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.