Homomorphism to Group Preserves Identity

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {T, *}$ be a group.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Then:

$\map \phi {e_S} = e_T$

Proof

By hypothesis, $\struct {T, *}$ is a group.

By the Cancellation Laws, all elements of $T$ are cancellable.

Thus Homomorphism with Cancellable Codomain Preserves Identity can be applied.

$\blacksquare$