# Verification report — KOU-15.75 (Kourovka Notebook Problem 15.75 b), B. I. Plotkin: two-variable identities for finite soluble groups)

Verification date: 2026-10-22.

**Verdict.** The note gives a **partial answer: the question reduces to the minimal simple groups; PSL(2,p) is
settled for all primes p ≡ ±1 (mod 16) and, in the Notebook's commutator convention, for all primes
p ≡ 0, ±1 (mod 7), and every minimal simple group of order less than |PSL(2,10007)| has a pair whose sequence
never reaches 1; the question stays open**. Every statement of the note is proved; the finite parts are
certificates re-checked by several independent programs. In detail:
- **Settled.** The reduction to Thompson's list (Theorem 2.4); the families L1 (p ≡ ±1 mod 16, both conventions)
  and L2 (p = 7 or p ≡ ±1 mod 7, Notebook convention), Theorems 3.4 and 3.5; certificates for PSL(3,3), the 626
  groups PSL(2,q) of Thompson's list with q ≤ 10^4, Sz(8), Sz(32), Sz(128), both conventions (Theorem 4.1);
  a finite non-soluble group satisfying some u_n = 1 has order at least |PSL(2,10007)| = 501050730168
  (Corollary 4.3).
- **Not new mathematics.** The reduction is the argument of Bandman, Greuel, Grunewald, Kunyavskii, Pfister and
  Plotkin for another sequence (Compositio 2006, Proposition 2.2), with Thompson's list as quoted there; the
  equations u_i = u_j, the trace coordinates and the Suzuki matrices come from the same paper (and from Bandman,
  Grunewald, Kunyavskii and Jones 2010). For this very sequence, in the Notebook's convention, Nikolova and
  Plotkin (2001) had stated the question as Plotkin's conjecture, outlined the reduction through relations
  u_k = u_n, and reported solutions found by computer (GAP) for PSL(3,3) and PSL(2,p), all primes 5 ≤ p < 100,
  without listing them; the corresponding certificates of Theorem 4.1 are therefore not new in convention K.
  The value is modest: computational progress and two infinite families.
- **Open.** Whether the following groups have a pair whose sequence never reaches 1: PSL(2,p) for primes
  p > 10^4, p ≡ ±2 (mod 5), p ≢ ±1 (mod 16), p ≢ ±1 (mod 7) (in the other convention without the condition mod 7);
  PSL(2,2^r) for primes r ≥ 17; PSL(2,3^r) for primes r ≥ 11; Sz(2^r) for primes r ≥ 11. Numerical evidence
  only: the relations u_i = ±u_j (j ≤ 6) appear to have finitely many solutions in the trace coordinates; if
  so, fixed relations cannot treat the characteristic-2, characteristic-3 and Suzuki families, and the
  Lang-Weil method of Bandman et al. does not apply directly.

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 copy) and version 49 (20 October 2026, the current version; arXiv API:
    `1401.0300v49`, updated 2026-10-20T09:15:18Z; PDF fetched and the pages with 15.75 read as text).
  - Problem 15.75 b) (B. I. Plotkin, 15th issue, 2002): the sequence u_1 = [x,y], ..., u_{n+1} = [[u_n,x],[u_n,y]];
    is an arbitrary finite group soluble if and only if it satisfies one of the identities u_n = 1? The text is
    the same in v48 and v49, in the main part, with no star, no sharp sign and no comment.
  - Part a) (some sequence of two-variable identities, each implying the next, characterizing the finite soluble
    groups) is in the Archive of solved problems, answered with other sequences by Bandman-Greuel-Grunewald-
    Kunyavskii-Pfister-Plotkin (Compositio 2006), Bray-Wilson-Wilson (BLMS 2005) and Ribnere (Monatsh. 2009).
  - Commutator: the Notebook writes [x,y] = x^-1 y^-1 x y (written out in Problem 8.11). Problem 16.77 (Plotkin)
    refers to sequences "satisfying 15.75"; it asks a different question (u-Engel elements and the soluble
    radical).
- **Corpus record.** ulamai/UnsolvedMath, KOU-15.75 (dataset version 1.8.0, snapshot d371ee9): the record
  contains part b) with the archived part a); its label is `partially_solved` (status_review: part a) solved in
  the Archive, the remaining part in the main section). Part b) itself is open. The record reproduces the v48
  text faithfully.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Convention K, [g,h] = g^-1 h^-1 g h (the Notebook's) | partial answer as in the verdict | whole note |
| Convention O, [g,h]' = g h g^-1 h^-1 (used by Bandman et al. and in their survey) | the same, without the family L2: PSL(2,p) settled for p ≡ ±1 (mod 16) and by the certificates | Theorem 3.4 (O), Theorem 4.1, Corollary 4.3 |
| The two conventions as one question | no: they give genuinely different sequences (A_5: 480 versus 720 pairs never reaching 1, so no automorphism of F_2 relates them) | Remark 2.6 |
| "satisfies one of these identities" = one n for all pairs, or for each pair some n | equivalent (monotonicity) | Lemma 2.1 |
| Index shift of the sequence | irrelevant | — |
| Infinite groups | not asked (finite groups only) | — |

