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  "body": "\n## Definition\n\nLet $X$ be a set.\nThe symmetric group $\\operatorname{Sym}(X)$ is the set of all bijections $\\sigma \\colon X \\to X$, with composition of maps as the operation.\nFor a positive integer $n$, the symmetric group $S_n$ is $\\operatorname{Sym}(\\{1, 2, \\dots, n\\})$.\nAn element of $S_n$ is a permutation of degree $n$.\n\n## Remarks\n\nThe identity map is the identity element, and the inverse of $\\sigma$ is the inverse map $\\sigma^{-1}$.\nIn this node, the product $\\sigma \\tau$ means first $\\tau$, then $\\sigma$.\nThe group $S_n$ is not [[abelian|082GHNPJ53]] for $n \\geq 3$, because $(1\\ 2)(2\\ 3) \\neq (2\\ 3)(1\\ 2)$.\nIf $X$ and $Y$ have the same cardinality, then $\\operatorname{Sym}(X)$ and $\\operatorname{Sym}(Y)$ are [[isomorphic|08MG7H8JW5]].\n\n## Explanations\n\n### L1 human\n\nThe symmetric group collects all the ways to shuffle the elements of a set.\nTwo shuffles combine into one shuffle when you do them one after the other.\nEach shuffle can be undone, so the shuffles form a group.\n",
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