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  "body": "\n## Statement\n\nLet $X$ be a finite set and let $\\sigma \\in \\operatorname{Sym}(X)$.\nThen $\\sigma$ is a product of cycles of length at least $2$ whose sets of moved points are pairwise disjoint.\nThis product is unique up to the order of the factors.\n\n## Remarks\n\nThe cycles come from the orbits of $\\langle \\sigma \\rangle$ on $X$: each orbit with more than one point gives one cycle.\nThe order of the factors does not matter, because [[disjoint cycles commute|07E6D8R4D6]].\nFor example, the permutation $1 \\mapsto 2$, $2 \\mapsto 1$, $3 \\mapsto 4$, $4 \\mapsto 5$, $5 \\mapsto 3$ in $S_5$ is $(1\\ 2)(3\\ 4\\ 5)$.\n\n## Explanations\n\n### L1 human\n\nStart at a point and follow the permutation until you return.\nThis gives one cycle; then start again at a point that you did not visit yet.\nWhen all points are visited, the permutation is the product of the cycles that you found.\n",
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