Group Isomorphism Preserves Identity/Proof 1

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:

$\map \phi {e_G} = e_H$


Proof

An group isomorphism is by definition a group epimorphism.

The result follows from Epimorphism Preserves Identity.

$\blacksquare$

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