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.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.2$. Some lemmas on homomorphisms: Lemma $\text{(ii)}$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.3: \ 3^\circ$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $22$