# Verification report — KOU-17.91 (Kourovka Notebook Problem 17.91, V. S. Monakhov: derived length of a finite soluble group and of its maximal subgroups)

Verification date: 2026-10-27.

**Verdict.** The note gives a **partial result**; parts a) and b) of the problem **remain open**. Every statement
it makes is proved, one of them with a computer-assisted proof, and the scope is as follows.
- **Proved.** If an absolute constant k with d(G) − d(M) ≤ k exists, then k ≥ 3 (Theorem 1.2): G = 3^{1+2}_+ ⋊ GL(2,3)
  = SmallGroup(1296, 2891) has d(G) = 6, and its maximal subgroup M = 3^{1+2}_+ ⋊ B of index 4 (B the Borel subgroup)
  has d(M) = 3. General restrictions (Theorem 1.4): d(G) − d(M) = d(G^(d(M))); G^(e+1) ≤ M_G with e = d(M/M_G);
  d(G) − d(M) ≤ 1 + min(e, d(M_G)); a difference 3 needs |G| ≥ 144, a difference 4 needs e ≥ 3, d(M_G) ≥ 3, d(G) ≥ 7
  and |G| ≥ 5184. The split-layer formula (Lemma 5.1), the construction behind the examples (Lemma 5.3), the flag
  lemma (Lemma 5.5) and the proposition that, in the construction of Lemma 5.3, a second abelian layer on which the
  first (irreducible) layer has no non-zero fixed points cannot raise the difference again; the two layers then have
  different characteristics (Proposition 5.6).
- **Computer-assisted.** 1296 is the least order of a finite soluble group with a maximal subgroup M such that
  d(G) − d(M) ≥ 3; exactly SmallGroup(1296, 2889), (1296, 2890) (non-split extensions 3^{1+2}_+ . GL(2,3)) and
  (1296, 2891) (split) have one (Theorem 1.3). A second pair with difference 3: G2 = W ⋊ (3^{1+2}_+ ⋊ SL(2,3)) of
  order 41472, W of order 2^6, which is PrimitiveGroup(64, 37), with a maximal subgroup of index 9 and derived length 3
  (Proposition 5.4; the module and two dimensions are computed, the rest is proved).
- **Computation only.** No difference 4 in the families of Remark 5.7. This is not a result about k.
- **Open.** Whether k exists (part a)), and its least value (part b)); in particular whether a difference 4 occurs.
- **Value.** Modest: a lower bound for k if k exists, the least order, and the restrictions.
- **Credit.** The group G = 3^{1+2}_+ ⋊ GL(2,3) is not new: it is a term, (S3 ⋉ Q8) ⋉ 3^{1+2}_+, of the extraspecial
  tower of Glasby and Howlett (J. Algebra 151 (1992) 236–260), who do not consider derived lengths of maximal
  subgroups; what is claimed is the difference 3, the least order and the restrictions.

The note is unrefereed. Three independent AI-assisted verification runs examined it (the third the final text).

