# Verification report — OWR-14299292-010 (Dona, Oberwolfach Report 27/2025: is A_n conjugacy-expansive? The alternating-group case of Kourovka Problem 19.12)

Verification date: 2026-10-23 (two rounds of verification; the second examined the text of the note).

**Verdict.** The note gives 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.
- **Proved.** Theorem A (5 <= n <= 24), computer-assisted, twice: by explicit certificates (pairs x in K, c in C with
  xc in pi_C(K), checked by permutation products only) and by an exact computation of the class multiplication
  coefficients. Theorem B (the classes of types (3), (2,2), (5), (3,3), (4,2) with fixed points, every n >= 5): for
  n >= 24 a proof without computation (swap lemma, gadgets, explicit partners), for n <= 23 by Theorem A.
  Theorem C (both classes of n-cycles, every odd n >= 5): elementary arguments and the quoted realizability
  theorem; since n <= 24 is covered by Theorem A, only n >= 25 depends on that theorem.
- **Known, not new.** S_n (n >= 3) and A_4 are not conjugacy-expansive (Halasi, Maróti, Sidki, Bezerra, J. Group
  Theory 15 (2012), Theorem 1.1; the note gives short direct proofs). The Hall reformulation is their Lemma 3.1, the
  determinant criterion their Proposition 5.1. A_5 and A_6 are covered by their Theorem 1.3, and further small A_n
  possibly by their computer check of the 138 simple groups of the GAP character table library, whose list is not
  printed.
- **Evidence, not theorems.** A_25, A_26, A_27 are conjugacy-expansive according to two exact computations
  (independent programs of two verification runs; no certificates for these n); the exceptional sets of
  Theorem B are exactly the tight sets for 12 <= n <= 31 (at least two computations up to n = 27); the rule
  "L in KC iff l(K) + l(L) <= n + 1" for the n-cycle classes holds for odd 7 <= n <= 31 (at least two
  computations up to n = 27); Corollary 5.2 and its converse hold for all pairs of partitions, 3 <= n <= 32.
- **Open.** n >= 25 for every class not covered by Theorems B and C, hence the alternating-group case of Kourovka
  Problem 19.12. The first two parts of the record (programmes) are not treated.

The note is unrefereed. Suggested label for the record: open, unchanged.

## Statement checked
- **Primary source.** D. Dona, "Alternating groups as products of three conjugacy classes", Oberwolfach Rep. 22
  (2025), no. 2, pp. 1431–1434 (Report 27/2025, Computational Group Theory, organisers H. Dietrich, B. Eick,
  G. Nebe, C. Roney-Dougal), doi:10.4171/OWR/2025/27 (Crossref: the report, pp. 1389–1450). Read in the text of
  the published report, pp. 1431–1434. The last paragraph (p. 1433) asks three things: to decide constructively
  whether C1C2 contains C3 for any three classes (the "dream goal"), effective large-class criteria (two sample
  targets with exponents delta and eta_i), and whether one can "settle the case G = Alt(n) of Problem 19.12 of
  the Kourovka Notebook", with the definition of the property spelled out.
- **Kourovka Notebook.** No. 21, arXiv:1401.0300v49 (20 October 2026; the text of 19.12 is the same in v48),
  Problem 19.12 (M. Bezerra, Z. Halasi, A. Maróti, S. Sidki, listed alphabetically): a finite group is conjugacy-expansive if NC contains at least as many classes as N for
  every normal subset N and class C; is every finite simple group conjugacy-expansive? No comment; not in the
  archive of solved problems.
- **Source of the notion.** Z. Halasi, A. Maróti, S. Sidki, M. Bezerra, "Conjugacy expansiveness in finite groups",
  J. Group Theory 15 (2012), no. 4, 485–496, doi:10.1515/jgt-2012-0004; read completely in the authors' version
  dated 1 February 2012 (the journal version was not compared).
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299292-010 (dataset version 1.8.0; status `open`). Its third
  sentence is a faithful paraphrase of the precise question; the first two are the programmes above.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Is A_n conjugacy-expansive for every n >= 5? (the precise part of the record) | partial: yes for 5 <= n <= 24; for every n the condition holds for the classes of support <= 6 and, for odd n, for the two classes of n-cycles; open otherwise | Theorems A, B, C |
| The same "for every n", n = 3, 4 included | A_3 is abelian; A_4 is not conjugacy-expansive (S = V_4, C a class of 3-cycles: SC is one class); so n >= 5, the range of Problem 19.12 (simple groups), is the intended one | Problem 1.2, Proposition 1.3(b) |
| Classes of S_n instead of A_n | not the question (S_n is not simple); S_n is not conjugacy-expansive for n >= 3 (known) | Proposition 1.3(a) |
| Character expansiveness (Halasi, Maróti, Petényi 2013) | a different notion; not treated | Section 1.3 |
| The first two parts of the record (constructive decision; effective large-class criteria) | programmes without a yes-or-no answer; not treated | Section 1.1 |

