Lemma 39.7.15. Let $G$ be a group scheme over a field. There exists an open and closed subscheme $G' \subset G$ which is a countable union of affines.

Proof. Let $e \in U(k)$ be a quasi-compact open neighbourhood of the identity element. By replacing $U$ by $U \cap i(U)$ we may assume that $U$ is invariant under the inverse map. As $G$ is separated this is still a quasi-compact set. Set

$G' = \bigcup \nolimits _{n \geq 1} m_ n(U \times _ k \ldots \times _ k U)$

where $m_ n : G \times _ k \ldots \times _ k G \to G$ is the $n$-slot multiplication map $(g_1, \ldots , g_ n) \mapsto m(m(\ldots (m(g_1, g_2), g_3), \ldots ), g_ n)$. Each of these maps are open (see Lemma 39.7.1) hence $G'$ is an open subgroup scheme. By Lemma 39.7.7 it is also a closed subgroup scheme. $\square$

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