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  "body": "\n## Statement\n\nLet $n \\geq 3$.\nThe [[alternating group|0Z5905Q73D]] $A_n$ is [[generated|0RP7VWQ0YN]] by the $3$-cycles in $S_n$.\n\n## Remarks\n\nEach $3$-cycle is even, so the $3$-cycles lie in $A_n$.\nAn even permutation is a product of an even number of transpositions, and each product of two transpositions is a product of $3$-cycles.\nFor distinct $a, b, c, d$, the formulas are $(a\\ b)(a\\ c) = (a\\ c\\ b)$ and $(a\\ b)(c\\ d) = (a\\ c\\ b)(a\\ c\\ d)$.\n\n## Explanations\n\n### L1 human\n\nAn even permutation is a chain of swaps, taken two at a time.\nEach pair of swaps can be written with $3$-cycles.\nSo the $3$-cycles are enough to build all even permutations.\n",
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