# Verification report — KOU-11.18 (Kourovka 11.18, R. Brandl: is G(a,b) finite?; with 17.80, O. Macedońska)

Verification date: 2026-10-22.

**Verdict.** The note gives partial answers: G(a,b) is infinite whenever m divides both a and b for some m in
{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.
- **Proved.** Theorem A: for m = 18, 20, 26, 28, 30, 42, 44 an explicit epimorphism G(m,m) → PSL(2,q)
  (q = 8, 41, 13, 29, 29, 43, 43) whose kernel (m = 18, 26) or preimage of the split torus (other m) has
  abelianization of free rank exactly 7, 2, 14, 4, 4, 22, 12 (certificates, two independent checkers each; the
  free ranks determined a second time without the certificate files by the final verification run).
  Corollary 1.1 (families, by the monotonicity Lemma 2.2). Theorem B: an Engel pair X = [X,_a Y], Y = [Y,_b X] in
  SL(2,K), K any field, has tr X = tr Y = s, tr XY = s(s − 1), is swapped by some g in GL(2,K), and satisfies the
  relations with d = gcd(a,b); d = 1 forces X = Y = I. Corollary 1.2: the same for homomorphisms into SL(2,K),
  PSL(2,K), GL(2,K), PGL(2,K); G(2,3) and every G(a,b) with coprime a, b have no nontrivial such image.
- **Not new.** M_18 infinite: Quek–Wong–Wong, J. Algebra Appl. 13 (2014) 1350135 (GAP proof credited to the
  referee), with the conclusion for M_18 drawn in A. Abdollahi's zbMATH review Zbl 1298.20045. The paper itself
  was not accessible; the note's certificate for m = 18 is an independent second proof.
- **Tested only** (Section 5). No Engel pair with unequal least periods, and none with a least period 2 or 3, in
  A5–A9, S7, M11, M12, PSL(3,3), PSL(3,4) (periods ≤ 60 or 80) and none with periods ≤ 3 in M22, PSL(3,5), A10, A11;
  hence G(2,3) has no nontrivial homomorphism into these groups. Coset enumeration of G(2,3) inconclusive. No
  certificate found for M_10, M_12, M_14, M_24 (Engel pairs in PSL(2,q), q ≤ 50).
- **Open.** G(2,3); for which (a,b) G(a,b) is finite; Problem 17.80 for the n not divisible by one of
  18, 20, 26, 28, 30, 42, 44 (smallest: M_3; M_4, M_5, M_6 are nontrivial).

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. Read in v48 (6 Oct 2026; local copy, text lines 1734–1739 and 4545–4548) and in
  v49 (20 Oct 2026, the current version; printed pages 49 and 119). Both versions print the two problems without
  any comment.
  - 11.18 (R. Brandl, 11th issue, 1990): G(a,b) = ⟨x, y | x = [x,_a y], y = [y,_b x]⟩; "Is G(a,b) finite?";
    G(1,b) = 1 is easy, G(2,2) = 1, nothing is known about G(2,3); motivation through two-variable words
    characterizing soluble groups (Brandl–Wilson, J. Algebra 116 (1988) 334–341).
  - 17.80 (O. Macedońska, 17th issue, 2010): [u,_n v] = [u, v, …, v] (n copies); is M_n = G(n,n) infinite for
    every n > 2?
  - Convention: [x,y] = x^-1 y^-1 x y (Problem 8.11), left-normed Engel commutators (17.80). All statements and
    certificates use this convention.
- **Corpus records.** ulamai/UnsolvedMath v1.8.0 (snapshot d371ee9): KOU-11.18 (problems.jsonl line 16758) and
  KOU-17.80, both `open`; the record text agrees with the Notebook.

