Image of the iden­tity un­der a group ho­mo­mor­phism is the identity

For any group homomorphism $f:G\to H$, we have $f\left(e_G\right)=e_H$ where $e_G$ is the identity of $G$ and $e_H$ the identity of $H$.

Indeed, $f\left(e_G\right)f\left(e_G\right)=f\left(e_G{e}_G\right)=f\left(e_G\right)$, so premultiplying by $f(e_G{)}^{-1}$ we obtain $f\left(e_G\right)=e_H$.