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  "body": "\n## Summary\n\nA permutation group is a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] of a set, that is, a set of [[permutations|05RWFKVTJ7]] closed under composition and inverses.\nGroups first arose as permutation groups, in the study of the roots of polynomial equations.\nBy [[Cayley's theorem|0151VYVX27]], every group is isomorphic to a permutation group.\nEvery [[group action|0WCY42EV9V]] gives a permutation group, the image of the group in the symmetric group of the set.\n\n## Explanations\n\n### L1 human\n\nA permutation group is a collection of ways to shuffle a set of objects that is closed under doing one shuffle after another and under undoing a shuffle.\nThe symmetries of a square, viewed as shuffles of its four corners, are a good picture.\nIf the corners are labelled $1, 2, 3, 4$ in order around the square, the quarter turn is the shuffle $1 \\to 2 \\to 3 \\to 4 \\to 1$.\nThe eight symmetries of the square give eight of the $24$ possible shuffles of the corners, and they form a permutation group.\n",
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