{
  "schema_version": 1,
  "problem_number": "OWR-14299292-010",
  "title": "Conjugacy Expansion in Alternating Groups: All Degrees up to 24 and Two Infinite Families of Classes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A finite group G is conjugacy-expansive if for every normal subset S and every conjugacy class C of G the normal set SC contains at least as many conjugacy classes of G as S does. The notion is due to Halasi, Maróti, Sidki and Bezerra, who asked whether every finite simple group is conjugacy-expansive (Kourovka Notebook, Problem 19.12); in Oberwolfach Report 27/2025, Dona asked for the case of the alternating groups. We give a partial answer: A_n is conjugacy-expansive for every 5 ≤ n ≤ 24 (exact computation with character-free certificates), and for every n ≥ 5 the defining condition holds for every class of support at most 6 and, for odd n, for both classes of n-cycles (the latter using the realizability theorem of Thom and Edmonds-Kulkarni-Stong); the general case n ≥ 25 stays open. The realizability theorem is quoted from Pervova and Petronio; the original papers of Thom and of Edmonds, Kulkarni and Stong were not read. By Hall's theorem, as observed by Halasi et al., the condition for a class C asks for a permutation π of the classes with π(K) ⊆ KC for every class K. For n ≤ 24 and every C we give explicit pairs x ∈ K, c ∈ C with xc ∈ π(K), which are checked by multiplying permutations; those for 19 ≤ n ≤ 24 were produced by an independent verification run and, because of their size, are regenerated by a program of the package. An exact character computation gives a second proof. For classes of small support the permutation is built from local configurations and a few explicit partners; for n-cycles it rests on a lemma saying that the two halves of a split class behave alike next to an n-cycle. We also give the short proofs of the known facts that S_n (n ≥ 3) and A_4 are not conjugacy-expansive. No printed answer to the question was found in the sources we searched; a search that finds nothing is not a proof of novelty, and no priority is claimed. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "conjugacy-expansive groups",
    "alternating groups",
    "products of conjugacy classes",
    "normal subsets",
    "Kourovka Notebook Problem 19.12",
    "Hall's marriage theorem",
    "character-free certificates",
    "branched coverings",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-14299292-010",
    "math.GR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299292-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299292-010/paper.pdf?v=cfb235292ea8",
  "doi": "10.5281/zenodo.23585754",
  "zenodo_record_url": "https://zenodo.org/records/23585754",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The general case n ≥ 25 stays open; computer-assisted for n ≤ 24; the realizability theorem is quoted from Pervova-Petronio (originals not read); unrefereed.",
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      "sha256": "cfb235292ea8b0688d8d85cdaf61d7d37e199ea2f8005a4561056a8c451c8e6e"
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    "source.zip": {
      "sha256": "e3e9210bef579731b03ea3dcd82f4dd7eb14464a424f9d176b4c6642f668250f"
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    "verification_report.md": {
      "sha256": "f83df8a18e354306464affb99698028d67151ea79de3f7b931e86b2dacd6ca34"
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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."
}