## Results in the paper
- **Preliminaries.** Lemma 2.1 (classes of A_n: splitting, conjugacy by even permutations, reality); Lemma 2.2
  (Hall: expansive for C iff a permutation pi of the classes with pi(K) in KC exists; BHMS Lemma 3.1); tight
  classes T(C); Remark 2.3 (determinant criterion; applies to only 1–4 classes of A_n for 5 <= n <= 20);
  Proposition 1.3 with proofs (S = A_n and transpositions, via the partition identity; S = V_4 in A_4).
- **Theorem A.** Certificates (i): 114,677 pairs (every non-identity class for 5 <= n <= 18; the classes of support
  <= 6 for 5 <= n <= 23). Certificates (ii): 1,653,478 pairs for 19 <= n <= 24, produced by verification run B, not
  included because of their size (61.6 MB compressed), regenerable with the recorded seed, SHA-256 listed. Second
  proof: exact computation (S_n characters by Murnaghan-Nakayama, Frobenius' rule for A_n, structure constants
  modulo primes, zero test by size), repeated by both verification runs, one of them by a method without square
  roots. Table 1: k(A_n) and max |T(C)| for 5 <= n <= 27.
- **Theorem B.** Lemma 4.1 (swap lemma), Definition 4.2 (gadgets), Lemma 4.3 (gadget lemma), Lemma 4.4
  (locality of long-cycle gadgets); Table 2 (8 long gadgets), Table 3 (42 small gadgets), Table 4 (partners);
  Step 1: T(C) is contained in E(mu, n) for n >= 24, with E((3),n) = E((5),n) = {1^n, 2^(n/2)},
  E((2,2),n) = E((3,3),n) = {1^n}, E((4,2),n) = {1^n, (3,1^(n-3)), 2^(n/2), (2^((n-1)/2),1), 3^(n/3)} (members
  present when they are even types); Step 2: injective partners; Step 3: the swap lemma.
- **Theorem C.** Theorem 5.1 (Thom; Edmonds-Kulkarni-Stong; as stated in Pervova-Petronio, Theorem 2.7) and
  Corollary 5.2 (via their Theorem 3.1 and the compatibility conditions); Lemma 5.3 (the two halves of a split
  class behave alike next to an n-cycle), Corollary 5.4, Lemma 5.5 (an injection on cycle types), Lemma 5.6
  (3-cycles change the half), the permutation of the proof (real and non-real cases); Remark 5.8 (the n-cycle
  classes are the classes with the fewest cycles, not the largest ones).
- **Section 6.** Evidence (a)–(c); why a general proof is harder (tight sets grow; non-real split classes).

