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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number that divides $|G|$.\nThen $G$ has an element of [[order|0WE2NHZJ7B]] $p$.\n\n## Remarks\n\nEquivalently, $G$ has a [[subgroup|0X1JXT47GA]] of order $p$, namely the cyclic subgroup generated by such an element.\nIt is a partial converse of [[Lagrange's theorem|3C53TD]]: a prime divisor of $|G|$ is always the order of a subgroup.\nOne proof lets $\\mathbb{Z}/p\\mathbb{Z}$ act by cyclic shifts on the set of tuples $(g_1, \\dots, g_p)$ with $g_1 g_2 \\cdots g_p = e$, and uses the [[fixed point lemma|098D2E7DEH]].\nThe [[first Sylow theorem|0V98WY94B8]] generalises the result to prime powers.\n\n## History\n\nAugustin-Louis Cauchy discovered the theorem in 1845.\n\n## Explanations\n\n### L1 human\n\nLagrange says that the order of an element divides the order of the group.\nCauchy says that, for prime numbers, each divisor really appears as the order of an element.\nFor example, a group of order $15$ has an element of order $3$ and an element of order $5$.\n",
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