## Results in the paper
- **Lemmas 2.1-2.3.** Monotonicity (F(1) = 1); u_n ∈ G^(n), so soluble groups satisfy some u_n = 1;
  certificates: u_M = u_{M+L} ≠ 1 implies that the pair never reaches 1; in PSL(2,q) the same with u_M ≠ ±1 and
  u_{M+L} = ±u_M in SL(2,q).
- **Theorem 2.4 (reduction).** Every finite non-soluble group has a minimal simple section; identities pass to
  sections; the answer is yes iff every group in Thompson's list (PSL(2,2^r), PSL(2,3^r), Sz(2^r), PSL(3,3),
  PSL(2,p) with p > 3, p ≡ ±2 mod 5) is settled. Remark 2.5: a pair never reaching 1 in a minimal simple group
  generates it; so the family pairs generate PSL(2,p) for p ∈ P.
- **Lemma 3.1 (trace algebra).** Products in the basis 1, x, y, xy over any commutative ring; u_n(x,y) =
  φ(U_n(tr x, tr y, tr xy)) with universal U_n ∈ Z[a,b,c]^4.
- **Lemma 3.2.** 2t^4 - 4t^2 + 1 has a root mod an odd prime p iff p ≡ ±1 (mod 16); t^3 + t^2 - 2t - 1 (and
  t^3 - t^2 - 2t + 1) has a root mod p iff p = 7 or p ≡ ±1 (mod 7).
- **Lemma 3.3.** For p ≥ 5, x = [[0,-1],[1,0]] or [[0,-1],[1,1]] and t ∈ F_p, some y ∈ SL(2,p) has tr y = tr xy = t
  (counting proof; a special case of Macbeath's theorem).
- **Theorem 3.4 (L1).** tr x = 0, tr y = tr xy = t a root of 2t^4 - 4t^2 + 1: u_3 = -u_2, tr u_2 = 0 (K);
  u'_4 = u'_3, tr u'_3 = 0 (O). **Theorem 3.5 (L2).** tr x = 1, tr y = tr xy = t a root of t^3 ± t^2 - 2t ∓ 1:
  u_4 = u_2, tr u_2 = 1 (K). The identities (F1), (F2) are divisibilities in Z[t], proved by computing in
  Q[t]/(m) and Gauss's lemma; the non-degeneracy follows from the traces. Remark 3.6: the resultant argument
  (gcds 2^10, 2^112, 1). Corollary 3.7: relative density 1/2 (K), 1/4 (O) of P by Dirichlet's theorem.
- **Theorem 4.1, Lemma 4.2, Corollary 4.3.** Certificates as listed; the Suzuki model of Bandman et al.
  (Section 4.1 there) has order q^2(q^2+1)(q-1) for q = 8, 32, 128 (product formula for U(a,b), action of M(c),
  T M(c) T = M(c^-1), and the finite condition T U(a,b) T ∈ BTU, which gives G = B ∪ BTU); the order bound
  501050730168 and the list of undecided groups.
- **Section 5.** The open list; numerical evidence on fixed relations (labelled as such); a remark on finitely
  many congruence families; a heuristic (labelled as such).