## Computations (programs and outputs in reproducibility/)
Theorem A rests on computations; the proofs of Theorem B for n >= 24 and of Theorem C do not.
- **check_note.py** (standard library, 24 s, TOTAL FAILS: 0; `check_note_output.txt`): the class test against
  conjugation orbits (n <= 8); brute-force supports and Hall matchings (5 <= n <= 8), A_4; Proposition 1.3(a); every
  certificate (i) (114,677 pairs); Tables 2–4, Lemma 4.4 for L <= 60, Step 1 for all 524,900 even types of
  24 <= n <= 40 (five mu) and Step 2 with witnesses; Lemmas 5.5, 5.6; Lemma 5.3, Corollary 5.4 and the proof of
  Theorem C by brute force for n = 7, 9; Corollary 5.2 for all pairs of partitions of n <= 9; Table 1. With
  `--certs-extra` on the six files of certificates (ii): all 1,653,478 pairs valid, every non-identity class
  certified for each n (`check_note_certs_extra_n19_24_output.txt`, 45 s).
- **Programs with which the results were obtained** (`original/`, NumPy, SciPy, NetworkX): the exact computation
  for n <= 24 (32 min), brute force n <= 9, re-check of the saved matchings, the certificates and their checker,
  the checks of Theorem B (Step 1 for 24 <= n <= 47) and of Theorem C (odd n <= 23), Remark 2.3, the partition
  identity (n <= 60).
- **Verification run A** (`independent_runs/run_A/`): own permutation checks of all gadgets, partners, Lemma 4.4,
  Lemma 5.6 and of Step 1 on 3,567,251 even types (24 <= n <= 50 and n = 60, 61, 64, 65); own certificate checker;
  own brute force (n <= 9); own character tables and structure constants (four primes) for n <= 24, with Table 1,
  T(C) = E(mu, n) for 12 <= n <= 24, Lemma 5.3, the n-cycle rule and the proof of Theorem C for odd n <= 23.
- **Verification run of the second round** (`independent_run_2/`, see below): own brute force (n <= 10, all
  structure constants), own exact supports for every n <= 27 (Table 1 for all 23 values of n), own checks of
  Tables 2-4 as read from the TeX source and of Step 1 for 24 <= n <= 52, own certificate checker on the
  certificates (i) and on the certificates (ii) regenerated from the archive, Corollary 5.2 for 3 <= n <= 32,
  and a re-run of the package (`independent_run_2/rerun_package_v1.txt`).
- **Verification run B** (`independent_runs/run_B/`): own method without square roots (five primes), supports
  identical to those of the note for every n <= 24 and every class; all structure constants equal to a brute force
  for n <= 10; deliberately wrong inputs detected; certificates (ii) produced and checked by its checker and by the
  note's; Theorem B exactly for n <= 31 and Theorem C exactly for odd n <= 31 (hook characters only); Corollary 5.2
  for all pairs of partitions of n <= 31; A_25, A_26, A_27 by one exact computation (seven primes); all programs of
  `original/` re-run from a copy with identical outputs.
- **Re-runs (2026-10-23, `RERUN_LOG.txt`).** 31 programs of all three parts from a fresh copy of the package
  (14 minutes): all outputs agree with the recorded ones up to running times and to lines that depend on how the
  recorded runs were split; the certificate generator regenerated the files for n = 19 and n = 20 with identical
  content (SHA-256 after decompression); the computation for n = 25 gave the recorded result.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two rounds of independent
verification, all AI-assisted, followed, each run with its own programs. In the first round, runs A and B examined
this first written version; in the second round a further run examined the text of the note (see the section
"Second round of verification" below).

| Item | Run A | Run B | When the note was written |
|---|---|---|---|
| Statement, range n >= 5, conventions, against the sources | CONFIRMED (OWR text, Kourovka v49, BHMS in the authors' version); asked to state n >= 5 and A_4 | CONFIRMED (Kourovka v49, zbMATH review of BHMS) | sources read again: OWR text, Kourovka v49, BHMS completely, Pervova-Petronio Sections 1–3, HMP 2013, Dona arXiv v1 |
| Lemmas 2.1, 2.2, 4.1, 4.3, 4.4 | PROVED (re-derived) | Lemma 4.4 re-derived; others used | — |
| Theorem A | CHECKED by own computation (four primes; own checker of certificates (i)) | CONFIRMED (own method without square roots; supports identical; certificates (ii) produced) | certificates (i) and (ii) checked with check_note.py |
| Theorem B | PROVED (case analysis re-derived; Step 1 by enumeration up to n = 65) | CONFIRMED (exact for n <= 31; gadget data recomputed) | check_note.py part [5] |
| Theorem C | PROVED modulo the cited theorem (its hypotheses checked against Pervova-Petronio) | CONFIRMED modulo the cited theorem (exact for odd n <= 31) | check_note.py part [6] |
| S_n and A_4 | PROVED | CONFIRMED | check_note.py parts [2], [3] |
| Novelty | not repeated beyond the sources read | supported by its search | searches repeated in part (Crossref, OpenAlex, one web search) |
| Classification PARTIAL | CORRECT | CORRECT | — |

