{
  "abstract": "In this article, we prove that the theory $T_3$ of groups of exponent $3$ has a model companion. Previous work established the existence of model companions for theories of groups of fixed finite exponent and nilpotency class at most $2$ (Saracino--Wood), and for theories of groups of prime exponent $p$ and nilpotency class at most $c<p$ (Maier). These results do not cover $T_3$, since groups of exponent $3$ may have nilpotency class $3$. Our theorem verifies the $n=3$ case of a conjecture proposed by the third author that, for each integer $n>1$, the theory $T_n$ of groups of exponent $n$ has a model companion if and only if every finitely generated group of exponent $n$ is finite, or equivalently, if and only if the Burnside problem has a positive solution for exponent $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30061",
  "author_total": 3,
  "authors": [
    {
      "name": "Yawara Ishida"
    },
    {
      "name": "Ryosuke Mizuno"
    },
    {
      "name": "Kota Takeuchi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jgymw0x177h2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30061",
  "fulltext_url": null,
  "id": "6jgymw0x177h2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30061",
  "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.GR"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214274637",
  "pdf_url": "https://arxiv.org/pdf/2609.30061",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jgymw0x177h2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30061",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30061"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Existence of a Model Companion for Groups of Exponent 3",
  "title_text": "Existence of a Model Companion for Groups of Exponent 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/6jgymw0x177h2",
  "venue": "arXiv",
  "versions": []
}