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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]].\nThe *commuting probability* of $G$ is\n$$\\mathrm{cp}(G) = \\frac{\\abs{\\set{(x, y) \\in G \\times G : xy = yx}}}{\\abs{G}^2}.$$\n\n## Remarks\n\nCounting the pairs by their first entry gives $\\mathrm{cp}(G) = \\frac{1}{\\abs{G}^2} \\sum_{x \\in G} \\abs{C_G(x)}$, where $C_G(x)$ is the [[centraliser|02YYV36BFM]] of $x$.\nSince the [[conjugacy class|0DEBXAJ5SA]] of $x$ has $\\abs{G}/\\abs{C_G(x)}$ elements, this sum gives $\\mathrm{cp}(G) = k(G)/\\abs{G}$, where $k(G)$ is the number of conjugacy classes of $G$.\nExample: $S_3$ has $3$ conjugacy classes, so $\\mathrm{cp}(S_3) = 3/6 = 1/2$.\nExample: the [[quaternion group|0NRJGW864D]] $Q_8$ has $5$ conjugacy classes, $\\{1\\}$, $\\{-1\\}$, $\\{\\pm i\\}$, $\\{\\pm j\\}$, $\\{\\pm k\\}$, so $\\mathrm{cp}(Q_8) = 5/8$.\nThe value $\\mathrm{cp}(G) = 1$ holds exactly when $G$ is [[abelian|082GHNPJ53]]; if $G$ is not abelian, then $\\mathrm{cp}(G) \\le 5/8$.\n\n## Explanations\n\n### L1 human\n\nThe commuting probability is the chance that two random elements of the group commute.\n",
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