# Verification report — KOU-15.67 (Kourovka Notebook 15.67, Ya. N. Nuzhin: adjoint Chevalley groups over the integers generated by three involutions, two of which commute; the type F_4)

Verification date: 2026-10-22.

**Verdict.** The note gives a **partial answer**. Status: partial answer: the adjoint Chevalley group F_4(Z) is
generated by three involutions two of which commute (computer-assisted proof with an exact certificate); the
types B_l, C_l (l >= 3) and D_l (l >= 4) stay open.
- **Settled.** The type F_4, positively: the explicit involutions a, b, c of Theorem 3.1, given in the Chevalley
  basis B_T (Definition 2.3; Table 1 lists its 20 extraspecial signs), satisfy a^2 = b^2 = c^2 = 1, ab = ba,
  a, b, c != 1 and <a, b, c> = F_4(Z). Equivalently (Remark 3.2), in the basis B_0 with all extraspecial signs +1
  the same holds with x_1110(-1) in place of x_1110(1) in a.
- **Computer-assisted.** The proof is complete and checkable: Proposition 4.1, Lemmas 2.1, 2.2 and 4.2 are proved
  in the text; the remaining finite part (four signs of structure constants and the 48 identities of a
  straight-line program with 1064 nodes) consists of identities between integral 52 x 52 matrices, verified
  exactly by `verify_certificate.py` (Python standard library, about 3.5 s) and confirmed by three independent
  programs.
- **Consequences.** For every prime p the reductions generate F_4(p) in the same way (Corollary 6.1); the least
  number of generating involutions of F_4(Z) with product 1 is 5 (Proposition 6.2).
- **Open.** The types B_l, C_l (l >= 3) and D_l (l >= 4). The same design was tried for B_3, C_3 and D_4 and
  failed (a non-result, Section 7); no obstruction from finite quotients is known (characteristic 2 by Nuzhin 1990
  as stated in its zbMATH review; odd characteristic not examined).
- **Novelty.** To our knowledge new; no printed answer for F_4 was found in the sources searched (below). No
  priority is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka
  Notebook, No. 21, arXiv:1401.0300.
  - Version 48 (6 October 2026): local text extraction, Problem 15.67 at lines 3597-3606.
  - Version 49 (20 October 2026): the current version according to the arXiv API on 2026-10-22 (`<id>`
    1401.0300v49, updated 2026-10-20T09:15:18Z); PDF (1,485,651 bytes, sha256
    `24cd6e5031baa7b1b4bb58f363b980c2f6f1769f9a8431d4fef0065dc777e196`), Problem 15.67 on page index 94.
  - The text of 15.67 and its comment are identical in both versions; the problem is not starred; the comment
    lists SL_n(Z), n > 13 (Tamburini-Zucca 1997); PSL_n(Z) if and only if n > 4 (Nuzhin 2008); PSp_4(Z) does not
    have the property, as PSp_4(3) does not (Archive 7.30); G_2 (Timofeenko 2015); E_l (Timofeenko 2017). F_4 is
    not mentioned. Problem 14.69 (with a) starred) and Archive 7.30 are also unchanged in v49.
- **Corpus record.** ulamai/UnsolvedMath, dataset version 1.8.0 (snapshot d371ee9), record KOU-15.67, status
  `open`; statement and comment as in the source (only TeX markup differs); proposer Ya. N. Nuzhin, 2002.
