# Verification report — KOU-18.117 (Kourovka Notebook 18.117, Shumyatsky: coprime commutators in finite simple groups)

Verification date: 2026-10-27.

**Verdict.** The note gives a **partial answer**, with computer-assisted proofs; the question stays open.
- **Settled (computer-assisted).** Theorem A: in the 16 sporadic groups M11, M12, M22, M23, M24, J1, J2, J3, HS,
  McL, He, Ru, Suz, Co3, Co2, Fi22, in the Tits group 2F4(2)' and in 36 groups of Lie type (L3(q), U3(q) for
  q = 3, 4, 5, 7, 8, 9, 11, 13; L4(3), L4(4), L5(2), U4(2), U4(3), U5(2), U6(2), S4(4), S4(5), S4(7), S6(2),
  S8(2), O7(3), O8+(2), O8-(2), O8+(3), 3D4(2), G2(3), G2(4), G2(5)) every element is [x, y] with x a 2-element
  and y of odd order. Theorem B: in O'N, Ly and J4 every element is a commutator of two elements of coprime
  orders. Corollary C: every nonabelian finite simple group of order < 10^9 has the property. Corollary D: 19 of
  the 26 sporadic groups and the Tits group have it.
- **Known and used, not reproved.** A_n (Shumyatsky, Forum Math. 27 (2015) 575–583, Theorem 3.1 of
  arXiv:1208.3177), PSL(2,q) (Pellegrini–Shumyatsky, Arch. Math. 99 (2012) 501–507, Theorem 1.1), the Suzuki
  groups (Zini, Bull. Aust. Math. Soc. 104 (2021) 448–452). Without coprimality: Ore's conjecture (Liebeck,
  O'Brien, Shalev, Tiep, J. Eur. Math. Soc. 12 (2010) 939–1008).
- **Unpublished or withdrawn claims, not used.** Pellegrini–Shumyatsky report a Magma check of all simple groups
  of order < 10^7 without data. The withdrawn preprint of Lucchini and Spiga (arXiv:2504.02330v1, withdrawn in v2)
  reported an unrecorded random Magma search for M11, M12, M22, M23, M24, J1, J2, J3, 2F4(2)', HS, McL, He, Ru,
  Suz, O'N, Co3, Co2, Fi22, Fi23; Fi23 is not settled by the note.
- **Open.** HN, Th, Co1, Fi23, Fi24', B, M (partial data only: Table 3 of the note, not a proof); the groups of
  Lie type of order ≥ 10^9 other than PSL(2,q), the Suzuki groups and the five groups S8(2), O7(3), U6(2),
  O8+(3), G2(5) of Theorem A. No counterexample was found.

The note is unrefereed. The results rest on GAP's exact algorithms (GAP 4.16.1), on the CTblLib character tables
(version 1.3.11), on the ATLAS lists of maximal subgroups as encoded in CTblLib, on the completeness of GAP's
computation of possible class fusions, and on the completeness of the primitive groups library (degree < 4096) and of
the perfect groups library (order 80640).

## Statement checked
- **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka
  Notebook, No. 21, arXiv:1401.0300v49 (20 October 2026), Problem 18.117 (18th issue, 2014), page 139,
  proposer P. Shumyatsky: is every element of a nonabelian finite simple group a commutator [x, y] with
  gcd(o(x), o(y)) = 1? The comment names A_n and PSL(2,q) as known and Ore's conjecture as the version without
  coprimality. The question first appears in Shumyatsky's Forum Math. paper (arXiv:1208.3177).
- **Corpus record.** ulamai/UnsolvedMath, KOU-18.117 (dataset version 1.8.0, status `open`); the statement is
  the notebook text.

