{
  "schema_version": 1,
  "problem_number": "KOU-11.18",
  "title": "Partial Answers to Kourovka Problems 11.18 and 17.80: Infinite Groups G(a,b) and Their Two-Dimensional Images",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "R. Brandl asked in the Kourovka Notebook (Problem 11.18, 1990) whether the group G(a,b) = ⟨x, y | x = [x,_a y], y = [y,_b x]⟩ is finite; he noted that G(1,b) = 1 and G(2,2) = 1 and that nothing is known about G(2,3). O. Macedońska asked (Problem 17.80, 2010) whether M_n = G(n,n) is infinite for every n > 2 (UnsolvedMath records KOU-11.18 and KOU-17.80). We give partial answers: G(a,b) is infinite whenever m divides both a and b for some m ∈ {18, 20, 26, 28, 30, 42, 44}, so M_n is infinite whenever one of these numbers divides n (for m = 18 this is in print); every homomorphism of G(a,b) into SL(2,K) or PSL(2,K) factors through G(d,d) with d = gcd(a,b); the case G(2,3) and the general questions stay open. That M_18 is infinite follows from a result of Quek, Wong and Wong (2014), proved there by a computation credited to the referee, as A. Abdollahi's zbMATH review points out; we know that paper only from the review, and we prove the result again. For each m we give an explicit epimorphism of G(m,m) onto PSL(2,q) with q ∈ {8, 13, 29, 41, 43} and integer vectors that define homomorphisms onto Z of its kernel or of the preimage of a split torus; these subgroups have abelianizations of free rank exactly 7, 2, 14, 4, 4, 22, 12 for m = 18, 20, 26, 28, 30, 42, 44. The certificates are machine-readable and are checked by independent programs; those for m = 20, 28, 30, 42, 44 are new in this note. Infiniteness passes to G(a,b) because G(a,b) is a quotient of G(a′,b′) whenever a | a′ and b | b′. The second result holds over every field: if X, Y ∈ SL(2,K) satisfy X = [X,_a Y] and Y = [Y,_b X], then tr X = tr Y, a conjugation in GL(2,K) swaps X and Y, and the relations hold with gcd(a,b) in place of a and b. Hence G(2,3), and every G(a,b) with coprime a, b, has no nontrivial homomorphism into GL(2,K) or PGL(2,K) for any field K. Tested evidence: G(2,3) has no nontrivial homomorphism into A_n, n ≤ 11. For the cases other than m = 18, and for the second result, no source was found in zbMATH, OpenAlex, arXiv and Crossref; two related papers of Quek, Wong and Wong were not accessible to us; we claim no priority. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Kourovka Notebook",
    "Engel pairs",
    "Engel commutators",
    "finitely presented groups",
    "infinite groups",
    "Reidemeister–Schreier method",
    "PSL(2,q)",
    "SL(2,K)",
    "trace identities",
    "partial answer",
    "UnsolvedMath",
    "KOU-11.18",
    "KOU-17.80",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-22",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-22",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-11-18/",
  "pdf_url": "https://eulersolve.org/papers/kou-11-18/paper.pdf?v=8b3d949298ef",
  "doi": "10.5281/zenodo.23557947",
  "zenodo_record_url": "https://zenodo.org/records/23557947",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "G(2,3) and the general questions stay open; the case m = 18 is in print (Quek-Wong-Wong 2014, known to us from its review); the cases 20, 26, 28, 30, 42, 44 are new, by machine-checked certificates.",
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    "source.zip": {
      "sha256": "625ba7d602b27f2694eb8f498458db4c862bab0913b3b124ba49523524be5ff9"
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      "sha256": "5f6e149e8d3d54e60baa28fc507a1af899abb177ab9215912f063477d6ff1e83"
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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."
}