$: Theorem $4$