## Readings
| Reading | Answer | Where |
|---|---|---|
| every element g = [x, y] with x, y in G and gcd(o(x), o(y)) = 1 | the reading used; the identity is [1, 1]; the property is a union of classes closed under inversion and Aut(G) | Section 1.1 |
| x, y allowed in Aut(G) or a cover | not the question; x, y lie in G | Section 1.1 |
| other commutator convention xyx^-1y^-1 | the same notion ([x^-1, y^-1]) | Section 1.1 |
| stronger form "x a 2-element, y of odd order" | proved in Theorem A; false in PSL(2,q), q ≡ 3 (mod 8) (Remark 3.1); for six classes of Theorem B (O'N 31A/B, Ly 33A/B, J4 35A/B) no certificate of this form is found, through any subgroup of Table 2; for Ly 11A/B and J4 7A/B the certificate printed in Appendix A (Lemma 2.1) is not of this form, but the routes through M11 and M22 give one (Remark 1.2) | Theorem A, Remarks 1.2, 3.1 |
| all nonabelian finite simple groups | open; partial answer | Sections 1.3, 6 |

## Results in the paper
- **Lemmas (Section 2), with complete proofs.** 2.1 (odd order, rational action: g^k ~ g, gcd(k−1, n) = 1,
  d = ord_n(k) prime to n ⇒ g = [g^j, c] with c a π(d)-element); 2.2 (real classes of odd order, k = −1);
  2.3 (fixed-point-free coprime action, with the table criterion |C_N(c)| = |C_H(c)|/|C_{H/N}(cN)| via
  Schur–Zassenhaus); 2.4 (involutions in an A4 from a positive class multiplication coefficient a(2X, 3Y, 3Z));
  2.5 (direct products); 2.6 (central elements: o(z) divides o(x) and o(y), also modulo a normal subgroup);
  2.7 (subgroup step against a set of possible fusions containing the true one, with an automorphism σ).
- **Theorem A (Section 3, Table 1).** Constructions (MathieuGroup; PrimitiveGroup(d, k); J3 from the AtlasRep
  generators shipped with the package), identification (order, simplicity, GAP's type; class numbers separate
  L3(4) from A8 and O7(3) from S6(3)); 2273 stored pairs; verify.g (membership, orders, classes by exact
  conjugacy tests among GAP's classes, all classes met, class number = CTblLib); the class-equation check;
  the M11 example; the search (not part of the proof); Remark 3.1 on PSL(2,q), q ≡ 3 (mod 8).
- **Theorem B (Section 4, Table 2, Appendix A).** Sources: maximal subgroups, subgroups of index 2 or 5 in
  maximal subgroups, a complement of and a factor of a complement of split maximal subgroups (M24 < 2^11:M24,
  M11 < 2 × M11 < 3^5:(2 × M11)) (Table 2 names each with its maximal overgroup and
  the number of possible fusions). Five constructed subgroups with identification arguments: 4_2.L3(4) <
  O'N (perfect groups library, two candidates, character tables), 2.A11 < Ly (Schur cover), 2^11:M24 < J4 and
  2^10:L5(2) < J4 (irreducibility and faithfulness from the CTblLib normal subgroups 1, 2^k, H; affine primitive
  groups; orbit lengths 1 + 276 + 1771 versus 1 + 759 + 1288, and 1 + 155 + 868), 11^(1+2):(5 × 2S4) < J4
  (faithful action, Burnside, GL(2,3) unique in GL(2,11), scalar C5, Schur–Zassenhaus). In each constructed
  subgroup the classes without a stored pair are provably not coprime commutators there (Lemma 2.6). Examples
  of every kind of certificate (Section 4.2), and the class-by-class tables (Tables 4–6).
- **Corollaries (Section 5).** C: SimpleGroupsIterator lists 42 groups of order < 10^9 outside A_n, PSL(2,q),
  Sz(q); all in Theorem A. D: 16 + 3 sporadic groups and the Tits group.
- **Partial data (Section 6, Table 3).** HN 38/54, Th 30/48, Co1 86/101, Fi23 85/98, Fi24' 87/108, B 84/184,
  M 92/194 classes certified (lemmas alone: 23, 20, 32, 27, 43, 30, 63); marked as partial data, not a proof.

