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  "body": "\n## Statement\n\nLet $n \\geq 2$.\nEach permutation in $S_n$ is a product of [[transpositions|0FWVRHBRWM]].\n\n## Remarks\n\nA $k$-cycle is a product of $k - 1$ transpositions, for example $(a_1\\ a_2\\ \\dots\\ a_k) = (a_1\\ a_k)(a_1\\ a_{k-1}) \\cdots (a_1\\ a_2)$.\nWith the [[disjoint cycle decomposition|0YWDAZPRPQ]], this gives the result, and the identity is $(1\\ 2)(1\\ 2)$.\nThe decomposition is not unique, but the parity of the number of factors is unique.\n\n## Explanations\n\n### L1 human\n\nYou can sort any arrangement of objects by swapping two objects at a time.\nSo each permutation is a chain of swaps.\nDifferent chains can give the same permutation, but they are all of even length or all of odd length.\n",
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