Example 39.5.6 (Constant group). Let $G$ be an abstract group. Consider the functor which associates to any scheme $T$ the group of locally constant maps $T \to G$ (where $T$ has the Zariski topology and $G$ the discrete topology). This is representable by the scheme

The morphism giving the group structure is the morphism

which maps the component corresponding to the pair $(g, g')$ to the component corresponding to $gg'$. For any scheme $S$ the base change $G_ S$ is a group scheme over $S$ whose functor of points is

as before.

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