Morphisms and closed subgroup schemes of group schemes

Let $G,H$ be group schemes of finite type over a field $k$, as in Group schemes of finite type over a field. A morphism of $k$-group schemes is a $k$-morphism $f:G\to H$ satisfying $f\circ m_G=m_H\circ (f\times f),f\circ e_G=e_H,i_H\circ f=f\circ i_G\mathrm{.}$ Consequently $G(T)\to H(T)$ is a group homomorphism for every $k$-scheme $T$, naturally in $T$.

A closed subgroup scheme of $G$ is a closed immersion $j:H\hookrightarrow G$ in the sense of Closed immersions of schemes, where $H$ is a group scheme and $j$ is a morphism of group schemes. Its multiplication, identity, and inverse are the restrictions of those of $G$. A closed immersion is a monomorphism: a factorization through its subscheme, when it exists, is unique, since the map of sheaves onto the subscheme's structure sheaf is surjective. Therefore these restricted structure morphisms are uniquely determined. A closed subscheme of $G$ is not assumed to be a subgroup merely because its $k$-rational points form one; all algebra-valued points, including points over nonreduced algebras, are relevant.