- **The group of the question.** Defined as in the proposer's school: Timofeenko (SEMR 14 (2017), p. 810) takes
  Phi(Z) to be the elements of the adjoint Chevalley group Phi(Q) whose coefficients with respect to the lattice
  Z(Phi), which he calls the lattice of roots, are integers; as Phi(Q) acts on L_Z (x) Q, the note reads this lattice
  as L_Z, so that Phi(Z) is the stabilizer of L_Z. His Lemma 3 states Phi(Z) = <x_r(1)>, citing p. 107 of the
  Russian edition of Steinberg's lectures; Kazakova and Nuzhin (2025, pp. 83-84) cite the same page for Chevalley groups over
  Euclidean rings. For F_4 the root lattice equals the weight lattice (Cartan determinant 1), so the simply
  connected and the adjoint group coincide, and Z is Euclidean; hence the group of the question is
  E(F_4, Z) = <x_r(1)>. Steinberg's text was not read; the theorem is used as these two papers quote it.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The group: the stabilizer of L_Z in F_4(Q) (the proposer's school), or the group generated by the x_r(t), t in Z | the same group, by Steinberg's theorem for Euclidean rings as quoted by Timofeenko and by Kazakova-Nuzhin | Section 2.3, formula (2) |
| Adjoint versus universal (simply connected) group | the same for F_4 | Sections 2.1, 2.3 |
| Explicit formula versus existence statement | the explicit formula holds in B_T (and, with x_1110(-1), in B_0); the existence statement holds for every Chevalley basis (Lemma 2.2) | Section 3 |
| a, b, c must be nontrivial (elements of order 2) | they are; for a non-soluble group such as F_4(Z) this follows from the other conditions, since a group generated by two elements of order at most 2 is soluble | Section 1.1, Theorem 3.1, Proposition 4.1(d) |
| The whole question (all types) | not settled; B_l, C_l (l >= 3), D_l (l >= 4) open | Section 7 |

## Results in the paper
- **Theorem 1.1 / 3.1.** a = x_1000(1) x_1100(1) x_1110(1) x_1111(1) h_b2(-1) h_b4(-1), b = n_b2 n_b4 h_b1(-1) h_b3(-1),
  c = n_b1 n_b3 h_b2(-1) in B_T are involutions, ab = ba, a, b, c != 1, and <a, b, c> = F_4(Z).
- **Lemma 2.1.** A table satisfying antisymmetry, the Jacobi identity and |N_rs| = p+1 (with [h_i, h_j] = 0 and
  [e_r, e_-r] = h_r) defines the simple Lie algebra of type F_4 with a Chevalley basis. **Lemma 2.2.** Two Chevalley bases are
  related by an isomorphism up to signs eps_r; this replaces x_r(t) by x_r(eps_r t) and preserves the group.
- **Definition 2.3, Table 1.** B_T: extraspecial signs +1 at 1100, 1342, 2342 and -1 at the other 17 (positive
  roots ordered by height, then lexicographically); B_0: all +1; the sign map eta is -1 exactly on 0011, 0110,
  1110, 0121, 1121, 1220, 1222, 1231, 1242, 1342, 2342.
- **Section 3.** The involution identities hold in all 128 sign patterns of the formulas; ab = ba holds in 2 of the
  8 torus classes; the certificate works only in the class of B_T. **Remark 3.2.** The B_0 form.
- **Proposition 4.1, Lemma 4.2.** The proof without computation of membership, a^2 = b^2 = c^2 = 1, a, b, c != 1,
  the reduction of ab = ba to four basis-dependent signs, and the reduction of generation to the 48 x_r(1).
- **Section 5.** The straight-line program (1064 nodes: 3 gen, 3 inv, 832 mul, 226 pow, exponents up to 327; 48
  targets; expanded words of lengths 3144 to about 5*10^14), the checker's steps 1-6 (inv nodes evaluated as the
  generators, which were verified to be involutions), the proof, the independent checks, the rejection tests;
  Remark 5.1 (the key step, not needed for the proof) and the search (not needed for the proof; it uses the
  commutators (a_i a_j)^2, of which 36 of the 66 with i < j are unipotent).
- **Corollary 6.1** (F_4(p) for every prime p), **Proposition 6.2** (n(F_4(Z)) = 5, by the soluble quotient
  <x, y, z | x^2, y^2, z^2, (xyz)^2>; the same two-step pattern as in the proof of the Corollary of Kazakova-Nuzhin
  2025, credited there), **Remark 6.3** (Scott's inequality, a consistency check: 140 >= 104).
- **Section 7.** Open types; non-results for B_3, C_3, D_4; no obstruction known; a human-readable proof for F_4
  would be desirable.

