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  "body": "\n## Statement\n\nLet $\\sigma$ and $\\tau$ be cycles in the [[symmetric group|0515A6D2Y1]] $\\operatorname{Sym}(X)$ whose sets of moved points are disjoint.\nThen $\\sigma \\tau = \\tau \\sigma$.\n\n## Remarks\n\nA point moved by $\\sigma$ is fixed by $\\tau$, and $\\sigma$ maps it to another point that $\\tau$ fixes. So both products send it to the same point.\nThe same argument works for the points moved by $\\tau$, and the points fixed by both are fixed by both products.\nCycles that are not disjoint need not commute, for example $(1\\ 2)(1\\ 3) \\neq (1\\ 3)(1\\ 2)$.\n\n## Explanations\n\n### L1 human\n\nTwo disjoint cycles move different points.\nEach cycle does its work without touching the points of the other cycle.\nSo the order in which you do them does not matter.\n",
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  "name": "Disjoint cycles commute",
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