# Verification report — KOU-11.16 (Kourovka Notebook, Problem 11.16 b), R. Brandl: finite simple groups of bounded Engel depth)

Verification dates: 2026-10-21 (two independent verification runs on the first written version of the results);
the note was written on 2026-10-22, after these runs, and a third independent verification run examined the
final text on 2026-10-22; its corrections are in the present version.

**Verdict.** The note gives a **partial answer to part b): for every r, all but finitely many nonabelian finite
simple groups in V_r are Suzuki groups, so b) holds if and only if the Engel depth of Sz(q) tends to infinity;
this last statement and part a) stay open**. Every statement it makes is proved, modulo the classification of
finite simple groups and standard facts on subgroups that are listed with their sources (their numbering in the
books and papers was not checked); the Suzuki computations are evidence only.
- **Proved.** Lemma 2.1 (G ∈ V_r iff the Engel depth μ(G) ≤ r), Lemma 2.2 (both commutator conventions give the
  same V_r), Lemma 2.3 (closure under sections), Lemma 2.4 (dihedral 2-groups; μ(A_m) ≥ ⌊log₂(m−2)⌋);
  Theorem 3.5: PSL(2,q) ∉ V_r for q > Q_r = ((d−1)(d−2)+2)², d = 3·2^(r−1) (d = 2^r for q even), and
  μ(PSL(2,q)) ≥ ⌊(log₂ q)/4⌋ for q ≥ 256 (Fricke trace recursion, Eisenstein's criterion, the Weil bound for
  singular plane curves, an elementary realisation lemma); Theorem 4.1 = Theorem 1.1: every V_r contains only
  finitely many nonabelian simple groups that are not Suzuki groups, with explicit bounds; Corollary 1.2:
  11.16 b) holds iff μ(Sz(q)) → ∞; Proposition 5.1: for an odd prime p every proper section of Sz(2^p) has
  Engel depth ≤ 3; Lemmas 5.2, 5.3 (the reductions used in the computations).
- **Taken from the literature.** The problem and the case r ≤ 2 (Brandl 1983; V_1 abelian by Gupta 1966); for
  q ≥ 4, PSL(2,q) ∈ V_3 iff q ∈ {4,5,8}, the conjecture and the table of minimal laws (Brandl–Nikolova 1986); the laws of A_5, A_6 (Nikolova 1984); the
  trace-map method and the recursion (Bandman–Greuel–Grunewald–Kunyavskii–Pfister–Plotkin 2006,
  Bandman–Grunewald–Kunyavskii–Jones 2010, Bandman–Garion–Grunewald 2012, eq. (3.3)); the Weil bound (Aubry–Perret
  1996, as stated in BGG 2012, (5.8)); Macbeath's theorem (special case re-proved); the classification of finite
  simple groups; the subgroup facts (C1)–(C8), (E1)–(E5) (Kleidman–Liebeck, Carter, Kleidman, Malle, Ward);
  Suzuki's maximal subgroups; the matrices of Sz(q) (BGGKPP 2006, Section 4.1 of the arXiv version).
- **Evidence only (finite computations, not proofs about large q).** μ(Sz(8)) = 62 and μ(Sz(32)) = 527 (all
  classes); certified lower bounds from explicit pairs: μ(Sz(128)) ≥ 3759, μ(Sz(512)) ≥ 24814,
  μ(Sz(2048)) ≥ 183810, μ(Sz(8192)) ≥ 792500; hence (Corollary 5.4) for 3 ≤ r ≤ 61 the Suzuki groups not decided
  are exactly the Sz(2^n) with gcd(n, 15015) = 1 (smallest Sz(2^17)). A random-mapping heuristic suggests
  μ(Sz(q)) ≈ c·q^(3/2); it is stated as a heuristic only.
- **Open.** Whether μ(Sz(q)) → ∞ (equivalently, part b)); the Notebook's remark that b) is true for r < 4 is
  not re-proved for the Suzuki groups at r = 3, and no published proof of that case was found; part a) (Fitting
  length) is not treated.

