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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime number, and let $k \\geq 0$ with $p^k \\mid |G|$.\nThen $G$ has a [[subgroup|0X1JXT47GA]] of order $p^k$.\nIn particular, $G$ has a [[Sylow p-subgroup|0053CQE9Y2]].\n\n## Remarks\n\nFor $k = 1$, this is [[Cauchy's theorem|0JM7PA288B]].\nLet $p^n$ be the largest power of $p$ that divides $|G|$. A proof by Wielandt lets $G$ act by left multiplication on the subsets of $G$ of size $p^n$.\nFor example, a group of order $12$ has a subgroup of order $4$ and a subgroup of order $3$.\n\n## History\n\nLudwig Sylow proved the Sylow theorems in 1872, and he published them in Mathematische Annalen.\n\n## Explanations\n\n### L1 human\n\nThe converse of Lagrange's theorem is false in general.\nBut it is true for prime powers: each prime power that divides the order of the group is the order of a subgroup.\nThe largest such subgroups are the Sylow subgroups.\n",
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