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  "body": "\n## Definition\n\nLet $S = \\{a_1, \\ldots, a_n\\}$ be a finite set with $n$ elements, listed in a fixed order, and let $\\ast$ be a [[binary operation|0M418FE7BB]] on $S$.\nThe Cayley table of $(S, \\ast)$ is the $n \\times n$ table whose rows and columns are labelled $a_1, \\ldots, a_n$.\nThe entry in row $i$ and column $j$ is $a_i \\ast a_j$.\n\n## Remarks\n\nThe operation is [[commutative|099E72KTF8]] exactly when the table is symmetric about its main diagonal.\nIn the Cayley table of a finite [[group|G5S1M9]], each element occurs exactly one time in each row and one time in each column.\nThus the Cayley table of a finite group is a Latin square.\n\n## Explanations\n\n### L1 human\n\nA Cayley table is like the multiplication table from school, but for any finite set with an operation.\nFor the integers modulo $2$ under addition, the table has the rows $0, 1$ and $1, 0$.\nYou can read many properties from the table, for example if the operation is commutative.\n",
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