## Computations (programs and outputs in reproducibility/)
- **Programs with which the results were first obtained:** `explicit/` (groups, search, verify, stored pairs
  for 53 + 5 groups, transfer to the CTblLib tables), `covers/`, `locals/` (constructions of 2.A11 and the
  11-local), `tables/` (engine.g, certify_all.g, outputs). The original table program certified all classes of
  O'N, Ly, J4 against all possible fusions; for O'N 4B it used the non-maximal CTblLib table (4^2x2).2^3.S3 of
  undocumented origin, replaced in the note.
- **Programs written with the note:** `note_tables.g` (Appendix A and Table 2, 1 s: O'N 30/30, Ly 53/53, J4
  62/62); `identify_note.g` (the facts of Section 4.1, 10 s); `check_note.g` (11 groups of Theorem A by the class
  equation, 4_2.L3(4), the O'N certificates; 2 s; ALL CHECKS PASSED); `run_full_check.sh` (verify.g for all 53
  groups, 2 min; 53 of 53 verified); `note_runs/` (partial data).
- **Re-runs on 2026-10-27 from the package folder** (`RERUN_LOG.txt`): the 53 verifications reproduced the
  recorded outputs (28 up to the order of the classes); certify_all.g reproduced out_ON.txt, out_Ly.txt,
  out_J4.txt byte for byte and the certified/open classes of all partial-data groups.

## Independent verification runs
The results were first obtained with the programs of `explicit/`, `covers/`, `locals/`, `tables/`. Three
independent verification runs, all AI-assisted, followed on 2026-10-27; each wrote its own programs (in
`reproducibility/verification_run_A/`, `verification_run_B/` and `independent_run_2/`). Runs A and B examined the
first results, and neither saw the other; the third run examined the final text of the note (section below).

| Item | Run A | Run B |
|---|---|---|
| Statement and readings | CONFIRMED | CONFIRMED |
| Lemmas R, K, F (with the table criterion), A4, D, S, Z | CONFIRMED (re-derived; brute-force tests of the F criterion on six groups and of A4) | CONFIRMED (re-derived) |
| Theorem A | CONFIRMED: own re-check of 10 groups (J3, U3(11), M24, L3(4), HS, 2F4(2)', S8(2), G2(4), O7(3), Co3); the 43 other outputs read | CONFIRMED: own re-check of all 2273 pairs in all 53 groups by the class equation, CTblLib invariants matched; wrong witness lists rejected in tests |
| Theorem B, table lemmas and subgroup steps | CONFIRMED: own program, all 145 classes, all possible fusions, Out(O'N) closure; the original pipeline reproduced byte for byte | CONFIRMED: own program, every step against all possible fusions; further routes through simple maximal subgroups found |
| Constructed subgroups | CONFIRMED (witnesses and identifications; GL(2,11) subgroups of order 48) | CONFIRMED (witnesses by class equation; module/orbit arguments for the J4 2-locals; 11-local rebuilt independently and isomorphic; the two uncovered classes not coprime commutators by Lemma Z) |
| Corollary C | CONFIRMED (list reproduced) | CONFIRMED (two enumerations) |
| Corollary D | CONFIRMED | CONFIRMED |
| Literature | accurate except Zini's volume (104) | accurate; Lucchini–Spiga v1 read in full |
| Partial data (Section 6) | not examined | not examined |

Neither run found an error in a theorem or a lemma.

**Corrections required by the two runs** (fourteen distinct items), all applied:
1. (A1, B3) O'N 4B is certified through maximal subgroups only (A7; also L2(31), 4^3.L3(2), L3(7), M11,
   4_2.L3(4)); the table (4^2x2).2^3.S3 is no longer used (Appendix A, Section 7).
2. (A2) Zini's paper is Bull. Aust. Math. Soc. 104 (2021) 448–452 (Crossref checked).
3. (A3) The certificates that are not 2-element-times-odd are named: Ly 33A/B, J4 35A/B, and also O'N 31A/B
   (found when the note was written: Lemma K with k = 2, d = 5, and through L2(31) a 31-element with a
   15-element) (Remark 1.2).
4. (A4) Section 6 states that certify_all.g recognizes 40 of the 53 groups of Theorem A (no effect on O'N, Ly,
   J4).
5. (A5) Irreducibility and faithfulness of the 2^10 (and 2^11) in J4 are derived from the CTblLib tables
   (normal subgroups 1, 2^k, H) (Section 4.1 (c), (d)).
6. (A6) The two uncertified classes of 11^(1+2):(5 × 2S4) are described as the classes 2A, 22A of H'; the
   classes outside H' contain no commutators; both are not coprime commutators in H (Lemma 2.6 in H/Q).
7. (A7) Shumyatsky's theorem is cited as Theorem 3.1 of the arXiv version; GAP's harmless syntax warnings are
   not shown (removed from the recorded outputs in the package, see the README).
8. (B1) The file set is frozen (`SHA256SUMS.txt`); the 74 files of stored pairs and generators have the same
   SHA-256 sums as the copies checked by run B; the working account is replaced by the note.
9. (B2) L2(31) has four classes of elements of order 16, mapped two to one onto {16A, 16B} or {16C, 16D}
   (Section 4.2).
10. (B4) The subgroups that are not maximal are named with their maximal overgroups (Table 2).
11. (B5) For L3(13) and U3(13) (no CTblLib table) the check used is stated: GAP's classes, with the class
    numbers 64 and 184 compared with the generic formulas (q^2+q+10)/3 and q^2+q+2, and the class equation.
12. (B6) The module arguments for 2^11:M24 (orbits 276 + 1771 versus 759 + 1288) and 2^10:L5(2) (155 + 868)
    are written out (Section 4.1).
13. (B7) The redundant routes through simple maximal subgroups are recorded (Appendix A, further routes).
14. (B8) The withdrawn Lucchini–Spiga preprint and its unrecorded search for 19 groups are reported in the
    credits; Fi23 is not settled.