## Statement checked
- **Primary source.** The Kourovka Notebook (eds. E. I. Khukhro, V. D. Mazurov), arXiv:1401.0300, version 49 of
  20 October 2026 (the latest version on 27 October 2026, checked with the arXiv API), Problem 17.91, proposed by
  V. S. Monakhov in the 17th issue (2010): with d(X) the derived length, a) is there an absolute constant k such that
  d(G) − d(M) ≤ k for every finite soluble group G and every maximal subgroup M; b) find the least such k. No editors'
  comment and no mark of a solution.
- **Corpus record.** ulamai/UnsolvedMath, KOU-17.91 (status `open`; proposer V. S. Monakhov, 2010). Its statement
  agrees with the Notebook text (all three verification runs compared them); its background cites version 48.
- **Earlier statement of the same question.** V. S. Monakhov, A. A. Trofimuk, "Invariants of finite solvable groups",
  Problemy Fiziki, Matematiki i Tekhniki 2010, no. 1(2), 63–81, Zbl 1246.20018 (in Russian). Read for the note in the
  file of the whole issue on the journal's website (https://pfmt.gsu.by/files/pfmt_2010_1_2.pdf, sha256
  `50d704e8d9284d837958bed841eb75de3b4007230e4f4bf9c0e42cc79e2638a0`): the title page, Section 4 (Theorems 4.6–4.9,
  Example 15, Question 9, pp. 77–78) and all other passages on derived length; the second verification run read the
  survey in full in the copy of Math-Net.Ru. Question 9 asks for the least natural number k, if it exists, with
  d(G) − d(M) ≤ k for every finite soluble G and every maximal subgroup M. It is posed after Doerk's theorem on the
  nilpotent length (Theorem 4.6 there, with Example 15: S4 and a Sylow 2-subgroup give the value 2) and its analogues
  for the F-length and the π-lengths (Theorems 4.7–4.9; Theorem 4.7, the F-length analogue, is incorrect in general
  by Murashka–Vasil'ev, Theorem 4). The survey gives no result, no bound and no example on d(G) − d(M). (The survey's second author is A. A. Trofimuk, as on its title page and in zbMATH.)
- **Conventions.** d(1) = 0; the difference is over all pairs (G finite soluble, M maximal); no "generic" weakening.
  The convention d(1) = 1 would change nothing for M ≠ 1.

## Readings
| Reading | Answer | Where |
|---|---|---|
| a) Is d(G) − d(M) bounded by an absolute constant? | open; if bounded, the bound is at least 3 | Theorem 1.2 |
| b) The least k | open; k ≥ 3 if k exists | Theorem 1.2 |
| Least order of a soluble group with a maximal subgroup of difference ≥ 3 | 1296, exactly three groups (computer-assisted) | Theorem 1.3, Table 1 |
| Restrictions on a difference ≥ 4 | e ≥ 3, d(M_G) ≥ 3, d(G) ≥ 7, \|G\| ≥ 5184 | Theorem 1.4(d) |
| The Fitting-height analogue (Doerk) | known: n(G) − n(M) ∈ {0, 1, 2}; not the question | Section 1.1 |
| Normal maximal subgroups; nilpotent G; index a prime | difference ≤ 1, ≤ 1, ≤ 2 | Corollary 2.4 |

## Results in the paper
- **Lemmas 2.1–2.3.** The product rule (AB)' = A'[A, B]B'; the standard facts on primitive soluble groups
  (V the unique minimal normal subgroup, P = V ⋊ H, C_P(V) = V, H faithful and irreducible on V, O_p(H) = 1),
  stated and proved so that no numbered statement of Doerk–Hawkes or Huppert is needed; P^(i) = V H^(i) for i ≤ e.
- **Theorem 1.4 and Corollary 2.4.** (a)–(d) as in the verdict; (i) normal maximal subgroups and nilpotent groups give
  difference ≤ 1, (ii) prime index gives ≤ 2, (iii) the necessary conditions for 3 and 4, (iv) a Miller–Moreno maximal
  subgroup gives difference ≤ 2 and d(G) ≤ 4, the derived-length part of Gol'chuk–Monakhov (2024), Theorem 1(a).
- **Lemmas 2.5, 2.6.** A soluble group of derived length ≥ 3 has order ≥ 24; a primitive soluble group with point
  stabiliser of derived length ≥ 2 (≥ 3) has order ≥ 24 (≥ 216), with elementary proofs (the case |V| = 8 via the
  normaliser of a subgroup of order 7 in GL(3,2)).
- **Theorem 1.2 (Section 3).** The explicit model E = F_3^2 × F_3 with the cocycle ω/2; G' = E SL(2,3), G'' = E Q8,
  G''' = E⟨−I⟩, G^(4) = E, G^(5) = Z(E), G^(6) = 1; M' = EU, M'' = the preimage of a line (abelian); maximality by
  the index-2 argument; core E⟨−I⟩ (order 54, derived length 3), G/core ≅ S4, e = 2. The bound of Theorem 1.4(c) is
  attained.
- **Theorem 1.3 (Section 4).** Reduction to non-prime-power orders 144 ≤ n ≤ 1296 and to soluble groups with
  d(G) ≥ 5; Table 1 lists the 15 soluble groups of non-prime-power order 144 ≤ n ≤ 1296 with d(G) ≥ 5 (below 1296:
  432#734 = AGL(2,3), 648#531–533, 864#4053, 864#4661 = C2 × AGL(2,3), all with difference ≤ 2), with their Fitting
  subgroups, splitting, and all classes of maximal subgroups; the third run found no further group of non-prime-power
  order ≤ 1296 with d(G) ≥ 5.
- **Section 5.** Lemma 5.1 (split layer: gap(F ⋊ X, F ⋊ Y) = gap(X, Y) + d(F^X_D) − d(F^Y_s)); Remark 5.2 (no
  extension of the pair of Theorem 1.2 by a normal subgroup inside the core, split or not, exceeds 3); Lemma 5.3 (the
  construction X = E ⋊ S, Y = Z(E) × S with E extraspecial, gap 2, and gap 2 + d(F) − d(F_t) after a layer F; the
  hypothesis "S^(t−1) fixed-point-free" of the first write-up was removed, as it is not used); the pair of Theorem 1.2
  as an instance (a term of the extraspecial tower of Glasby–Howlett); Proposition 5.4 (the pair of order 41472);
  Lemma 5.5 (flag lemma; the standard argument for the unitriangular group of a flag); Proposition 5.6 (two abelian
  layers); Remark 5.7 (computations).

