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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number.\nWrite $|G| = p^n m$ with $n \\geq 0$ and $p \\nmid m$.\nA Sylow $p$-subgroup of $G$ is a [[subgroup|0X1JXT47GA]] of $G$ of order $p^n$.\nThe set of the Sylow $p$-subgroups of $G$ is written $\\operatorname{Syl}_p(G)$, and its size is written $n_p$.\n\n## Remarks\n\nA Sylow $p$-subgroup is a maximal [[p-subgroup|06QYRTX9KQ]] of $G$.\nIf $p$ does not divide $|G|$, then the only Sylow $p$-subgroup is $\\{e\\}$.\nIn $S_3$, the Sylow $3$-subgroup is $A_3$, and there are three Sylow $2$-subgroups, each of order $2$.\n\n## Explanations\n\n### L1 human\n\nA Sylow subgroup takes all the factors $p$ of the order of the group at once.\nIt is as large as a $p$-subgroup can be by Lagrange's theorem.\nThe Sylow theorems say that such subgroups exist and that they are all alike.\n",
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