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  "body": "\n## Statement\n\nLet $G$ be a [[group|G5S1M9]].\nThen $G$ is [[isomorphic|08MG7H8JW5]] to a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] $\\operatorname{Sym}(G)$.\nExplicitly, the map $g \\mapsto \\ell_g$, where $\\ell_g(x) = gx$, is an injective [[group homomorphism|0JYTG5K9Y4]] from $G$ to $\\operatorname{Sym}(G)$.\n\n## Remarks\n\nThe map comes from the [[action by left multiplication|0K5TWJAGPD]], and it is injective because $\\ell_g(e) = g$.\nIf $|G| = n$ is finite, then $G$ is isomorphic to a subgroup of $S_n$.\nOften a smaller symmetric group is enough, for example $S_3$ is isomorphic to a subgroup of $S_3$ and not only of $S_6$.\n\n## History\n\nArthur Cayley introduced abstract groups in 1854.\nAt that time, groups were usually groups of permutations, and the theorem shows that the abstract notion gives nothing more, up to isomorphism.\n\n## Explanations\n\n### L1 human\n\nEach element of a group shuffles the elements of the group when you multiply on the left.\nDifferent elements give different shuffles, and the product of elements gives the composition of shuffles.\nSo each group is a group of permutations, up to isomorphism.\n",
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