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  "body": "\n## Statement\n\nLet $\\sigma, \\tau \\in S_n$.\nThen $\\sigma$ and $\\tau$ are [[conjugate|0DXD4QSZ8A]] in $S_n$ if and only if they have the same [[cycle type|0J3K4Y1KPD]].\n\n## Remarks\n\nThe key formula is $\\pi (a_1\\ a_2\\ \\dots\\ a_k) \\pi^{-1} = (\\pi(a_1)\\ \\pi(a_2)\\ \\dots\\ \\pi(a_k))$ for each $\\pi \\in S_n$.\nSo conjugation keeps the cycle type, and a suitable $\\pi$ sends the cycles of $\\sigma$ onto the cycles of $\\tau$.\nThus the number of conjugacy classes of $S_n$ is the number of partitions of $n$.\nFor example, $S_4$ has $5$ conjugacy classes.\n\n## Explanations\n\n### L1 human\n\nConjugation by $\\pi$ renames the points, but it keeps the shape of each cycle.\nConversely, two permutations of the same shape differ only by a renaming of the points.\nSo in $S_n$, conjugate means \"same shape\".\n",
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