## Computations (programs and outputs in reproducibility/)
- **Program of the note** (`check_note.g`, GAP 4.16.1, about 3 minutes; 51 checks, ALL CHECKS PASSED): both pairs
  rebuilt from their definitions (orders, IdGroup, every term of the derived series, maximality, cores, the three
  modules, the two dimensions, PrimitiveGroup(64, 37) and its maximal subgroups); the small facts of Lemmas 2.5 and
  2.6; the search of Theorem 1.3 for all orders 144–1296 except 768, 1152 and 1280 (130,577 groups, Table 1
  reproduced); the largest difference over all 28,954 soluble groups of non-prime-power order ≤ 400 without the
  filter d(G) ≥ 5 (it is 2); the structure of the three groups of order 1296.
- **Programs of the first computation** (`scripts/`, recorded outputs in `outputs/`): the certificates (1 s each);
  the complete search over all non-prime-power orders 144–1296, about 1.4 hours of CPU time (961 orders in 355 s,
  order 768 in 1106 s + 1169 s, order 1280 in 1146 s + 1222 s, order 1296 in 1.4 s); the explorations of Remark 5.7.
  At the writing stage the certificates and `run_gap3_1296.g` were run again from the package copies: identical
  outputs up to running times.
- **Programs of the third verification run** (`independent_run_2/`, see its README): `run2_pair1296.py` (Python, no
  GAP; the pair of Theorem 1.2 in two models, one of them from UT(3,3) and a complement of Inn(E) in Aut(E) found by
  search, all nine complements being conjugate; every displayed subgroup, maximality, core; 49 checks), `run2_tables.g`
  (every row of Table 1, with all classes of maximal subgroups computed as complements of chief factors, not with
  `MaximalSubgroupClassReps`; the three groups of Theorem 1.3; the caption identifications; Lemmas 2.5, 2.6; 50
  checks), `run2_pair41472.g` (Proposition 5.4 for each of the three modules; Remark 5.7 for (3^{1+2}_+, SL(2,3))
  over F_2; 26 checks), an own complete search over all 2,496,072 library groups of non-prime-power order 2–1296
  (orders 2–143 without the filter d(G) ≥ 5; 4722 s of CPU time) with `run2_parse.py` (13 checks), all with 0
  failures; and `rerun_compare.py`, the comparison of a re-run of the whole package with the recorded outputs
  (`independent_run_2/rerun_package_log.txt`).
- **Dependence.** Theorem 1.3 depends on the SmallGroups library (SmallGrp 1.7.0; all groups of order ≤ 2000 except
  1024, which is a prime power and excluded by Corollary 2.4(i)) and on `IsSolvableGroup`, `DerivedLength` and
  `MaximalSubgroupClassReps` in GAP 4.16.1.

