Group Isomorphism Preserves Identity

Theorem
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be a group isomorphism.

Let:
 * $e_S$ be the identity of $\left({S, \circ}\right)$
 * $e_T$ be the identity of $\left({T, *}\right)$.

Then:
 * $\phi \left({e_S}\right) = e_T$
 * $\forall x \in S: \phi \left({x^{-1}}\right) = \left({\phi \left({x}\right)}\right)^{-1}$

Proof
An group isomorphism is by definition a group epimorphism.

The result follows from Epimorphism Preserves Identity and Epimorphism Preserves Inverses.