## Computations (sanity checks; programs and outputs in reproducibility/)
The reduction, Lemmas 2.1-2.3 and 3.1-3.3 and the logical part of all proofs need no computation. Computation
enters in the identities (F1), (F2), in condition (5) of Lemma 4.2 and in the certificates.
- **Program written with the note** (`check_note.py`, standard library, about 100 s; TOTAL failures: 0). (A) (F1),
  (F2) and the traces in the trace algebra over Q[t]/(g), Q[t]/(h), Q[t]/(h(-t)), with controls, and the
  multiplication table against 720 matrix products; (B) Lemma 3.2 for the 302 odd primes ≤ 2000 and the families
  with explicit matrices (288 + 288 + 584 instances); (C) the constants: 626 = 6 + 3 + 617 groups, exactly the q
  of the certificate files; 10007; the orders; all 629 minimal simple groups of order < |PSL(2,10007)| covered;
  densities 48/96 and 24/96; (D) A_5: 480 and 720; controls 0; (E) PSL(2,q) certificates for q ≤ 49 and 40 other
  groups (five extension fields and 35 random primes), both conventions, own field arithmetic, corrupted periods
  rejected; (F) the six PSL(3,3) certificates; (G) the Suzuki formulas, condition (5) for q = 8, 32, 128, the
  closure of order 29120, and all 425 Sz(8) certificates.
- **Programs with which the results were first obtained** (`original/`, Python with numpy and sympy): the exact
  computation over Z[t] (`line_family.py`; gcds, factors, exact division, resultants), the matrix check of the
  families for p ≤ 3000 (`family_check.py`), the certificate searches (`psl2_certify.py`, `psl33_cert.py`,
  `sz_certify.py`) with two independent verifiers (`verify_psl2_certs.py`: 626 + 626 OK; `verify_sz_certs.py`:
  425 + 188 + 15 OK), the exhaustive counts (`exhaustive_small.py`), the trace-triple counts of Section 5
  (`trace_sweep.py`, evidence only). At the writing, `line_family.py` (three lines), `family_check.py 3000`,
  `test_trace_algebra.py 60` and `sz_check.py` were re-run with identical outputs, and the random seeds of the
  certificate files were identified (PSL(2,q): 1 and 2; PSL(3,3): 1; Suzuki: 1, 4, 5, 6, 7, 8).

| Item | Value |
|---|---|
| PSL(2,q) certificates | 626 groups × 2 conventions; first random pair succeeded for 580 (K) / 574 (O) groups; at most 8 pairs; largest M + L = 3,765,841 (q = 9677, K), 3,469,335 (q = 9697, O) |
| PSL(3,3) | exhaustive: 14,657,760 (K) and 14,062,464 (O) of 31,539,456 pairs never reach 1; 3 + 3 explicit certificates |
| Sz(8), Sz(32), Sz(128) | 300, 100, 8 random pairs per convention; certified 218 / 207, 97 / 91, 7 / 8; all others reached 1 |
| Families, explicit matrices | p ≤ 3000 (original), p ≤ 2000 (check_note), p ≤ 1500 (run A), p ≤ 20000 for the root criteria (run B), p ≤ 3000 and 48 primes up to about 10^18, root criteria for p < 2·10^5 (run 3) |

## Independent verification runs
The results were first obtained, with proofs and with the programs in `original/`. Two independent verification
runs, both AI-assisted, followed (2026-10-22); each re-derived the proofs and wrote its own programs; the second
run wrote its programs before reading the original ones.

| Item | Run A | Run B |
|---|---|---|
| Statement, conventions, record, v48 and v49 | CONFIRMED (both versions read) | CONFIRMED (v49 read) |
| Lemmas 2.1-2.3, Theorem 2.4, Remark 2.6 | PROVED (re-derived) | PROVED (re-derived) |
| Thompson's list | CHECKED against Bandman et al., Theorem 2.1 (Thompson 1968 not read) | the same |
| Lemma 3.1 (trace algebra) | PROVED; own implementation tested against matrices | not used (another method) |
| Lemmas 3.2, 3.3 | PROVED; criteria checked for p ≤ 1500 | PROVED; criteria checked for p ≤ 20000 |
| (F1), (F2) | reproduced exactly (own sympy code: gcds, divisibility, resultants 2^4, 2^10, 2^112, 1) | proved by another method (generic pair over Q[s][t,α], degree bound in s), with controls |
| Non-degeneracy | resultant argument re-derived | trace argument found (tr u_2 = 0, tr u'_3 = 0, tr u_2 = 1), used in the note |
| Families with matrices | 232 + 232 + 458 instances, p ≤ 1500 | split primes, random conjugates, extension fields F_p[t]/(f) |
| PSL(2,q) certificates | all 1252 OK (C program, own field arithmetic) | all 1252 OK (own code); own certificates for 57 groups |
| PSL(3,3) | 6 OK; exhaustive counts reproduced | own certificates; exhaustive counts reproduced; corrupted example rejected |
| Suzuki | own generators from the Suzuki-Tits ovoid, identical with Bandman et al. §4.1; all 628 certificates OK; condition (5) for q = 8, 32, 128 | own construction with certified orders (Schreier generators: 12,300 and 262,160 for q = 32, 128); generators of Bandman et al. generate the same group; all 628 certificates OK; own certificates |
| Table 1 | all rows except PSL(2,8), PSL(2,9) | all rows, independent models |
| Corollaries 3.7, 4.3 | PROVED (bookkeeping recomputed) | PROVED (bookkeeping recomputed) |
| Literature | not stale; no answer found | not stale; 68 citing works screened; Popović-Wilson 2022 added |
| Overall | CONFIRMED as partial, minor fixes | CONFIRMED as partial, minor fixes |