## Independent verification runs
The results were first obtained with proofs and programs. Three independent verification runs, all AI-assisted,
followed; each wrote its own programs. Runs A and B examined the first write-up of the results (not the note); the
third run examined the note (see below). Run A
concentrated on the statement and the proofs, run B on independent computation and the literature.

| Item | Run A | Run B |
|---|---|---|
| Statement against the Notebook (v49) and the corpus record | CONFIRMED | CONFIRMED (v49 is the latest arXiv version) |
| Lemmas 2.1–2.3, Theorem 1.4(a)–(c), Corollary 2.4(i)–(iii) | CONFIRMED (re-derived) | CONFIRMED (Lemma 2.3 and Theorem 1.4(c) re-derived) |
| Theorem 1.4(d) (144, 5184) | CONFIRMED (re-derived; small facts re-run) | CONFIRMED (own scan of the primitive groups of degree ≤ 300) |
| Theorem 1.2 (pair of order 1296) | CONFIRMED (re-derived; certificate re-run, identical) | CONFIRMED (own construction from Aut(SmallGroup(27, 3)); explicit generators; primitive action on the cosets) |
| Theorem 1.3 (least order 1296) | CONFIRMED (coverage of all id ranges parsed; all search programs except order 1296 re-run, identical; groups of Table 1 below 1296 recomputed; no difference 3 without the filter for all soluble groups of non-prime-power order ≤ 500) | CONFIRMED (own program on all 2,496,072 library groups of non-prime-power order 2–1296; maximal subgroups of the 15 groups by a second algorithm; DerivedLength by a second code path on 1615 groups; `run_gap3_1296.g` re-run, identical) |
| The three groups of order 1296: split / non-split | — | 2891 split (3 classes of complements), 2889 and 2890 non-split |
| Lemmas 5.1, 5.3, Remark 5.2 | CONFIRMED (re-derived; one unused hypothesis noted) | CONFIRMED (read) |
| Proposition 5.4 (pair of order 41472) | CONFIRMED (re-derived given the module; certificate re-run) | CONFIRMED (own construction for each of the three modules; G2 = PrimitiveGroup(64, 37)) |
| Lemma 5.5, Proposition 5.6 | CONFIRMED (re-derived, including the faithfulness steps) | no error found on reading; not proved independently |
| Remark 5.7 (searches for difference 4) | outputs read; TESTED only | NOT CHECKED |
| Literature | not its part | Question 9 of Monakhov–Trofimuk (2010) found; Doerk (1994) and Murashka–Vasil'ev (2025) read in full; Gol'chuk–Monakhov (2024) from the zbMATH review |

**Fixes required by runs A and B, and what was done:**
1. (A) State the dependence of the least-order result on the SmallGroups library and GAP, with versions: done in
   Section 4 and Section 6 (GAP 4.16.1, SmallGrp 1.7.0; the library contains all groups of order ≤ 2000 except 1024,
   which is excluded by the prime-power lemma; `DerivedLength`, `MaximalSubgroupClassReps`).
2. (A) The unwritten "hand analysis" of a same-characteristic layer over the pair of order 41472: left out.
3. (A) The unused hypothesis "S^(t−1) fixed-point-free on E/Z(E)" in the construction: removed (Lemma 5.3).
4. (A) Spell out the statements taken from Doerk–Hawkes and Huppert: stated and proved in Lemma 2.2.
5. (A, optional) GAP syntax warnings in the exploration scripts: not changed, as the recorded outputs are kept; the
   README says that they do not affect the results.
6. (B) Cite and read Monakhov–Trofimuk (2010), Question 9: read in the journal's own file; cited in the abstract, the
   introduction and "Scope and priority", with what it contains.
7. (B) Doerk: pages 19–21, Zbl 0817.20017; Gol'chuk–Monakhov: Zbl 1544.20031, DOI 10.1134/S0001434624050389; both
   checked on zbMATH Open, the DOI on Crossref; zbMATH internal ids are not used.
8. (B) List the groups of derived length ≥ 5 below 1296, the split/non-split nature of the three groups, and
   G2 = PrimitiveGroup(64, 37): Table 1, Theorem 1.3, Proposition 5.4.
9. (B) Say that the least-order result is computer-assisted and that an independent re-run over all 2,496,072 library
   groups agrees: Theorem 1.3 is labelled computer-assisted; Section 4 and the Verification paragraph.