**Changes made when the note was written.** The certificates of Theorem B were recomputed by `note_tables.g`
from the subgroups of Table 2 only (the first certificate of some classes changed, e.g. O'N 10A through J1, J4
4A through U3(11)); the identification facts were recomputed by `identify_note.g`, which also confirms the
completeness argument for 4_2.L3(4) among the perfect groups of order 80640; `check_note.g` re-checks 11 groups
of Theorem A, 4_2.L3(4) and the O'N certificates; the full re-check of Theorem A and the table programs were run
again from the package folder. All results agree with those of the two runs.

### Third verification run (on the final text of the note; `reproducibility/independent_run_2/`)
An AI-assisted run on 27 October 2026, the second stage of the verification. It examined the text of the note
(not the working account of the first results) and wrote all its programs itself; they read only the stored pairs
and generators, the GAP libraries and the LaTeX source, and do not call the programs of the package or of runs A
and B. The stored pairs and generators were first checked to be pure data (74 files; SHA-256 equal to those of the
package).

| Item | Third run |
|---|---|
| Statement, record, sources | CONFIRMED: notebook v49 text (identical in v48, which the record cites); Shumyatsky arXiv v2 (Theorem 3.1, the conjecture), Pellegrini–Shumyatsky arXiv v1 (definition, Theorem 1.1, Section 3, Magma remark) and Lucchini–Spiga v1 (Section 4, the 19 groups) read again, v2 withdrawal page, Zini's Crossref and zbMATH records; all eight DOIs checked with Crossref |
| Lemmas 2.1–2.7 | CONFIRMED: proofs re-derived line by line; brute-force tests of 2.1–2.4 and 2.6 on small groups (322286 tests, no failure) |
| Theorem A | CONFIRMED: own program; all 2273 pairs in all 53 groups by the class equation (pairwise non-conjugate commutators, 5633 exact conjugacy tests, GAP's ConjugacyClasses not used); constructions, orders (against the order formulas) and isomorphism types (against the library tables) of Table 1; Table 1 and the counts 507, 855, 670, 188 of Section 3 compared automatically |
| Theorem B | CONFIRMED: own program; real classes, power maps and class multiplication coefficients from the character values; every possible fusion re-checked; σ for O'N; the positions of Table 2 against the CTblLib lists of maximal subgroups; Tables 2 and 4–6 (all 145 rows: centralizer orders, lemmas with their parameters, the complete set of routes and the asterisks) compared entry by entry; the examples of Section 4.2 |
| Constructed subgroups | CONFIRMED: all facts of Section 4.1 (a)–(e) recomputed; stored pairs by the class equation; the 11-local built again from the formulas printed in (e), isomorphic to the stored group |
| Corollaries C, D | CONFIRMED: own enumeration from the order formulas (277 simple groups of order < 10^9; the 42 groups of Table 1) |
| Partial data (Section 6) | third column of Table 3 and the classes 27B, 27C of Th CONFIRMED by own program; columns four and five reproduced by the re-run of the package |
| Package | all commands of the package README, the programs of `tables/aux/` and the two constructions run again from `source.zip`; all recorded outputs reproduced (`independent_run_2/package_rerun/RERUN_RUN2.txt`) |
| Novelty | arXiv, zbMATH, Crossref, OpenAlex (works citing Shumyatsky, Pellegrini–Shumyatsky, Zini) and one web search: nothing in print beyond A_n, PSL(2,q), 2B2(q) |

