Homomorphism to Group Preserves Identity

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

Let $$\left({T, *}\right)$$ be a group.

Let $$\left({S, \circ}\right)$$ have an identity $$e_S$$.

Then:


 * $$\phi \left({e_S}\right) = e_T$$;


 * If $$x^{-1}$$ is an inverse of $$x$$ for $$\circ$$, then $$\phi \left({x^{-1}}\right)$$ is an inverse of $$\phi \left({x}\right)$$ for $$*$$.

Proof

 * If $$\left({T, *}\right)$$ is a group, then all elements of $T$ are cancellable, and Homomorphism with Cancellable Range Preserves Identity applies.
 * If $$\left({T, *}\right)$$ is a group, then $$\left({T, *}\right)$$ has an identity and Homomorphism with Identity Preserves Inverses applies.