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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]] and let $(H_i)_{i \\in I}$ be a family of [[subgroups|0X1JXT47GA]] of $G$, indexed by a nonempty set $I$.\nThen the intersection $\\bigcap_{i \\in I} H_i$ is a subgroup of $G$.\n\n## Remarks\n\nThe identity is in each $H_i$, and if $a, b$ are in each $H_i$, then $ab^{-1}$ is in each $H_i$.\nThe union of two subgroups is in general not a subgroup: in $(\\mathbb{Z}, +)$, $2 \\in 2\\mathbb{Z}$ and $3 \\in 3\\mathbb{Z}$, but $5$ is in neither.\nThe union of two subgroups $H$ and $K$ is a subgroup only when $H \\subseteq K$ or $K \\subseteq H$.\n\n## Explanations\n\n### L1 human\n\nIf several subgroups each pass the subgroup test, then their common part also passes it.\nFor example, $2\\mathbb{Z} \\cap 3\\mathbb{Z} = 6\\mathbb{Z}$ in the integers.\nThis fact lets us define the smallest subgroup that contains a given subset.\n",
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