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:

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

Proof

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

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 7.2$: Lemma $\text{(ii)}$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 12$: Theorem $12.3: \ 3^\circ$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.10$: Theorem $22$