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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$.\nThe subgroup generated by $S$, written $\\langle S \\rangle$, is the intersection of all [[subgroups|0X1JXT47GA]] of $G$ that contain $S$.\n\n## Remarks\n\nBy [[the intersection property|0ZTGMDNVTB]], $\\langle S \\rangle$ is a subgroup, and it is contained in each subgroup that contains $S$.\nIts elements are the products $s_1^{\\varepsilon_1} s_2^{\\varepsilon_2} \\cdots s_k^{\\varepsilon_k}$ with $k \\geq 0$, $s_i \\in S$ and $\\varepsilon_i \\in \\{1, -1\\}$, where the empty product ($k = 0$) is the identity.\nIn particular, $\\langle \\emptyset \\rangle = \\{e\\}$.\n\n## Explanations\n\n### L1 human\n\nStart with some elements of a group, and add everything that you can make from them with products and inverses.\nThe result is the subgroup generated by these elements.\nFor example, in $(\\mathbb{Z}, +)$, the subgroup generated by $\\{4, 6\\}$ is the set $2\\mathbb{Z}$ of even integers.\n",
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