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  "body": "\n## Summary\n\nThe commuting probability of a finite group is the probability that two elements chosen independently and uniformly at random commute.\nIt equals $1$ exactly for abelian groups, and it equals the number of conjugacy classes divided by the order of the group.\nA non-abelian finite group has commuting probability at most $5/8$.\nSo the value measures how far a group is from being abelian.\n\n## Explanations\n\n### L1 human\n\nPick two elements of a finite group at random, with repetition allowed, and ask whether $xy = yx$.\nIn an abelian group the answer is always yes.\nIn the symmetric group $S_3$, of the $36$ ordered pairs exactly $18$ commute, so the probability is $1/2$.\nA group cannot be \"almost abelian\" in this sense: once some pair fails to commute, at least $3/8$ of all pairs fail.\n",
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