## Readings
| Reading | Answer in the note | Where |
|---|---|---|
| 11.18 universal: is G(a,b) finite for all a, b ≥ 1? | no; already in print for (18,18) (QWW 2014 + review) | Theorem A, Section 1.3 |
| 11.18 classification: for which (a,b) is G(a,b) finite? | infinite when m \| a and m \| b, m ∈ {18, 20, 26, 28, 30, 42, 44}; otherwise open | Corollary 1.1, Section 6 |
| 11.18 named case G(2,3) | open; no nontrivial image in SL/PSL/GL/PGL(2,K) (proved), none in A_n, n ≤ 11, and further groups (tested) | Corollary 1.2, Section 5 |
| 17.80: M_n infinite for every n > 2? | yes for n divisible by one of 18, 20, 26, 28, 30, 42, 44; open otherwise | Corollary 1.1 |
| Brandl's motivation (minimal simple quotients) | not addressed by infinite examples | Section 1.1 |
| Other commutator convention [u,v] = uvu^-1v^-1 | not treated (the groups may differ) | Section 1.1 |

## Results in the paper
- **Lemmas 2.1–2.4.** Periodicity of u_k = [u,_k v]; monotonicity G(a',b') ↠ G(a,b) for a | a', b | b';
  perfectness, and "least period 1 forces the trivial pair" (G(1,b) = 1); the Reidemeister–Schreier criterion with
  exact upper and lower bounds for the free rank of H/[H,H]; composition rules for the rewritten relators.
- **Theorem A (Section 3).** Five exact finite steps for each row of Table 1: Engel relations in SL(2,q) (least
  periods m, also in PSL(2,q)); surjectivity (the generated group has |PSL(2,q)| elements); the coset space;
  r independent integer vectors killing all rewritten relators (gcd 1, so H maps onto Z); rank of R modulo
  2^31 − 1 equal to |E'| − r (498, 1721, 1079, 867, 867, 1871, 1881).
- **Theorem B (Section 4).** Trace identities from Cayley–Hamilton; Lemma 4.1 (Fricke–Vogt; Horowitz 1972, Magnus
  1980) with a self-contained proof; equal traces; a swapping g via the trace form, Burnside's theorem and an
  elementary Skolem–Noether step, with g in GL(2,K); equal least periods; the gcd. Corollary 1.2 by the sign
  argument for PSL, det for GL, K^×/(K^×)^2 for PGL.
- **Section 5 (tested).** Scans of finite simple groups, coset enumeration, pairs for M_4 and M_6 (verified
  exactly by check_note.py), the second run's search for further certificates.

## Computations (sanity checks; programs and outputs in reproducibility/)
Theorem A rests on exact finite computations; Theorem B, the corollaries and the lemmas do not depend on
computation.
- **Programs written with the note.** `check_note.py` (standard library, about 3 s; TOTAL failures: 0): all
  certificates (Steps 1–4, including the extra pair for m = 30), all Engel pairs of SL(2,q) for the ten prime powers
  q ≤ 16 with no bound on the periods (counts 1, 1, 11, 6, 9, 28, 21, 21, 82, 31 including the trivial pair; every
  nontrivial pair satisfies the conclusions of Theorem B), relator lengths 786431 / 524323, the M_4 and M_6 pairs.
  `check_ranks.py` (numpy, about 2.5 min; TOTAL failures: 0): free ranks exactly 7, 14, 2, 4, 4 (and 10), 22, 12.
- **First run** (`first_run/`): verify_cert.py for m = 18, 26; check_matrices.py; check_thmB.py (240 Engel pairs
  in SL(2,p), p ≤ 23); pairscan.c (SL(2,q), q ≤ 49, periods ≤ 120); permscan.c (Section 5); tc.c (coset
  enumeration; jobs.log).
- **Verification run A** (`verification_run_A/`): own certificate checker (G(18,18) relators traced letter by
  letter from all 504 points); Theorem B over F_q for ten q ≤ 27 with an F_q-rational swapping g.
- **Verification run B** (`verification_run_B/`): check_newcert.py for the five new certificates (and the second
  m = 30 pair); step5_scan_pairs.py (SL(2,q) exhaustively for q ≤ 32, by traces for q ≤ 128: no unequal periods,
  no trace violation).
- **Final verification run** (`independent_run_2/`, programs written from the definitions): Theorem A for all
  seven rows and the second m = 30 pair with its own numbering, its own spanning tree, ranks modulo 10^9+7 and
  998244353 and its own integer null vectors checked exactly (free ranks 7, 2, 14, 4, 4, 22, 12 and 10); every
  vector of the ten certificate files checked exactly in the files' coordinates, with negative controls; all
  Engel pairs, with no bound on the periods, in SL(2,q) for q <= 16, GL(2,q) for q <= 9, PSL(2,q) for odd
  q <= 13, PGL(2,q) for odd q <= 11 (conclusions of Theorem B and Corollary 1.2 confirmed for each); no pair of
  least period 3 in SL(2,q) for q <= 32; the M_4 and M_6 pairs and the relator lengths.
- **Re-runs (2026-10-22).** All programs with a recorded output were run again from a copy of the package, the
  long scans on a part of their range; outputs identical up to running times (`reproducibility/RERUN_LOG.txt`).
  The final verification run repeated this from an extracted copy of the archive.

