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  "slug": "bol-prime-cube-classification",
  "title": "Counting Bol loops at every prime cube",
  "shortTitle": "Bol loops at every prime cube",
  "url": "https://evidencepress.org/releases/bol-prime-cube-classification/",
  "oneLine": "An explicit classification gives 11 centrally nilpotent right Bol loops at order eight and p + 10 at every odd prime-cube order, including groups.",
  "abstract": "The candidate classifies centrally nilpotent right Bol loops of order p cubed up to ordinary isomorphism, for every prime. A five-parameter central extension reduces to three normalised parameters and a residual change-of-generators action. Explicit orbit representatives give six nonassociative classes at two and p+5 at odd primes; adding the five groups gives 11 and p+10. Characteristics two and three require different power laws. Structural corollaries and a small-prime recognition program make the tables reusable. The general trivial-centre branch and isotopy are excluded.",
  "datePublished": "2026-10-09",
  "dateModified": "2026-10-09",
  "version": "3.1.0-candidate",
  "doi": "10.5281/zenodo.23270849",
  "doiUrl": "https://doi.org/10.5281/zenodo.23270849",
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    "Anonymous"
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  "provenance": {
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  "problem": {
    "name": "Grishkov–Kinyon–Vojtěchovský Problem 5.2",
    "url": "https://arxiv.org/html/2504.14646v2"
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  "keywords": [
    "Bol loops",
    "central nilpotence",
    "prime-cube order",
    "isomorphism classification"
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  "keyResults": [
    "Complete centrally nilpotent classification: 11 classes at p=2; p+10 at every odd prime, including five groups.",
    "Explicit multiplication and pairwise inequivalent representatives, with separate power laws at two and three.",
    "Bruck subclass, nuclei, exponents and a generator-based recognition procedure."
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  "relatedWorks": [
    {
      "citation": "Grishkov, Kinyon and Vojtěchovský (2026), Bol loops of order 27. Theorem 4.2 supplies the centre reduction; Problem 5.2 asks for the all-prime count.",
      "url": "https://doi.org/10.1142/S0218196726500049",
      "doi": "10.1142/S0218196726500049"
    },
    {
      "citation": "Chein and Goodaire (2007), Bol Loops of Nilpotence Class Two. Existing collection, commutator and associator identities underpin the model.",
      "url": "https://doi.org/10.4153/CJM-2007-012-7",
      "doi": "10.4153/CJM-2007-012-7"
    },
    {
      "citation": "Daly and Vojtěchovský (2009), Enumeration of nilpotent loops via cohomology. General extension-isomorphism machinery.",
      "url": "https://doi.org/10.1016/j.jalgebra.2009.03.042",
      "doi": "10.1016/j.jalgebra.2009.03.042"
    },
    {
      "citation": "de Barros, Grishkov and Vojtěchovský (2012), Commutative automorphic loops of order p³. A methodological predecessor in a different variety.",
      "url": "https://doi.org/10.1142/S0219498812501009",
      "doi": "10.1142/S0219498812501009"
    },
    {
      "citation": "Bianco and Bonatto (2021), On connected quandles of prime power order. Existing Bruck-subclass classification.",
      "url": "https://doi.org/10.1007/s13366-020-00501-y",
      "doi": "10.1007/s13366-020-00501-y"
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