{
  "abstract": "We give an affirmative answer to the abstract group problem for the category $\\YB$ of all set-theoretic solutions of the Yang--Baxter (YB) equation: every group is isomorphic to the automorphism group of a YB-solution, which can be chosen on a finite set if the group is finite. In the terminology of László Babai, this means that the category $\\YB$ is universal and finitely universal, which is an analogue of Frucht's classical theorem for graphs. Furthermore, the YB-solutions can be chosen within three very small classes: the solutions of the Frobenius-Separability equation, the solutions of the Baaj--Long--Skandalis equation and the recently introduced bipartite solutions. The proof is based on new constructions which associate to any graph a YB-solution in each of these classes, with the same automorphism group as the graph. Explicit YB-solutions with automorphism groups $C_n$, $S_n$, $D_n$ and $Q_8$ are given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39228",
  "author_total": 1,
  "authors": [
    {
      "name": "G. Militaru"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e114sa7w2bj38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39228",
  "fulltext_url": null,
  "id": "e114sa7w2bj38",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39228",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39228",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e114sa7w2bj38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39228",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39228"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Every group is the automorphism group of a set-theoretic solution of the Yang--Baxter equation",
  "title_text": "Every group is the automorphism group of a set-theoretic solution of the Yang--Baxter equation",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/e114sa7w2bj38",
  "venue": "arXiv",
  "versions": []
}