## Computations (programs and outputs in reproducibility/)
- **The checker** `verify_certificate.py` (standard library, /usr/bin/python3 3.9.6, 3.5 s): CERTIFICATE VALID;
  Jacobi identity on 140608 basis triples, |N_rs| = p+1, a^2 = b^2 = c^2 = 1, ab = ba, all 48 targets equal x_r(1);
  largest matrix entry 408096. Input sha256: certificate `5a1d1ff2...59ff33b7`, table `5bab4e71...b3d852729`.
- **Rejection tests** `rejection_tests.py` (36 s): 11 corruptions rejected, the original accepted.
- **The program of the note** `check_note.py` (standard library, 7.6 s; TOTAL failures: 0): (A) the certificate;
  (B) the table of B_T reproduced, entry by entry, from its 20 extraspecial signs by the extraspecial-pair
  recursion; (C) the table of B_0, the sign map eta, and the B_0 certificate (2079 nodes, structural inverses for
  the 22 targets with eta = -1) accepted by the checker; (D) the facts of Proposition 4.1, gcd 1 of the entries of
  a-1, b-1, c-1, a != b and a b (ab) c c = 1, the order 12 of bc, x_r(1)^2 = x_r(2), Remark 5.1; (E) the 128 sign
  patterns: all involutions, ab = ba exactly in the classes (1,1,1) and (1,-1,-1), and the target -0121 (shortest
  word, 3144 letters) is x_r(1) in the class of B_T and not x_r(+-1) in each of the other seven classes; (F)
  rank(g-1) = 28 for g = a, b, ab, c; (G) node counts and word lengths.
- **The search** `search_certificate.py` (numpy, 21.6 s) regenerates `certificate_F4.json` byte for byte;
  `export_structure_constants.py` regenerates the table byte for byte; `structure_check.py`,
  `hand_step_check.py`, `scott_check.py` reproduce their recorded outputs.
- **Exploratory** (`exploratory/`, non-results): all seven searches for B_3, C_3, D_4 reproduced (identical up to
  times); `free_product_ball.py` gives the baseline 3641 quoted in Section 7.
- **Programs of the third verification run** (`independent_run_2/`, standard library): `run2_check.py` (18 s,
  71 checks, TOTAL FAILURES: 0), `run2_patterns.py` (47 s), `run2_negative.py` (24 s), `run2_unipotent.py` (1 s);
  see the third run below.
- **Re-runs.** On 2026-10-22 every program of the package was run from the package folder; the outputs agree with
  the earlier outputs up to running times (and file paths, now printed as file names). The third verification run
  ran every program again from an extracted copy of the source archive of the version it examined, with the same
  result (`independent_run_2/rerun_package.txt`). No `__pycache__` folder is part of the package.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first write-up of the result (not the final
text); each wrote its own programs, which are in `reproducibility/independent_runs/` with their outputs. A third,
also AI-assisted, examined the text of the note before its final revision and wrote its own programs, which are in
`reproducibility/independent_run_2/` with their outputs.
- **First run.** Re-derived without computation the membership of a, b, c, the torus characters, a^2 = b^2 = c^2
  = 1, the reduction of ab = ba to four signs, the reduction to the 48 root elements, the argument of Lemma 2.1,
  Corollary 6.1 and Proposition 6.2; read Timofeenko 2017 and Kazakova-Nuzhin 2025 (journal PDF files), the zbMATH
  reviews of Nuzhin 1990 and 1997 II, and version 49 of the Notebook; its exact program (root system by
  reflections, Jacobi identity as ad[x,y] = [ad x, ad y], checks of N_sr = -N_rs, N_-r,-s = -N_rs and of the
  Killing form, genuine inverses) confirmed every identity; 5 corruptions rejected. Verdict: confirmed as a
  partial answer. Required fixes: (1) Timofeenko's definition of Phi(Z) and the role of Steinberg's theorem; (2)
  the zbMATH review as the source for characteristic 2, odd characteristic labelled as not verified; (3) the
  evaluation of the inv nodes stated; (4) novelty "to our knowledge".
