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  "body": "\n## Summary\n\nCycle notation writes a permutation as a list of cycles.\nThe cycle $(a_1\\ a_2\\ \\dots\\ a_k)$ sends $a_i$ to $a_{i+1}$ for $i \u003c k$, sends $a_k$ to $a_1$, and fixes all other points.\nA cycle with $k$ entries is a $k$-cycle, and a $1$-cycle is the identity, so it is usually not written.\n\n## Remarks\n\nThe same cycle has $k$ different notations, because you can start the list at any entry: $(1\\ 2\\ 3) = (2\\ 3\\ 1) = (3\\ 1\\ 2)$.\nThe inverse of $(a_1\\ a_2\\ \\dots\\ a_k)$ is $(a_k\\ \\dots\\ a_2\\ a_1)$.\nEach permutation of a finite set is a product of [[disjoint cycles|0YWDAZPRPQ]], so cycle notation can show each permutation.\nFor example, in $S_5$ the permutation with $1 \\mapsto 3$, $3 \\mapsto 1$, $2 \\mapsto 4$, $4 \\mapsto 5$, $5 \\mapsto 2$ is $(1\\ 3)(2\\ 4\\ 5)$.\n\n## Explanations\n\n### L1 human\n\nCycle notation follows each symbol to where it goes, until you come back to the start.\nEach round trip is a cycle, and you write it in parentheses.\nThis is shorter than a table, and it shows the structure of the permutation directly.\n",
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