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, Problem 11.16 (R. Brandl, 11th issue, 1990), read in version 48 (6 October
  2026; local text extraction) and in version 49 (20 October 2026; text extraction of the PDF downloaded on
  21 October 2026). The text is the same in both versions; no comment, star or archive mark follows it.
  - Definition: V_r is the class of all finite groups G satisfying a law [x, _r y] = [x, _s y] for some
    s = s(G) > r, where [x, _1 y] = [x, y], [x, _(i+1) y] = [[x, _i y], y].
  - a) Is there a function f such that every soluble group in V_r has Fitting length < f(r)? (For r < 3 see
    Brandl, Bull. Austral. Math. Soc. 28 (1983).)
  - b) Is it true that V_r contains only finitely many nonabelian simple groups? "This is true for r < 4."
- **Corpus record.** ulamai/UnsolvedMath, KOU-11.16 (dataset version 1.8.0, snapshot d371ee9, record id 700709,
  status `open`, proposed by R. Brandl, 1990): the statement field reproduces the Notebook text with both parts.
- **Conventions.** [x,y] = x⁻¹y⁻¹xy as in the Notebook; Lemma 2.2 shows that [x,y]′ = xyx⁻¹y⁻¹ gives the same
  classes V_r. The tail of (x,y) is the least k with [x, _k y] on a cycle of z ↦ [z,y], and μ(G) is the
  largest tail; G ∈ V_r iff μ(G) ≤ r (Lemma 2.1), and for G ≠ 1 μ(G) is Brandl's (Engel) depth. The value of s
  may depend on G but not on (x,y); for finite G this is equivalent to periodicity pair by pair (Lemma 2.1).
  "Finitely many" means finitely many isomorphism types.
- **Normalisation checks.** The exact values μ(PSL(2,q)) = 3, 4, 3, 4, 6, 7 for q = 4, 7, 8, 9, 11, 13, and the
  minimal laws with s = 63, 172, 129, 124, 1986, 2191, agree with the table of Brandl–Nikolova (1986, p. 246),
  which has no column q = 5 (PSL(2,5) ≅ PSL(2,4)); the laws of A_5 and A_6 agree with Nikolova (1984).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Part b) as stated (every r, all nonabelian finite simple groups) | open; equivalent to μ(Sz(q)) → ∞ | Corollary 1.2 |
| Part b) for the simple groups that are not Suzuki groups | yes, with explicit bounds | Theorems 1.1, 4.1 |
| Part b) for PSL(2,q) | yes: PSL(2,q) ∉ V_r for q > Q_r | Theorem 3.5 |
| r ≤ 2 | known: groups in V_2 are soluble (Brandl 1983) | Section 1.2 |
| r = 3 | known for PSL(2,q) (Brandl–Nikolova 1986); finiteness for the other non-Suzuki families by Theorem 1.1; for the Suzuki groups not proved here (undecided Sz(2^n), gcd(n,15015) = 1), no published proof found | Sections 1.2, 5.1 |
| s independent of G (a variety) | trivially yes by the theorem of Heineken–Neumann quoted by Brandl–Nikolova; not the question, since s = s(G) | Section 1.1 |
| Part a) (Fitting length of soluble groups in V_r) | not treated | — |

## Results in the paper
- **Section 2.** Engel tails, μ(G); Lemma 2.1 (law ⇔ periodicity at index r; for G ≠ 1, μ(G) is Brandl's depth);
  Lemma 2.2 (conventions); Lemma 2.3 (sections); Lemma 2.4 (nilpotent groups; tail(ρ,τ) = n in the dihedral group
  of order 2^(n+1); μ(A_m) ≥ ⌊log₂(m−2)⌋ for m ≥ 6). A verification run computed μ(A_7) = 15.