No run of the first round found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) The range n >= 5 is stated, and A_4 is shown not to be conjugacy-expansive (S = V_4, C a class of
   3-cycles): Problem 1.2 and the paragraph after it, Proposition 1.3(b).
2. (Run A) The n-cycle classes are described as the classes with the fewest cycles, not as the largest classes:
   Remark 5.8 (|C+| = n!/(2n) < n!/(2(n-2)), the size of the class of type (n-2,1,1)).
3. (Run A) The odd conjugator of the small gadget ((5,3), (1 2 3 6)(4 8)) for mu = (4,2) is noted (it is not used:
   split types are covered by even long gadgets): Step 1 of the proof of Theorem B, and check_note.py part [5].
   The bounds of two even long gadgets are given as L >= 6 (Table 2); an earlier text had L >= 7, which is also
   true, since these gadgets are used only for odd L: Verification paragraph and reproducibility/README.md.
4. (Run A, optional) Boccara, Discrete Math. 38 (1982) 129–142, could not be read (the publisher refused automated
   access; the zbMATH record has no review), so it is not used; it is mentioned in "Scope and priority". That the
   originals of Thom (1965) and of Edmonds, Kulkarni and Stong (1984) were not read is said in the abstract, in the
   introduction next to the status sentence, at Theorem 5.1 and in "Scope and priority".
5. (Run B) Dona's paper is cited in its journal version, J. Combin. Theory Ser. A 225 (2027), Paper 106253,
   doi:10.1016/j.jcta.2026.106253 (verified with Crossref), with the arXiv number; "only version" now refers to arXiv
   (bibliography; "Scope and priority": read in the arXiv version v1, the journal version not compared).
6. (Run B, optional) The DOI of Halasi et al., 10.1515/jgt-2012-0004, verified with Crossref, is cited.
7. (Run B, optional) The character-free certificates for 19 <= n <= 24 of run B are part of the proof of Theorem A
   (certificates (ii)), with credit. Because of their size (61.6 MB compressed) they are not in the package; the
   generator, the checkers, the SHA-256 checksums (also of the decompressed content) and the commands are in
   `independent_runs/run_B/CERTIFICATES_n19_24.txt`; the files for n = 19, 20 were regenerated with identical
   content, and check_note.py accepted all six files.
8. (Run B, cosmetic) familyB.py, check (6), prints "contained in" although equality holds: explained in
   reproducibility/README.md (the program is included unchanged; it prints a "note:" line whenever the sets differ,
   and none occurs).

**Changes made when the note was written, after runs A and B.**
- The standard-library program check_note.py was written; it checks the finite and exact parts of the note and,
  with `--certs-extra`, the certificates (ii).
- The package was run again from a fresh copy (31 programs), including the regeneration of the certificates for
  n = 19, 20 and the computation for n = 25.
- The bibliography was checked again with Crossref (nine DOIs; the DOI of Halasi-Maróti-Petényi through the DOI
  handle service); the literature search was repeated in part.
- The results were renumbered; the correspondence with the numbering of the programs is in
  reproducibility/README.md.
None of these changes alters a theorem.

## Second round of verification
In the second round a further independent run, AI-assisted, examined the text of the note itself, line by line and as
if nothing had been checked, before any change was made; its programs and outputs are in `independent_run_2/`.

| Item | Second round |
|---|---|
| Statement, range n >= 5, conventions | CONFIRMED against the record, Dona's report (page 1433 of the published PDF), Kourovka 19.12 in v48 and v49 (identical, no comment), Halasi et al. (authors' version, read completely) |
| Lemmas 2.1, 2.2, Remark 2.3, Proposition 1.3 | PROVED (re-derived); Lemma 2.1 checked against conjugation orbits for n <= 10 |
| Theorem A | CONFIRMED: own exact supports for every n <= 27 (perfect matching for every class; Table 1 for all 23 values of n), all structure constants equal to an own brute force for n <= 10; certificates (i) and the regenerated certificates (ii) accepted by an own checker |
| Theorem B | PROVED (swap lemma, gadget lemma, locality, Steps 1-3 re-derived); Tables 2-4 as read from the TeX source recomputed; Step 1 as written checked on all even types of 24 <= n <= 52; T(C) = E(mu,n) and the permutation of Remark 4.5 on own exact data for 12 <= n <= 27 |
| Theorem C | PROVED modulo Theorem 5.1, whose hypotheses as stated by Pervova and Petronio (compatibility conditions (1)-(5), Theorems 2.7 and 3.1) are met; Lemma 5.3, the rule of Remark 5.7 and the permutation of the proof on own exact data for odd 7 <= n <= 27; Corollary 5.2 and its converse for all pairs of partitions, 3 <= n <= 32 |
| Package | re-run from an extracted copy of the archive (27 programs, 41 runs programs, among them the exact computation of `original/` for every n <= 24 and the generator of the certificates (ii) for every 19 <= n <= 24): all outputs agree with the recorded ones up to running times and printed paths; the six regenerated certificate files have the recorded checksums |
| Novelty | searches repeated in part (OpenAlex, Crossref, zbMATH, arXiv API, one web search): nothing new |
| Mathematical error | none |

