Constant finite group scheme

To let the abstract Galois group $G=\operatorname{Gal}(K/F)$ act on $\operatorname{Spec}K$ over $\operatorname{Spec}F$, regard $G$ as a scheme: put one copy of the base at every group element, with multiplication dictated by the multiplication table of $G$.

Let $G$ be a finite group and $S$ a scheme. The constant finite group scheme $G_S$ is

with multiplication, identity, and inverse induced by those of $G$. If $S=\operatorname{Spec}R$, then

It is finite étale over $S$, and its $T$-points are the locally constant maps from $T$ to the finite set $G$. It supplies the group scheme in the Galois extension as étale torsor construction.