## Independent verification runs
The results for m = 18, 26, Theorem B and the tested statements were first obtained with the programs in
`first_run/`. Three independent verification runs, all AI-assisted, followed (2026-10-22); each wrote its own
programs. Runs A and B examined the first write-up of the results; the final run (run C below) examined the note
as written.

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, readings | CONFIRMED (v48) | CONFIRMED (v48 and v49) |
| Certificate m = 18 (kernel of G(18,18) → PSL(2,8)) | CONFIRMED (own checker; letter-by-letter from all 504 points) | CONFIRMED (three methods; free rank exactly 7) |
| Certificate m = 26 | CONFIRMED (own checker) | CONFIRMED (free rank exactly 14) |
| Monotonicity, gcd reduction | CONFIRMED (general form required) | CONFIRMED |
| Theorem B and Corollary 1.2, all characteristics | CONFIRMED (line by line; g in GL(2,K); sign lifting suggested) | CONFIRMED computationally (q ≤ 32); Steps 1–2 re-derived |
| New infinite cases m = 20, 28, 30, 42, 44 | — | FOUND and PROVED (certificates, torus preimages) |
| Sources (zbMATH reviews, Crossref) | CONFIRMED; one wording too strong | CONFIRMED; v49 read |
| Tested statements | A5, A6, A7 lists reproduced | A5–A8, S7, PSL(3,3), PSL(3,4) reproduced; PSL(2,q) searches |

**Final verification run (run C)** on the note as written:

| Item | Run C |
|---|---|
| Statement, conventions, corpus records | CONFIRMED (v48 and v49 re-read; records unchanged) |
| Five new certificates m = 20, 28, 30, 42, 44 (first check by another run) | CONFIRMED (own checker; every vector exact; negative controls rejected) |
| Free ranks of Table 1 | CONFIRMED without the certificate files (own tree, two primes, own exact vectors) |
| Certificates m = 18, 26 (spot check) | CONFIRMED (files' vectors exact; ranks 7 and 14) |
| Monotonicity; Theorem B and Corollary 1.2 | CONFIRMED (line by line; exhaustive scans SL/GL/PSL/PGL(2,q)) |
| M_4, M_6 pairs; relator lengths | CONFIRMED (nontriviality only, as worded) |
| Sources and novelty | CONFIRMED; related work BWW05, BGGKPP06, BGK10 to be cited |
| Package re-run from the archive | identical outputs (up to running times) |

No run found a mathematical error.