- **Section 3.** Lemma 3.1 (Fricke's identity with a two-line proof; tr(a_k y) = s(t_k − 1) and
  t_(k+1) = t_k² − s²t_k + 2s² − 2; conjugation to h ↦ h² + ch, c = 4 − s²); Lemma 3.2 (SL versus PSL: tail
  ≥ r+1, and ≥ r+2 if q is even or s ≠ 0); Lemma 3.3 (realisation of (tr y, tr[x,y]) = (s,t) for all s² ≠ 4,
  self-contained; a special case of Macbeath's theorem); Lemma 3.4 (R_r(h,c) = g_c^r(h) + c is Eisenstein at c,
  and R_r(h, 4−s²) at s − 2 for p odd: absolutely irreducible of degrees 2^r and 3·2^(r−1); equivalently, all
  roots have valuation 2^(−r)); Theorem 3.5 and Remark 3.6; Table 1 (exact μ(PSL(2,q)), q ≤ 64).
- **Section 4.** CFSG; facts (C1)–(C8) (alternating sections A_l of classical groups, l ≥ 5, and sections PSL_2(q)),
  (E1)–(E5) (sections PSL_2(q) of exceptional groups other than ²B_2), with the remark that their numbering in the
  references was not checked; Theorem 4.1; proof of Theorem 1.1 and Corollary 1.2.
- **Section 5.** The matrices of Sz(q), subfield subgroups, Suzuki's maximal subgroups; Proposition 5.1;
  Lemmas 5.2 (class dynamics Ψ_y(u) = y⁻¹uyu⁻¹y) and 5.3 (orbits of a subgroup of C(y), with the certificate
  O_(μ+λ) = O_μ, O_(μ−1+λ) ≠ O_(μ−1)); exact values and Table 2; Corollary 5.4; the heuristic; Section 5.2 on what a
  proof would need (a Frobenius-stable absolutely irreducible component of the chain variety and a Lang–Weil
  estimate for the twisted Frobenius; a sufficient condition [x, _j y](∞) = 0 giving tail j + 2).
- **Section 6.** The package; Conclusion; Verification; Scope and priority.

## Computations (sanity checks; programs and outputs in reproducibility/)
- `check_note.py` (standard library, 21 s, 28 checks, all passed; output `check_note_output.txt`): Fricke's
  identity and Lemma 3.1 on 2850 random pairs over 19 fields (q ≤ 1009); the construction of Lemma 3.3 for all
  5648 pairs (s,t) over 16 fields (q ≤ 32); exact μ(PSL(2,q)), μ(SL(2,q)) for q ≤ 16 with the minimal laws of
  Brandl–Nikolova and Nikolova; Lemma 2.4(b),(c); Lemma 3.4 (Eisenstein conditions, Newton polygons modulo small
  primes, degrees); Theorem 3.5 (constants, counting inequality, construction in 38 cases, 22 with q > Q_r); the
  order 29120 for q = 8; the witness pairs for Sz(8), Sz(32); the certificates of all six pairs of Table 2
  (Bruhat-form membership and exact tail certificates), compared with the recorded outputs; Corollary 5.4.
- Programs of the first version (`original/`): Table 1 (83 s, numpy); Lemmas 3.1, 3.3 and the construction of
  Theorem 3.5 (19 s); exact μ(Sz(8)), μ(Sz(32)) by Lemma 5.2 over all classes (0.4 s, 4 min 40 s; numpy); a
  brute-force check for Sz(8) (30 s); the random pairs of Table 2 (20 s to 5 min per run). Re-run at the writing
  from a copy: `psl2_checks.py 7`, `sz_exact.py 3 --all`, `sz8_bruteforce_check.py`,
  `sz_sample.py 7 150 31 5000000 --quot --crosscheck`: identical outputs up to running times. The third
  verification run re-ran `check_note.py` and all eleven documented commands of `original/` from an extracted
  copy of the source archive: identical outputs up to running times.
