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.