**Changes made at the writing stage, after the two runs.** Lemmas 2.5 and 2.6 received proofs that do not use the
libraries (the first write-up quoted the list of groups of order < 24 and the irreducible soluble subgroups of
GL(3,2)); Corollary 2.4(iv) was added; Lemma 5.1 states, with proofs, the two sufficient conditions used later (an
element without fixed points on an abelian layer; a subgroup of coprime order without fixed points on an elementary
abelian layer, by Maschke's theorem); Proposition 5.6 assumes of W2 only that it is non-zero (the first write-up
assumed it irreducible, which is not used); `check_note.g` was written. Runs A and B did not examine these changes;
the third run examined them first.

**Third run (verification of the written note).** It examined the text of the note line by line, the changes just
listed first, and read the sources again: the problem in the Notebook (v49, still the latest version), pp. 63 and
76–78 of the survey of Monakhov–Trofimuk in the journal's file, Satz 1 of Doerk (1994), the zbMATH review of Gol'chuk–Monakhov,
Theorems 1–4 of Murashka–Vasil'ev with the proof of Theorem 4, and Section 1 of Glasby–Howlett (arXiv version). It
re-derived all proofs: in particular the elementary proofs of Lemmas 2.5 and 2.6 (including the case |V| = 8 through
the normaliser of a subgroup of order 7 in GL(3,2)), Corollary 2.4(iv), the two sufficient conditions of Lemma 5.1,
Proposition 5.6 under the hypothesis W2 ≠ 0, both derived series of Section 3, and the logic of the least-order
search. Its own programs (above) reproduced every number of the note, Table 1 included, and its own complete search
found no soluble group of non-prime-power order ≤ 1296 with d(G) ≥ 5 other than those of Table 1, and difference ≤ 2
for every maximal subgroup of every soluble group of non-prime-power order ≤ 143. It re-ran the package from an
extracted copy of the source archive (every program except the sampling program `B_dl_check.g` of run B, about 29 minutes of CPU time;
all 33 compared outputs agree with the recorded ones up to running times, GAP syntax warnings and, in `explore3.g`, the
order in which the MeatAxe lists the modules; `independent_run_2/rerun_package_log.txt`). It checked the five DOIs with Crossref and the Zbl numbers
with zbMATH Open, and searched again (arXiv API, zbMATH Open, OpenAlex, Crossref, one web search). Verdict: no
mathematical error; the result is not found in print; the result is partial. Required corrections, all made: (1) the
three GAP functions named consistently; (2) Table 1 and Section 4 restricted to the groups actually searched
(non-prime-power orders); (3) the reference to Glasby–Howlett, whose extraspecial tower contains the group of
Theorem 1.2 and whose construction step is that of Lemma 5.3; (4) Theorem 4.6 of the survey is Doerk's theorem, and
its Theorem 4.7 (the F-length analogue of Ballester-Bolinches and Pérez-Ramos) is incorrect in general by
Murashka–Vasil'ev, Theorem 4; (5) a narrower statement of what is new (Lemmas 2.1–2.3 and 5.5 are standard, the group
G is not new); (6) the layer F in Lemmas 5.1 and 5.3 soluble; (7) the hypothesis [W, Z(E)] = W stated in the proof of
Proposition 5.4; (8) the sentence before Lemma 5.5 restricted to the case covered by Proposition 5.6; (9) the reason
for d(M) = 2 in Corollary 2.4(iv); (10) two phrases of the abstract; (11) the description of run A (it recomputed the
groups of Table 1 below order 1296 and the three groups of Theorem 1.3) and the list of changes in the Verification
paragraph; (12) Section 6 (four parts of the package); (13) the README explains the words used in the comments of the
recorded programs.

