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  "body": "\n## Definition\n\nLet $\\Omega$ be a set, and let $\\operatorname{Sym}(\\Omega)$ be the [[symmetric group|0515A6D2Y1]] of $\\Omega$: the group of all bijections $\\Omega \\to \\Omega$ under composition.\nA *permutation group* on $\\Omega$ is a [[subgroup|0X1JXT47GA]] $G$ of $\\operatorname{Sym}(\\Omega)$.\nThe *degree* of $G$ is the cardinality $\\abs{\\Omega}$.\n\n## Remarks\n\nExample: label the corners of a square $1, 2, 3, 4$ in cyclic order.\nThe eight symmetries of the square give the set $G = \\set{e, (1\\,2\\,3\\,4), (1\\,3)(2\\,4), (1\\,4\\,3\\,2), (1\\,3), (2\\,4), (1\\,2)(3\\,4), (1\\,4)(2\\,3)}$.\nIt is closed under composition and inverses, because the composite of two symmetries of the square is again one.\nSo $G$ is a permutation group of degree $4$, isomorphic to the [[dihedral group|0JZ27DMJ5D]] of order $8$.\n\nNon-example: the set $\\set{e, (1\\,2), (1\\,3), (2\\,3)}$ in $S_3$.\nIt contains $(1\\,2)$ and $(1\\,3)$ but not their product, which is a $3$-cycle, so it is not a subgroup.\n\nA [[group action|0WCY42EV9V]] of a group $H$ on $\\Omega$ is the same as a homomorphism $H \\to \\operatorname{Sym}(\\Omega)$, and its image is a permutation group on $\\Omega$.\nThe image is isomorphic to $H$ exactly when the action is [[faithful|0P7G9NGG2G]].\nBy [[Cayley's theorem|0151VYVX27]], the action of a group $H$ on itself by left multiplication is faithful, so $H$ is isomorphic to a permutation group on $H$.\n",
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