Image of the identity is the identity

#group_theory

Suppose $G_{1}$ and $G_{2}$ are groups, with identity elements $e_{1}$ and $e_{2}$ , respectively. Prove that if $ϕ : G_{1} \to G_{2}$ is a homomorphism^[1], then $ϕ (e_{1}) = e_{2}$ .

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Suppose $G_1$ and $G_2$ are groups, with identity elements $e_1$ and $e_2$, respectively. Prove that if $\phi:G_1\to G_2$ is an isomorphism, then $\phi(e_1)=e_2$.

The version of this problem that appeared on the Spring 2019 exam assumed $ϕ$ was an isomorphism, but that assumption was unnecessarily strong. ↩︎