Kernel of a group homomorphism

Question

Is it true that for a group homomorphism $\phi: G\to H$, $\phi(e_G)$ necessarily equals $e_H$?

Answer 1

Notice that since $\phi$ is a homomorphism, $$\phi(e_{G})=\phi(e_{G}\star_{G} e_{G})=\phi(e_{G})\star_{H}\phi(e_{G})$$ Can you do it from here?

Answer 2

Yes.

$\phi(x)=\phi(x*_1e_G)=\phi(x)*_2\phi(e_G) ,\ \forall x\in G$

Hence $ \phi(e_G) $ is identity in H.

Thus $\phi(e_G)=e_H$