Corrections required by the runs, all made in the note: (A1) the Suzuki generators are verbatim those of Bandman
et al., Section 4.1, who define Sz(q) = ⟨U, M, T⟩, and the order argument for q = 32, 128 is given (Lemma 4.2);
(A2) "by Section 5.2" in the proof of the family theorem, and the basis property is used only for u_2 ≠ ±1 (now
Remark 3.6, since the main proof uses traces); (A3) the hedge "numerical evidence" on every statement that fixed
relations cannot treat the characteristic-2, characteristic-3 and Suzuki families; (A4) generation of PSL(2,p)
by the family pairs (Remark 2.5, automatic for p ∈ P); (A5) p = 7 among the covered primes in Corollary 4.3;
(A6) v49 cited as the current version; (B1) the O-relation of L1 is u'_4 = u'_3, never "u_3 = -u_2 in both
conventions"; (B2) the GKNP survey cited as J. Math. Sci. 116 (2003) 2972-2981 (translation of Zap. Nauchn.
Sem. POMI 272 (2000), as recorded in zbMATH); (B3) the trace-based non-degeneracy proof (now the main proof);
(B4) Popović-Wilson, J. Algebra 607 (2022), as related work on a different map.

Changes made at the writing (after the runs): the non-degeneracy rests on traces (resultants kept as Remark
3.6); Lemma 3.3 has an elementary counting proof (the runs checked a character-sum count); the product formula
of U(a,b) and the action of M(c) are proved, not only tested; the status wording says "of order less than
|PSL(2,10007)|" (PSL(2,10007) itself is not settled). These changes are checked by `check_note.py`.

### Third verification run: the final text (2026-10-22)
A third verification run, AI-assisted, examined the final text and the package; its programs, written before
those of the package were read, and their outputs are in `reproducibility/independent_run_2/` (with a README).

| Item | Run 3 |
|---|---|
| Statement, record, sources | CONFIRMED: v49 still current (arXiv API, 2026-10-22), 15.75 b) unchanged in v48 and v49; Problem 8.11; record text and label; every cited passage of Bandman et al. (2006), the 2014 survey and BGKJ (2010) compared again |
| Proofs | all re-derived line by line, including the counting proof of Lemma 3.3 and the two formulas in the proof of Lemma 4.2; no error |
| (F1), (F2) | proved again: full polynomials in Z[t] (own trace algebra, tested against 560 matrix products), exact division by g, h, h^- with integral quotients; degrees 69, 914, 315 as stated; resultants of Remark 3.6 recomputed (16, 1, 2^10, 2^112, 1) |
| Lemmas 3.2, 3.3 and the families | Lemma 3.2 for all 17983 odd primes < 2*10^5; Lemma 3.3 for all t, 5 <= p <= 700; explicit (also randomly conjugated) matrices for all primes p <= 3000 and 48 primes near 10^6, 10^9, 10^12, 10^18 (1072 + 2032 instances) |
| Certificates | all 1252 PSL(2,q) certificates (own field arithmetic, Rabin irreducibility test), the six PSL(3,3) certificates and all 628 Suzuki certificates OK; own certificates for 30 groups PSL(2,q), PSL(3,3), Sz(8), Sz(32), Sz(128), both conventions |
| Suzuki model | formulas of Lemma 4.2, T^2 = 1, TMT = M(1/c), form J = T preserved, condition (5) for q = 8, 32, 128; Sz(8): 29120 elements with the element orders of Sz(8) |
| Table 1 | all counts reproduced with own models (A_5 and A_6 also as permutation groups); fraction for PSL(2,8), K, is 0.315 |
| Package | `check_note.py` (Python 3.9.6) and the quick programs of `original/` re-run from `source.zip`: outputs identical to the recorded ones |
| Literature | found Nikolova-Plotkin 2001 (prior computer results for this sequence, not cited before), Kunyavskii-Plotkin-Shklyar 2003 (review only) and the author's version of the Grunewald-Kunyavskii-Plotkin survey (read; no answer); no printed answer to 15.75 b) |
| Overall | CONFIRMED as partial; no mathematical error; credit and minor fixes required |