- Programs of the third verification run (`independent_run_2/`, numpy, scipy, sympy; about nine minutes): see
  below and `independent_run_2/README.md`.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first written version of the results; each
re-derived the proofs and wrote its own programs (in `reproducibility/independent_runs/`). Neither found an error
in a proof; both confirmed the scope (partial answer). A third independent run, also AI-assisted, examined the
final text (below); it found no error in a theorem, proposition or lemma and confirmed the scope.

- **First run.** Compared the statement with the corpus record and v48, and the definition of depth with Brandl
  (1983, p. 101) and Brandl–Nikolova (1986, p. 245); re-derived Lemmas 2.1–2.4, 3.1–3.4, Theorem 3.5, the facts
  (C1)–(E5) (not their numbering), Proposition 5.1 (including the Frobenius structure of C_(q±2^(m+1)+1):C_4),
  Lemmas 5.2, 5.3 and Corollary 5.4; checked against the sources BGG (3.3) and (5.8), BGKJ Theorem 3.4 and
  Section 4, and the proof structure of Brandl–Nikolova. Own programs: μ(PSL(2,q)) for q ≤ 32 and q = 37, 41;
  Lemma 3.3 for q = 8, 9, 16, 25, 27, 32, 49; the construction of Theorem 3.5 for (r,p) = (3,11), (4,1009),
  (5,1013), (6,2003), (4,10007), tails exactly r + 2; curve point counts; Sz(8) from scratch (order 29120,
  perfectness, 11 classes, μ = 62); all 34 nontrivial classes of Sz(32) with its own representatives (numpy,
  μ = 527, equal class by class); all printed pairs (q = 32 to 8192), with checks of form preservation and of the
  Suzuki–Tits ovoid (full for q ≤ 512, sampled for 2048, 8192). Required fixes: the table of Brandl–Nikolova
  has no column q = 5; BGKJ cites only an unrelated paper of Brandl (the third run found two); the unchecked numbering of the subgroup
  facts and the proviso l ≥ 5; the q = 128 row of Table 2 mixes two runs; (optional) the citation of Serre for
  binary quadratic forms.
- **Second run.** Programs written before reading the original ones: μ for all 25 values of Table 1, SL(2,q)
  (q ≤ 13), A_5, A_6, A_7 (μ(A_7) = 15, minimal law (15, 35295)), with both commutator conventions for small
  groups; minimal laws equal to the table of Brandl–Nikolova and to Nikolova's laws; Sz(8) and Sz(32) in its own
  construction (μ = 62, 527); Lemma 3.1 on 300 random pairs per field for 19 fields; Lemma 3.3 exhaustively for
  q ≤ 19; the construction of Theorem 3.5 in 26 cases; thresholds far below Q_r; curve counts; the six pairs of
  Table 2 with its own arithmetic and Bruhat-form membership certificates (Sz(512) also with full storage, period
  148701 = 291·511); the torus-orbit method against full storage on 68 pairs; eleven deliberately wrong inputs
  rejected; Proposition 5.1 for q = 8, 32; re-run of the eleven documented commands of the original programs
  (identical outputs). Literature: Crossref check of the bibliography, OpenAlex, Semantic Scholar, zbMATH, arXiv;
  it found Nikolova (1984) and Brandl, Engel cycles (1983). Required fixes: "s ≠ 0" in Remark 3.6(1) (with s = 0
  the least prime for r = 1 is 5; corrected by the third run: it is 3); the English title of Malle (1991); the pages 1–8 of Aubry–Perret; BGG
  Proposition 4.9 uses the fixed-field pre-period n_0(q); cite Nikolova (1984) and Brandl, Arch. Math. 41 (1983).
