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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity element $e$.\nA subgroup of $G$ is a subset $H \\subseteq G$ with these three properties.\nThe identity $e$ is in $H$.\nFor all $a, b \\in H$, the product $ab$ is in $H$.\nFor each $a \\in H$, the inverse $a^{-1}$ is in $H$.\nThe notation $H \\leq G$ means that $H$ is a subgroup of $G$.\n\n## Remarks\n\nA subgroup $H$ is a group with the operation of $G$ restricted to $H$, and its identity is the identity of $G$.\nEach group $G$ has the subgroups $\\{e\\}$ and $G$.\nThe even integers $2\\mathbb{Z}$ form a subgroup of $(\\mathbb{Z}, +)$.\nThe natural numbers do not form a subgroup of $(\\mathbb{Z}, +)$, because $-1$ is not a natural number.\n\n## Explanations\n\n### L1 human\n\nA subgroup is a smaller group that lives inside a bigger group and uses the same operation.\nTo check that a subset is a subgroup, you check that it contains the identity and that it is closed under products and inverses.\nFor example, the rotations of a square form a subgroup of all the symmetries of the square.\n",
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