## Relation to the literature, novelty and scope
- **Searches (27 October 2026).** arXiv (derived length and maximal subgroups, the Notebook, the papers of Monakhov),
  zbMATH Open (derived length and maximal subgroups, Kourovka, Monakhov, Trofimuk, Doerk), OpenAlex (English and
  Russian queries; works citing Doerk 1994 and Gol'chuk–Monakhov 2024), Crossref, Math-Net.Ru, Numdam, the journal's
  website, and five web searches (one of them by the third run); Semantic Scholar answered with HTTP 429 and was not
  used; the publisher's page of Gol'chuk–Monakhov answered with a bot challenge and was not retried. The third run
  repeated searches in the arXiv API, zbMATH Open, OpenAlex and Crossref; the arXiv query "derived length" and
  "maximal subgroup" returned Glasby–Howlett, which is now cited.
  - No printed answer to the question, and no example with difference 3, was found.
  - Related known results, presented exactly in the note: Doerk (1994), Satz 1: n(U) = n(G) − i with i ∈ {0, 1, 2}
    for the nilpotent length; Gol'chuk–Monakhov (2024), Theorem 1(a): a soluble group with a Miller–Moreno maximal
    subgroup has |π(G)| ≤ 3 and d(G) ≤ 4 (d(G) ≤ 3 with a Sylow tower if |π(G)| = 3); Murashka–Vasil'ev
    (arXiv:2503.24335v1), Theorem 1: h*(G) − h*(M) ≤ 2, λ(G) − λ(M) ≤ 1, λ_p(G) − λ_p(M) ≤ 1, and Theorem 4: a
    hereditary saturated formation F for which n_F(G) − n_F(M) is unbounded, with M normal of prime index in their
    construction (so it says nothing about derived length); by their Theorem 4 the F-length analogue of Doerk's
    theorem (Ballester-Bolinches–Pérez-Ramos 1994, Theorem 4.7 of the survey) fails for some F. Glasby–Howlett
    (J. Algebra 151 (1992) 236–260, DOI 10.1016/0021-8693(92)90142-9; arXiv:1405.7228): extraspecial towers
    GL(2,3) = S3 ⋉ Q8, (S3 ⋉ Q8) ⋉ 3^{1+2}_+, ...; the second is the group of Theorem 1.2 (all complements of the inner
    automorphisms in Aut(3^{1+2}_+) are conjugate); they do not consider derived lengths of maximal subgroups. None of
    them answers the question.
- **What was read.** The Notebook problem (v49); the corpus record; Monakhov–Trofimuk (2010) as described above;
  Doerk (1994) in full (Numdam); Murashka–Vasil'ev in full (arXiv v1); of Glasby–Howlett, Section 1 of the arXiv
  version; of Gol'chuk–Monakhov (2024) the zbMATH review and the Crossref metadata only. The proofs of the cited
  theorems were not checked.
- **Caveats.** Theorem 1.3 is computer-assisted (library and GAP). The existence of the modules of Proposition 5.4 and
  the two dimensions are computations. A search that finds nothing is not a proof of novelty; the bound k ≥ 3 may be
  known. No priority is claimed. Claimed as new, to our knowledge: the difference 3 of the two pairs, the least order
  1296 with the three groups, Theorem 1.4(c),(d), Lemma 5.3 and Proposition 5.6; Lemmas 2.1–2.3 and 5.5 are standard.
- **Scope.** Neither part of Problem 17.91 is answered.

## 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/