- **All required fixes were made** (see the Verification paragraph of the note). The note was written after the
  runs, with these changes: Lemma 3.3 now has a self-contained proof (it had rested on Macbeath's theorem and on
  a citation for binary quadratic forms); Lemma 3.4 is proved by Eisenstein's criterion (the runs checked the
  equivalent Newton-polygon argument); Lemma 3.2(c) adds the case of even q; the explicit bound
  μ(PSL(2,q)) ≥ ⌊(log₂ q)/4⌋ (q ≥ 256) was added to Theorem 3.5; and Section 5.2 gives the eigenvalue argument
  for the sufficient condition [x, _j y](∞) = 0. All but the last are checked by `check_note.py`; all five were
  examined by the third run.
- **Third run (2026-10-22, on the final text; programs in `reproducibility/independent_run_2/`).** It examined
  the five changes first and used none of the earlier programs. It re-derived every proof line by line; read
  Brandl (1983) and Brandl–Nikolova (1986) again (the table p. 246, the Theorem for q ≥ 4, the conjecture, the
  remark on Sz(8), the proof with Schmidt's bound for q > 169, the direct checks for 7 < p < 168 and the separate
  arguments for the characteristics 2, 3, 5), BGG (3.1)–(3.3), Proposition 4.9 and (5.8), BGKJ Theorem 3.4,
  Section 4 and its reference list, BGGKPP Sections 2.4 and 4.1; checked all DOIs with Crossref; repeated the
  searches (zbMATH, OpenAlex, arXiv API, one web search). Own programs: the construction of Lemma 3.3 as written
  for all 28425 pairs (s,t) over the 27 fields with q ≤ 64 and 1200 random pairs in large fields; the Eisenstein
  conditions and degrees of Lemma 3.4 (sympy, r ≤ 6); Lemma 3.2 (a)–(c) on all pairs of SL(2,q), q ≤ 13 (about
  6.8 million pairs with q even or tr y ≠ 0, no violation; for tr y = 0 the conclusion of (c) fails, so the
  hypothesis is needed); Table 1 with the minimal laws, μ(PSL(2,2)) = μ(PSL(2,3)) = 2, μ(A_7) = 15 with s = 35295;
  the construction of Theorem 3.5 in 24 cases with q > Q (r ≤ 5, q up to 2^20) and, for the bound
  ⌊(log₂ q)/4⌋, in 11 fields with 256 ≤ q ≤ 2^20 + 7, exact tails r + 2 by iteration; Remark 3.6(1) (least primes;
  all primes 1000 < p < 3000; the 3306 odd prime powers q ≤ 30000 for r = 3); Sz(8) in full (order 29120, 11
  classes, μ = 62 with the class maxima 2, 11, 8, 62, 56, 27, period 90); Proposition 5.1 for q = 8 without
  Suzuki's list (⟨x,y⟩ computed for every double coset ⟨y⟩x⟨y⟩; all 202617 pairs with tail ≥ 4 generate Sz(8));
  the eigenvalue argument of Section 5.2 on all pairs with y = M(c) in Sz(8) (61824 pairs with some a_j
  exchanging the two fixed points, all with tail j + 2) and on random pairs in Sz(32), Sz(128); the field
  representation of the printed pairs determined from the data; own Bruhat certificates and exact tails for all
  pairs of Table 2 (3759 and 3554 with period 52324 in G; 24814 with period 148701 in G and 291 on H-orbits;
  183810 and 792500 on H-orbits) and for the 10 + 34 recorded class witnesses of Sz(8), Sz(32). It found no error
  in a theorem, proposition or lemma. Required fixes (all made): Remark 3.6(1), with s = 0 the least prime for
  r = 1 is 3 (point (h,s) = (1,0)), not 5; "for q ≥ 4" in the abstract; the priority sentence excludes r ≤ 2
  (Brandl 1983); Table 2 ratios ≥ 2.59 and ≥ 1.06 (they had been rounded upwards); credit to Gupta for V_1;
  BGKJ cites two unrelated papers of Brandl; the theorem number of Suzuki (1962) marked unchecked; two
  justifications in Section 5.2 (the eigenvalue c^(±2(1+2^m)) ≠ 1; φ_y bijective on U_∞); the description of
  the proof of Brandl–Nikolova in Remark 3.6(2); the field representation in Table 2 and in the README; this
  record. Not checked by this run: the exactness of the Sz(32) class maxima (only the witnesses), the sources
  listed as not read below.

