{
  "schema_version": 1,
  "problem_number": "KOU-16.46",
  "title": "Lower Bounds for the Finite Quotients of Conder's Group on the Cubic Tree (Kourovka Notebook Problem 16.46)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 16.46 of the Kourovka Notebook, posed by M. Conder and going back to his paper of 1992, concerns the free product G = G₃ ∗_{G₁} G₄ of two of the Djoković–Miller groups that act arc-transitively on the cubic tree, amalgamated along the modular group G₁, and the normal subgroup K of index 8 of G, which has a presentation with six generators and 22 relators: does K have a non-trivial finite quotient? Conder found no proper subgroup of finite index. We do not decide the question. We prove, without computation, that K is perfect, that in every non-trivial image of K the generators keep their orders and the symmetric group S₄ embeds, and that K is the amalgamated free product of four copies of L = S₄ ∗_{D₈} S₄ over a common subgroup C₃ ∗ C₃, glued along its four sign automorphisms; a non-trivial finite quotient of K gives one of the subgroup J generated by two of these copies. By exhaustive computer searches we show that J, and hence K, has no non-trivial action on at most 36 points, so K has no subgroup of index between 2 and 36, and, with GAP, that no non-abelian finite simple group of order less than 10⁸ receives a non-trivial homomorphism from J. Hence every non-trivial finite quotient of K has order at least 10⁸. Jambor (2015) proved, by a machine computation, that K has no quotient PSL₂(q) or PGL(2,q); together with a search of the twelve other simple groups of order between 10⁸ and 10⁹, every non-trivial finite quotient of K has order at least 10⁹. In contrast, the amalgam of two copies differing by the twist (A,B) ↦ (A⁻¹,B⁻¹) maps onto A₆ and PSL(2,7). The bounds are computer-assisted; the question itself remains open. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "Kourovka Notebook",
    "Conder",
    "cubic tree",
    "arc-transitive groups",
    "Djoković–Miller groups",
    "amalgamated free products",
    "Bass–Serre theory",
    "finite quotients",
    "finite simple groups",
    "low-index subgroups",
    "computer-assisted proof",
    "partial answer",
    "UnsolvedMath",
    "KOU-16.46",
    "math.GR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-16-46/",
  "pdf_url": "https://eulersolve.org/papers/kou-16-46/paper.pdf?v=35deb5ba6dad",
  "doi": "10.5281/zenodo.23624375",
  "zenodo_record_url": "https://zenodo.org/records/23624375",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The question stays open; the order bounds are computer-assisted (Python, GAP 4.16.1), and the bound 10^9 also uses Jambor's machine computation; unrefereed.",
  "files": {
    "paper.pdf": {
      "sha256": "35deb5ba6dadaa56a1c4f2603e68a15fb679a4448e27b22bf763b0c212dca121"
    },
    "source.zip": {
      "sha256": "34e6c8d3e53eb004c4ebcfe093fe5201ce1e3035e0a59d70993b07f759bfe82d"
    },
    "verification_report.md": {
      "sha256": "fdef0cfd838aabc0630e549b25852288bec1173e3c08f2c0bbb4a43a3076e975"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}