Corrections required by the third run, all made: (C1) credit to Nikolova-Plotkin 2001 in the abstract, Sections
1.1, 1.3, 4.1 and "Scope and priority"; (C2) mention of Kunyavskii-Plotkin-Shklyar 2003; (C3) the reading of the
survey of Grunewald-Kunyavskii-Plotkin and the searches of this run in "Scope and priority"; (C4) the
undecided groups in the abstract restricted to Thompson's list (p ≡ ±2 mod 5, r prime); (C5) Table 1:
0.315 for PSL(2,8), convention K; (C6) Section 5: the primes 5 ≤ q ≤ 13 for the relation u_1 = ±u_2 were
counted separately (solutions only for q = 7); (C7) Corollary 3.7: "covered by Theorems 3.4 and 3.5"
instead of "settled"; (C8) Lemma 4.2: θ is a field automorphism (multiplicativity is used); (C9) after
Theorem 3.5: the image of u'_3 in convention O, and "modulo h and h^-"; (C10) the abstract names the
convention of the family L2; (C11) this run recorded in the note and the package.

## Relation to the literature, novelty and scope
- **Searches (21-22 October 2026, at the first derivation and by the two runs).** arXiv API (Plotkin and solvable
  groups, Engel-like identities, two-variable identities, sequences of words, characterizations of finite solvable
  groups, minimal simple groups and identities, iterated identities, Kourovka 15.75), zbMATH Open, OpenAlex
  (68 distinct works citing Bandman et al. 2006, the Bandman-Garion-Kunyavskii survey 2014, Bandman-Grunewald-
  Kunyavskii-Jones 2010, Ribnere 2009, Grunewald-Kunyavskii-Plotkin 2013 or Grunewald-Kunyavskii-Nikolova-Plotkin
  2003; titles screened, plausible abstracts read), Semantic Scholar, Crossref and four web searches. The third
  verification run (22 October 2026) repeated the search: arXiv API (eight queries; two refused by the server
  after one retry), zbMATH Open (five queries), OpenAlex (works citing Bandman et al. 2006, the 2014 survey
  and BGKJ 2010: 46 distinct), Crossref (all DOIs, two searches) and one web search, which found the two
  open copies below.
  - No printed answer to Problem 15.75 b) was found. The survey of Bandman, Garion and Kunyavskii (Open Math. 12
    (2014), Remark 3.5 and Section 7) states that the authors do not know the answer and that the question
    resists their method for computational reasons; it is a case of Question 6.12 of Bandman, Grunewald,
    Kunyavskii and Jones (2010).
  - Prior partial results on this sequence: D. Nikolova, E. Plotkin, "Matrix equations in linear groups",
    Mathematics and Education in Mathematics 2001 (Proc. 30th Spring Conf. Union of Bulgarian Mathematicians,
    Borovets), 194-199 (open copy on math.bas.bg, read in full): [x,y] = x^-1 y^-1 x y, the question stated as
    Conjecture 3.1 (B. Plotkin), the reduction through relations u_k = u_n, and GAP solutions for PSL(3,3) and
    PSL(2,p), 5 ≤ p < 100 (with a table of minimal relations for p < 50, e.g. u_1 = u_2 at p = 7, 37, 43). Not
    cited by the other sources read. Kunyavskii, Plotkin, Shklyar, Proc. Latvian Acad. Sci. B 57 (2003)
    97-101 (Zbl 1052.20013): a strategy for such sequences; only the review was seen.
  - Related but different: Popović and Wilson (J. Algebra 2022; orbits of another map and the soluble radical);
    Erschler (2015) and Belov-Erschler (2019) on iterated identities in another sense.
- **What was read.** The Notebook (v48, v49); Bandman et al. 2006 in the arXiv version math/0303165v1 (Sections 1,
  2 and the beginning of 4.1); the 2014 survey (arXiv v1: Section 3 to Remark 3.5, Section 7); Bandman-Grunewald-
  Kunyavskii-Jones (arXiv v2: Theorem 3.4, Section 6.2); Nikolova-Plotkin 2001 (in full); the survey of
  Grunewald-Kunyavskii-Plotkin in the author's version (Sections 1-3, 6 and 7: the Lie analogue, Theorem 6.18,
  and Question 7.1 = Question 6.12 of BGKJ; no answer to 15.75 b); not compared with the journal version). Of
  the survey of Grunewald-Kunyavskii-Nikolova-Plotkin only the bibliographic data (and its zbMATH entry), of
  Kunyavskii-Plotkin-Shklyar 2003 the zbMATH review were read.
  Thompson 1968, Macbeath 1969 and Suzuki 1962 were not opened. All DOIs were checked with Crossref on
  2026-10-22.
- **Caveats.** The numerical evidence on fixed relations is not a proof. A search that finds nothing is not a
  proof of novelty; no priority is claimed.
- **Scope.** A partial answer to part b); part a) is not treated (archived). The undecided groups are listed in
  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/