Group schemes of finite type over a field

Let $k$ be a field. A group scheme of finite type over $k$ is a finite-type $k$-scheme $G$ with $k$-morphisms $m:G{\times }_{k}G\to G,e:\mathrm{Spec}k\to G,i:G\to G$ satisfying the following identities of scheme morphisms. Multiplication is associative, $m\circ (m\times \mathrm{id})=m\circ (\mathrm{id}\times m)$ on $G^3$; $m\circ (e\times \mathrm{id})=\mathrm{id}=m\circ (\mathrm{id}\times e)$ under the canonical identifications; and $m\circ (i,\mathrm{id})=e\circ p=m\circ (\mathrm{id},i)$, where $p:G\to \mathrm{Spec}k$ is the structure map. The products exist by Existence of all scheme fibre products; the base and finite-type conventions are Schemes and morphisms over a base and Locally finite type and finite type morphisms.

For every $k$-scheme $T$, put $G(T)={\mathrm{Hom}}_k(T,G)$. The three structure morphisms give a group law on $G(T)$, naturally under precomposition in $T$. In particular for every commutative unital $k$-algebra $R$, $G(R)$ means $G(\mathrm{Spec}R)$ and is a group, including for algebras with nilpotents. The definition imposes neither reducedness nor smoothness, and allows finite nonreduced group schemes. A group scheme is called commutative if $m$ agrees with its composition with the factor-exchange map.