Definition 39.4.3. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.

1. A closed subgroup scheme of $G$ is a closed subscheme $H \subset G$ such that $m|_{H \times _ S H}$ factors through $H$ and induces a group scheme structure on $H$ over $S$.

2. An open subgroup scheme of $G$ is an open subscheme $G' \subset G$ such that $m|_{G' \times _ S G'}$ factors through $G'$ and induces a group scheme structure on $G'$ over $S$.

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