Zariski closure of an algebraic linear group

Question

Let $G$ and $H$ algebraic linear groups and $\phi : G \to H$ a regular group homomorphism. I wonder if $\overline{\phi(G)}$ (the Zariski closure of $\phi(G)$) is again a subgroup and how this could be proven. First of all, I know Zariski closure $\overline{Z}$ of a set $Z$ is defined as $\overline{Z} = V(I(Z))$. I imagine this definition is also valid for algebraic linear groups, since $G$ is an affine variety. However, I can’t seem to find a strategy for this proof: any ideas?