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  "body": "\n## Statement\n\nLet $\\sigma \\in S_n$, and let $\\sigma = c_1 c_2 \\cdots c_r$ be its decomposition into disjoint cycles of lengths $k_1, \\dots, k_r$.\nThen the [[order|0WE2NHZJ7B]] of $\\sigma$ is $\\operatorname{lcm}(k_1, \\dots, k_r)$.\n\n## Remarks\n\nA $k$-cycle has order $k$.\nThe disjoint cycles commute, so $\\sigma^m = c_1^m \\cdots c_r^m$, and this is the identity exactly when each $c_i^m$ is the identity.\nFor example, $(1\\ 2)(3\\ 4\\ 5)$ has order $6$.\n\n## Explanations\n\n### L1 human\n\nA permutation returns to the identity when each of its cycles is back at the start.\nA cycle of length $k$ is back at the start after each multiple of $k$ steps.\nSo you need the first common multiple of all the cycle lengths.\n",
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