The third run found no wrong theorem, lemma or table entry. **Corrections required by the third run** (six items),
all applied:
1. Remark 1.2: for the six classes O'N 31A/B, Ly 33A/B, J4 35A/B no certificate "2-element times odd" is found
   through any subgroup of Table 2; the certificates printed for Ly 11A/B and J4 7A/B (Lemma 2.1 with o(c) = 5, 3)
   are not of this form either, and the remark now says that these classes have such certificates through M11 and
   M22 (Table 2), which Appendix A does not list for classes certified by a lemma.
2. Table 2: M11 is a subgroup of the maximal subgroup 3^5:(2 × M11) of Ly (a direct factor of the complement
   2 × M11), not a complement; the entry and the sentence after Table 2 were corrected.
3. Section 4.1(e) and Table 2: H' = Q:SL(2,3) is the union of 13 classes of H (as a group it has 82 classes); the
   11 stored pairs meet 11 of these 13 classes.
4. Abstract and Section 7: the trusted base names the ATLAS lists of maximal subgroups (abstract) and the
   completeness of the libraries of perfect groups of order 80640 and of primitive groups of degrees 1024 and 2048
   (Section 7).
5. Section 3: the program of the first version compares the isomorphism type for the groups of Lie type only (order
   and simplicity suffice for the sporadic groups and the Tits group); the third run compared it for all 51 groups
   that have a library table.
6. The run is recorded in the note (Sections 3, 5, 7: Verification, Scope and priority), in this report and in the
   package README; `SHA256SUMS.txt` was recomputed. The appendix heading typesets O'N, Ly, J4 as in the text, and
   the appendix starts on a new page so that its tables precede the references.

## Relation to the literature, novelty and scope
- **Searches (27 October 2026; repeated in part by the three verification runs).** arXiv API (coprime commutators,
  commutators of elements of coprime orders, the authors of the known results, Ore conjecture with
  coprimality), zbMATH Open, Crossref, the OpenAlex lists of works citing Shumyatsky (2015), Pellegrini–Shumyatsky
  (2012) and Zini (2021), and three web searches. Semantic Scholar answered with rate-limit errors and was not used.
  MathSciNet and Google Scholar were not searched.
  - No printed answer beyond A_n, PSL(2,q) and the Suzuki groups was found. The citing works treat other
    questions (conciseness of commutator words, nilpotency and solubility criteria, profinite groups, products of
    classes in SL(2,q)/PSL(2,q), even permutations as commutators; Kundu–Mishra refute a stronger coset
    conjecture of Pellegrini–Shumyatsky, not 18.117).
  - The only earlier report on further simple groups is the withdrawn Lucchini–Spiga v1 (no data).
  - A search that finds nothing is not a proof of novelty; the note makes no priority claim.
- **What was read.** The notebook (v49) at Problem 18.117; Shumyatsky (2015) in the arXiv version (Theorem 3.1,
  the statement of the question); Pellegrini–Shumyatsky (2012) in the arXiv version (Theorem 1.1, Section 3, the
  Magma remark); of Zini (2021) the Crossref and zbMATH records with the abstract; Lucchini–Spiga v1 in full and
  the v2 abstract page. The third run read again Shumyatsky's arXiv v2, Pellegrini–Shumyatsky's arXiv v1 and
  Lucchini–Spiga v1 with the v2 page, and Zini's records. The proofs of the cited theorems were not checked. The other cited works (ATLAS, the
  generic character tables of Simpson–Frame, Kimmerle–Lyons–Sandling–Teague on orders of simple groups,
  Coutts–Quick–Roney-Dougal, Holt–Plesken, Isaacs, the GAP packages) are cited for standard facts or for software
  and data and were not read for the note.
- **Bibliographic data.** DOIs checked with Crossref on 2026-10-27 (Shumyatsky, Pellegrini–Shumyatsky, Zini,
  Liebeck–O'Brien–Shalev–Tiep, Simpson–Frame, Coutts–Quick–Roney-Dougal, Isaacs, Kimmerle et al.).
- **Scope.** Partial answer; the corpus status should stay `open`, with a note on the partial result.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/