{
  "id": "enwiki-001318019909",
  "slug": "commuting-probability",
  "title": "Commuting probability",
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  "lead": "In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite groups equipped with a suitable probability measure, and can also be generalized to other algebraic structures such as rings.",
  "body": "In mathematics and more precisely in group theory, the commuting probability (also called degree of commutativity or commutativity degree) of a finite group is the probability that two randomly chosen elements commute. It can be used to measure how close to abelian a finite group is. It can be generalized to infinite groups equipped with a suitable probability measure, and can also be generalized to other algebraic structures such as rings. p(G) := \\frac{1}{\\# G^2} \\#\\!\\left\\{ (x,y) \\in G^2 \\mid xy=yx \\right\\} where \\# X denotes the cardinality of a finite set X . If one considers the uniform distribution on G^2 , p(G) is the probability that two randomly chosen elements of G commute. That is why p(G) is called the commuting probability of G . The finite group G is abelian if and only if p(G) = 1 . One has p(G) = \\frac{k(G)}{\\# G} where k(G) is the number of conjugacy classes of G . If G is not abelian then p(G) \\leq 5/8 (this result is sometimes called the 5/8 theorem ) and this upper bound is sharp: there are infinitely many finite groups G such that p(G) = 5/8 , the smallest one being the dihedral group of order 8. There is no uniform lower bound on p(G) . In fact, for every positive integer n there exists a finite group G such that p(G) = 1/n . If G is not abelian but simple, then p(G) \\leq 1/12 (this upper bound is attained by \\mathfrak{A}_5 , the alternating group of degree 5). The set of commuting probabilities of finite groups is reverse-well-ordered, and the reverse of its order type is \\omega^\\omega .",
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    "title": "Commuting probability",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/commuting-probability",
    "issued": {
      "date-parts": [
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          2026,
          11,
          15
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          2026,
          11,
          15
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    "DOI": "eo1:enwiki-001318019909",
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