- **Second run.** Built its own model of F_4 over Z from the Cartan matrix by the extraspecial-pair recursion
  (the basis B_0), mapped the table of the certificate to it by the sign map eta, evaluated the program in its own
  model (modulo a prime and exactly), examined all 128 sign patterns (ab = ba in 2 of 8 classes; the certificate
  only in the class of B_T), made 13 negative tests, checked the reductions modulo 2, 3, ..., 31 and 101, re-ran
  the programs of the first write-up (certificate and table regenerated byte for byte), and checked the
  bibliography with Crossref. Verdict: confirmed, no error. Recommendations: the description of B_T by
  extraspecial signs and the B_0 form; the basis dependence in the main text; the DOI of Timofeenko 2017 and
  version 49 of the Notebook; the remark on a human-readable proof.
- **Changes made.** All required fixes and all recommendations were applied in the note (Sections 2.3, 2.5, 3,
  5.2, 6, 7 and 8); the program `check_note.py` was added, with Parts B, C and E new.
- **Third run (second round; it examined the text of the note).** Read the note line by line before its final
  revision; read the sources again (Problems 15.67, 14.69 and Archive 7.30 in v48 and v49 of the Notebook, with v49
  confirmed as the current arXiv version on 2026-10-22; Timofeenko 2017 and Kazakova-Nuzhin 2025 in the journal files;
  the zbMATH review Zbl 0725.20015) and re-derived every proof (Lemmas 2.1, 2.2, 4.2, Proposition 4.1, Corollary 6.1,
  Proposition 6.2). Its programs (`independent_run_2/`, Python standard library) build F_4 in a way independent of
  the search program and of the first two runs: the 27-dimensional minuscule module of the Lie algebra of type E_6
  is built from its Cartan matrix (Chevalley generators acting by 0/1 matrices) and folded to type F_4, and the root
  vectors are normalized by the 20 extraspecial signs. In this model the tables of B_T and B_0 equal the data files
  entry by entry; the triple of Theorem 3.1 and all 48 targets of the program were checked exactly in the adjoint
  representation and again in the 27-dimensional module; in the adjoint representation also the four signs (5), the
  facts of Proposition 4.1, the B_0 form with its certificate (2079 nodes, all 48 targets), Remark 5.1, the gcd of
  Corollary 6.1, the facts of Proposition 6.2 and the ranks of Remark 6.3; the statements on the 128 sign patterns
  were confirmed (the program modulo a prime in the seven other classes: no target is x_r(+-1) there). Eight
  corrupted inputs were rejected. The whole package was re-run from the source archive (identical outputs up to
  running times); the ten DOIs were confirmed with Crossref; the literature search was repeated (below). Verdict:
  confirmed as a partial answer; no error in a theorem or a proof. Required corrections (all made): (1) an incorrect
  side remark in Section 1.1 (a, b, c != 1 is automatic only for groups not generated by two involutions, e.g.
  non-soluble groups; S_3 = <1, (12), (23)> shows that "non-abelian" is not enough); (2) Timofeenko's own wording
  ("lattice of roots Z(Phi)") in Section 2.3; (3) the bracket [h_i, h_j] = 0 in Section 2.2; (4) Section 5.5 and the
  docstring of `search_certificate.py`: only 36 of the 66 commutators (a_i a_j)^2 are unipotent; (5) credit to
  Kazakova-Nuzhin 2025 for the pattern of Proposition 6.2; (6) the comparison with Timofeenko's triples (his first
  involution has two root elements); (7) the second run re-ran the programs of the first write-up; (8) the record of
  this run (abstract, Sections 1.2, 5.4, 8, this report, `independent_run_2/`).

## Relation to the literature, novelty and scope
- **Known cases.** A_l: yes iff l >= 4 (Tamburini-Zucca 1997 for SL_n(Z), n > 13; Nuzhin 2008 for PSL_n(Z));
  B_2 = C_2: no (PSp_4(Z), via PSp_4(3); Notebook, Archive 7.30); G_2: yes (Timofeenko 2015; also Kazakova-Nuzhin
  2025, remark); E_6, E_7, E_8: yes (Timofeenko 2017; E_7 in the adjoint group). Timofeenko 2017 names F_4 as the
  remaining exceptional type; Kazakova-Nuzhin 2025 (the proposer a coauthor) name A_n, E_n and G_2 as the only
  solved types. The shape of the triple follows Timofeenko's construction for E_l (lifts of the two halves of the
  bipartite Coxeter element), which uses Nuzhin's method for groups of Lie type over finite fields.