**Corrections required by the run of the second round**, all applied in the note:
1. The authors of the paper of 2012 are named in its order (Halasi, Maróti, Sidki, Bezerra; "Halasi et al."); the
   Kourovka Notebook lists them alphabetically (abstract, Sections 1.1 and 1.3; Zenodo description; this report).
2. The quotation "all the 138 simple groups" is cited by section ([HMSB12, Section 1]) instead of a page of the
   authors' version, since the bibliography gives the journal pages.
3. Section 3.2: the complex conjugate in the character formula is the value at the inverse class,
   conj(psi(x_L)) = psi(x_L^-1), which is what the modular computation uses.
4. Section 5: compatibility means conditions (1)-(5) of Pervova and Petronio, Section 1 (the note had (1)-(3));
   (4) and (5) concern non-orientable base surfaces and are void here.
5. Abstract: the certificates for 19 <= n <= 24 are regenerated by a program of the package (they are not in it).
6. Step 1 of the proof of Theorem B: "even has an odd conjugator" replaced by "has in fact an odd conjugator"
   (the word "even" clashed with "even gadget").
7. Section 3.1: the checking programs are called separate, not "written independently of each other"; the checker of
   the second round is added (five programs for the certificates (i), four for (ii)).
8. Section 3.1(ii): the generator of the certificates (ii) reads the exact supports as a guide for a fallback
   search, which was never used (instead of "the exact supports were never needed").
9. The evidence is updated: Table 1 (n = 25, 26, 27 by two computations), Section 6(a)-(c), Section 3.2.
10. This round is recorded in Section 7 (five parts of the package, `independent_run_2/`, its re-run), in the
    Verification paragraph and in Scope and priority, in this report and in the README files.
11. Wording: "after his theorem" replaced by "starting from his theorem" (Section 1.1).
None of these changes alters a theorem, a proof or the status.

## Relation to the literature, novelty and scope
- **Searches (22 and 23 October 2026).** Works citing Halasi et al. in OpenAlex and Semantic Scholar (one:
  Halasi-Maróti-Petényi 2013; Crossref counts none); OpenAlex phrase and title searches; works citing Dona's
  paper (Semantic Scholar: one, arXiv:2609.31453 on a robust Thompson conjecture for A_n, about squares of large
  classes, not class counts) and the OWR report (none); zbMATH (titles with "expansive" or "expansiveness" in group theory,
  the review of Halasi et al.); arXiv (phrase searches for conjugacy expansiveness, for normal subsets and products
  of conjugacy classes of alternating groups, and for the Kourovka problem); five web searches in all. At the
  time the note was written the arXiv API was rate-limited (status 429) and was not queried again. The run of
  the second round repeated the searches in part on 23 October 2026 (OpenAlex: one work citing Halasi et al.'s
  paper, none citing Dona's journal paper; Crossref: no citing works; zbMATH: titles with "expansive" or
  "expansiveness" in group theory; arXiv API: the 100 newest math.GR abstracts with "expansive"; one web
  search; Semantic Scholar and three arXiv queries refused, retried once, not pursued): nothing new.
  - No answer to the question was found, and no paper containing Theorems A–C.
  - Related work concerns products of large classes or normal subsets of A_n (Dona 2027; Garonzi and Maróti 2021),
    squares of classes and growth of products; none of it counts the classes in SC.
- **What was read.** Halasi, Maróti, Sidki and Bezerra completely (authors' version of 1 February 2012); Halasi-Maróti-Petényi in the
  journal's open file; Dona's OWR contribution in the published report; Dona's paper in the arXiv version v1; Problem
  19.12 in the Kourovka Notebook v49; Pervova-Petronio, arXiv v3, Sections 1–3 (the definition of branch data and
  compatibility, Theorem 2.7 with its attribution, Theorem 3.1). Not read: Thom (1965); Edmonds, Kulkarni and Stong
  (1984; the publisher's page refused automated access to a verification run); Boccara (1982); the web page with the
  computations of Halasi et al. (not reachable, again on 23 October 2026). The run of the second round read the
  sources again (except Halasi-Maróti-Petényi), Problem 19.12 also in v48.
- **Caveats.** The 138 groups of the computer check of Halasi et al. are not listed, so Theorem A for small n may
  overlap with that check. Theorem C for odd n >= 25 rests on the quoted realizability theorem. The certificates (ii)
  are not in the package but can be regenerated and checked. This negative search is not a proof of priority.
- **Scope.** Partial answer, as in the verdict; no priority is claimed.