**Corrections required by the runs**, all applied in the note:
1. (A1) No "not in print" for the new cases: "no source found (zbMATH, OpenAlex, arXiv, Crossref; the 2014 and 2016
   papers not accessible)".
2. (A2) The general monotonicity (a | a', b | b') and the exact families: Lemma 2.2, Corollary 1.1.
3. (A3) The elementary sign lifting for PSL(2,K), characteristic 2, g in GL(2,K), images in GL(2,K) and PGL(2,K):
   Theorem B(ii), Corollary 1.2.
4. (A4) Citation of Horowitz (1972) and Magnus (1980) for the trace polynomial; the unused Fricke identity dropped.
5. (A5) The referee's proof that G_18 is infinite known only through Zbl 1298.20045: abstract, Section 1.3.
6. (B1) 786431 letters before free reduction, 524323 after: Section 2, Verification paragraph.
7. (B2) Kourovka v49 (2026-10-20) cited as the current version.
8. (B3) Exact free ranks 7 and 14, and those of the new certificates: Theorem A, Table 1.
9. (B4) The five new certificates with check_newcert.py, credited to the second run, new in this note.
10. (C1) Related work: Bray-Wilson-Wilson (2005) and Bandman-Greuel-Grunewald-Kunyavskii-Pfister-Plotkin (2006)
    (two-variable sequences characterizing finite soluble groups, with other words; known only from BGK10 and
    Crossref) and Bandman-Grunewald-Kunyavskii (2010; trace map; periodic points of the one-sided Engel map
    (x,y) -> ([x,y],y) in SL(2,q) for every q): Sections 1.1, 1.3, "Scope and priority".
11. (C2) "the presentations may define other groups" for the other commutator convention: Section 1.1.
12. (C3) The case 18 | n of 17.80 follows from QWW14 with Lemma 2.2: Section 1.3.
13. (C4) Section 5 heading "Engel pairs in other finite groups" (the list contains S7).
14. (C5) Skolem-Noether not attributed to the two textbooks; the step is proved in the text.
15. (C6) The final run recorded in the note (Verification paragraph, Section 7), here, in the README, the RERUN_LOG
    (with "first 14 lines" for step5_scan_pairs.py 13 13 120) and in `independent_run_2/`.

## Relation to the literature, novelty and scope
- **Known results used or reported.** Quek–Wong–Wong 2013 (G(n,n) with yxy = xyx: trivial unless 5 | n, ≅ SL(2,5)
  if 5 | n; Brandl's review Zbl 1282.20029 with his conjecture G(5,5) ≅ SL(2,5)); 2014 (quotient G_n; G_18 infinite
  by the referee's GAP proof; Abdollahi's review draws the conclusion for M_18); 2016 (Engel pairs in SL(2,F);
  zbMATH summary Zbl 1337.20055). Trace identities: Fricke–Vogt, Horowitz (1972), Magnus (1980); the trace map
  for periodic points of word maps, among them the one-sided Engel map, in Bandman–Grunewald–Kunyavskii (Groups
  Geom. Dyn. 4 (2010) 607–655, Section 5.4 of arXiv:0809.0369v2), which does not mention G(a,b). Brandl's
  motivation: two-variable sequences characterizing finite soluble groups were later given, with other words, by
  Bray–Wilson–Wilson (Bull. LMS 37 (2005)) and Bandman–Greuel–Grunewald–Kunyavskii–Pfister–Plotkin (Compositio 142
  (2006)). Burnside (Lang) and Reidemeister–Schreier (Lyndon–Schupp) from textbooks; the Skolem–Noether step is
  proved in the note.
- **What was read.** The problem texts in v48 and v49; the three zbMATH entries; Crossref metadata of all cited
  articles (all DOIs verified on 2026-10-22); of Bandman–Grunewald–Kunyavskii 2010 (arXiv v2) Sections 1, 2 and
  5.4, the whole text searched for the Notebook and for G(a,b). Not read: the papers of 2014 and 2016 (publisher
  refused automated access, HTTP 403; not worked around), the 2013 paper, Brandl–Wilson 1988, Bray–Wilson–Wilson
  2005, the Compositio paper of 2006, S. G. Quek's thesis (2015).
- **Searches** (2026-10-18; repeated 2026-10-22 by the verification runs): zbMATH Open, OpenAlex (citing works),
  arXiv API, Crossref, Semantic Scholar citation lists (its search refused with HTTP 429), three web searches. The only
  work citing QWW 2014 is QWW 2016. No source was found for Theorem A with m ≠ 18, for Theorem B or for
  Corollary 1.2. A search that finds nothing is not a proof of novelty; no priority is claimed.
- **Scope.** The note answers neither question in full; see the verdict.

## 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/