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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $P$ be a [[Sylow p-subgroup|0053CQE9Y2]] of $G$.\nThen $n_p$ divides the [[index|0DTXBQPA2S]] $[G : P]$.\n\n## Remarks\n\nBy [[this formula|09ZJ6CAK7S]], $n_p = [G : N_G(P)]$, and $[G : P] = [G : N_G(P)] \\cdot [N_G(P) : P]$.\nIf $|G| = p^n m$ with $p \\nmid m$, then $n_p$ divides $m$.\nFor example, in a group of order $15$, $n_3$ divides $5$ and $n_3 \\equiv 1 \\pmod 3$, so $n_3 = 1$.\n\n## Explanations\n\n### L1 human\n\nThe number of Sylow subgroups is the size of a conjugation orbit.\nOrbit sizes divide the order of the group, and here even the part of the order without $p$.\nThis gives a strong limit on $n_p$.\n",
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