Lemma 39.4.4. Let $S$ be a scheme. Let $(G, m, e, i)$ be a group scheme over $S$.

1. A closed subscheme $H \subset G$ is a closed subgroup scheme if and only if $e : S \to G$, $m|_{H \times _ S H} : H \times _ S H \to G$, and $i|_ H : H \to G$ factor through $H$.

2. An open subscheme $H \subset G$ is an open subgroup scheme if and only if $e : S \to G$, $m|_{H \times _ S H} : H \times _ S H \to G$, and $i|_ H : H \to G$ factor through $H$.

Proof. Looking at $T$-valued points this translates into the well known conditions characterizing subsets of groups as subgroups. $\square$

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