## Relation to the literature, novelty and scope
- **What was read.** Brandl (1983) and Brandl–Nikolova (1986) in full (open publisher files); Problem 11.16 in
  versions 48 and 49 of the Notebook; arXiv versions: BGG (1008.1397v2), Sections 3–5 in part with (3.3),
  Proposition 4.9 and (5.8); BGKJ (0809.0369v2), Theorem 3.4 with proof and the beginning of Section 4; BGGKPP
  (math/0303165v1, 2003, "Engel-like identities characterizing finite solvable groups"), Sections 1, 2 and the
  beginning of 4.1. Nikolova (1984): zbMATH review only (Zbl 0557.20022). The surveys
  Grunewald–Kunyavskii–Nikolova–Plotkin (2003) and Grunewald–Kunyavskii–Plotkin (2013): abstracts only (read by
  the verification runs). Brandl, Engel cycles (1983), Khukhro–Shumyatsky (2016), Heineken–Neumann (1967):
  bibliographic data only (the last as quoted by Brandl–Nikolova). Macbeath (1969), Aubry–Perret (1996),
  Ghorpade–Lachaud (2002), Leep–Yeomans (1994): not read; what is used is stated in BGKJ and BGG. The books
  (Carter, Kleidman–Liebeck, Wilson) and the papers on maximal subgroups (Kleidman 1988 ×2, Malle 1991, Ward 1966,
  Suzuki 1962) were not opened; their numbering was not checked (one web search did not confirm the number
  "Theorem 9" of Suzuki 1962). Gupta (1966): bibliographic data only, as quoted by Brandl (1983). All DOIs were
  checked with Crossref.
- **Searches** (21–22 October 2026, by the three verification runs and at the writing): zbMATH Open ("Engel depth":
  only Brandl 1983 and Brandl–Nikolova 1986), OpenAlex (7 works citing Brandl 1983, 6 citing Brandl–Nikolova),
  Semantic Scholar, the arXiv API ("Engel depth": no result; Engel and Suzuki; Engel words; Engel sequences;
  Engel and Kourovka; recent math.GR papers on Engel conditions in simple groups), Crossref, and a few web
  searches; mathnet.ru refused automated requests. Closest work: Brandl–Nikolova (r = 3, PSL(2,q)); the trace-map
  papers (BGG Proposition 4.9 uses the pre-period n_0(q) over a fixed field, not its growth in q); the Engel-sink
  papers of Khukhro and Shumyatsky (the periodic part, not the pre-period).
- **Novelty.** No printed answer to the question was found in the sources searched; a search that finds nothing
  is not a proof of novelty, and no priority is claimed. For r ≤ 2, Theorems 3.5 and 4.1 follow from Brandl
  (1983), and for r = 3 the conclusion of Theorem 3.5 is contained in Brandl–Nikolova (1986); Theorem 3.5 for
  r ≥ 4, Theorem 4.1 for r ≥ 3, Corollary 1.2, Proposition 5.1 and Table 2 were not found in print. Theorem 3.5
  is a direct extension of Brandl–Nikolova's method and may be regarded as routine by experts.
- **Scope.** Part b) is answered for all families except the Suzuki groups and reduced to μ(Sz(q)) → ∞; part a)
  is not treated.
- **Suggested label for the corpus.** open, unchanged: partial answer to part b): for every r, all but finitely
  many nonabelian finite simple groups in V_r are Suzuki groups, so b) holds if and only if the Engel depth of
  Sz(q) tends to infinity; this last statement and part a) stay open. Evidence: unrefereed_claim; at most
  "partially solved" at the maintainers' discretion; never "solved".

## 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/