- **Searches (22 October 2026; repeated in part by the second and the third run).** zbMATH Open (authors Nuzhin, Timofeenko;
  titles with "three involutions", involutions with Chevalley or integers, F_4 with involutions, "generating
  triples"; involutions 2025-2026), arXiv API (the same phrases; authors Nuzhin, Timofeenko, Shaipova, Gvozdev,
  Vsemirnov), OpenAlex (English and Russian searches; works citing Tamburini-Zucca 1997), Crossref, Semantic
  Scholar (one query answered, others refused), two web searches (English, Russian). The third run repeated the
  search on 2026-10-22: arXiv API (7 queries, phrases and authors), Crossref (4 searches: the phrase since 2025, F4
  with integers and involutions since 2024, authors Nuzhin since 2025 and Timofeenko since 2017), OpenAlex (records of
  Tamburini-Zucca 1997, Timofeenko 2015 and 2017, Kazakova-Nuzhin 2025 and Nuzhin 2025; the 8 works citing
  Tamburini-Zucca 1997; searches in English and Russian since 2023; no work citing Timofeenko 2015, 2017 or
  Kazakova-Nuzhin 2025 is listed), zbMATH API ("three involutions" 2023-2026, author Nuzhin 2024-2026, author
  Timofeenko, F_4 with involutions and Chevalley; two queries refused with HTTP 502) and one web search (Russian).
  Later work of the school found: GL_n(Z), PGL_n(Z), groups over Z + iZ, finite groups, carpets, regular polytopes;
  nothing on F_4(Z), B_l(Z), C_l(Z), D_l(Z).
- **What was read.** Timofeenko 2017 and Kazakova-Nuzhin 2025 completely (text extractions of the journal PDF
  files); Problems 15.67, 14.69 and Archive 7.30 in v48 and v49 of the Notebook; zbMATH entries of Nuzhin 2008
  (Zbl 1324.20027), Nuzhin 1990 (Zbl 0725.20015, with the review used for characteristic 2) and Scott 1977 (Zbl
  0399.20047). Not read: Tamburini-Zucca 1997, Nuzhin 2008, Timofeenko 2015 (only the results quoted in the
  Notebook and in the two papers above are used), Nuzhin 1997 I and II, Scott 1977, Frenkel-Kac 1980, the books of
  Steinberg and Carter, and Chevalley 1955. The section numbers given for Carter's book were not checked against
  the book.
- **Bibliography.** DOIs checked with Crossref on 2026-10-22: Tamburini-Zucca (10.1006/jabr.1997.7055),
  Timofeenko 2015 (10.17516/1997-1397-2015-8-1-104-108), Kazakova-Nuzhin 2025 (10.46698/d7840-8893-1360-c),
  Nuzhin 1990 (10.1007/BF02001358), Nuzhin 1997 I (10.1007/BF02671953), II (10.1007/s10469-997-0066-3), Chevalley
  1955 (10.2748/tmj/1178245104), Steinberg (10.1090/ulect/066), Scott 1977 (10.2307/1970920), Frenkel-Kac 1980
  (10.1007/BF01391662). Timofeenko 2017 (10.17377/semi.2017.14.068) is registered with mEDRA, not Crossref; the DOI
  resolves to the journal's PDF.
- **Caveats.** The identification of the group of the question with <x_r(1)> rests on Steinberg's theorem as
  quoted by Timofeenko and by Kazakova-Nuzhin; Steinberg's text was not read. The consistency with the
  odd-characteristic classification (Nuzhin 1997) was not examined. A search that finds nothing is not a proof of
  novelty; unindexed conference abstracts or theses of the Krasnoyarsk school cannot be excluded.
- **Scope.** One more type of the question settled; the classical types untouched; no priority claimed.

## 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/