= Solution
A <group homomorphism> $\phi:H\to G$ is a map preserving multiplication:
$$
\boxed{\phi(h_1h_2)=\phi(h_1)\phi(h_2)\quad\text{for all }h_1,h_2\in H.}
$$
It necessarily sends $e_H$ to $e_G$ and sends inverses to inverses, by applying this relation to $e_H^2$ and $hh^{-1}$. Its <kernel of a group homomorphism> is
$$
\boxed{\ker\phi=\{h\in H:\phi(h)=e_G\}.}
$$
The kernel is a <normal subgroup> of the domain: if $k\in\ker\phi$ and $h\in H$, then $\phi(hkh^{-1})=\phi(h)e_G\phi(h)^{-1}=e_G$. The kernel’s identity condition refers to the identity in the target group.
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