# Verification report — OWR-793-006 (Praeger–Seress: minimal subdegrees of primitive groups of twisted wreath type, Question 4)

Verification date: 2026-10-11.

**Verdict.** The note gives a **negative answer to Question 4** of C. E. Praeger and Á. Seress (Oberwolfach
Report 12/2005, p. 692): there is a finite primitive permutation group of twisted wreath type with
MinSubDeg(G) = k · MinSubDeg(H).
- **The example.** P = PGL(3,4) on the 21 points of PG(2,4), Q the stabiliser of a point y, T = PSL(2,4) = A_5,
  φ the conjugation action of Q through its action on F_4^3 / y. The group G = T twr_φ P, acting on its base
  group (degree 60^21), is primitive of type TW with k = 21, primitive component H = A_5 × A_5,
  MinSubDeg(H) = 12 and MinSubDeg(G) = 252 = 21 · 12 (Theorem 1.1).
- **Proved without computation.** The reduction to the top group (Lemmas 2.1, 2.3, 2.4, known in substance),
  Theorem 1.1(a), (b), and Proposition 5.2.
- **Proved in two ways.** Theorem 1.1(c) is equivalent to Proposition 3.3: every subgroup S of PGL(3,4) of
  order greater than 240 induces, at every point d, a group of order at least 6 on the five lines through d.
  (i) Computer-assisted: seven complete enumerations, by four methods, with programs of four independent
  stages. (ii) A written argument from the list of the maximal subgroups of PSL(3,4) and PGL(3,4) in the table
  of Connor and Leemans.
- **Computer-assisted only.** Theorem 1.2: equality also for the top groups PΓL(3,4) (315 = 21 · 15) and
  PSL(3,5) (465 = 31 · 15), strict inequality for PSL(3,4) and PΣL(3,4); Table 3 (five classical examples,
  all strict).
- **Limits.** The full paper of Giudici, Li, Praeger, Seress and Trofimov (2006) and its companion paper
  (2007) were not accessible and were not read; no statement of the note depends on their content. No answer to
  Question 4 was found in the literature accessible to us; a search that finds nothing is not a proof of
  novelty, and no priority is claimed. The note does not claim that the example is the smallest one, the cases
  of equality are not classified, and the question remains open under additional hypotheses (for instance
  ker φ = 1).

The note is unrefereed.

## Statement checked
- **Primary source.** C. E. Praeger, Á. Seress, "On minimal subdegrees of finite primitive permutation
  groups", Oberwolfach Reports 2 (2005), no. 1, 690–693 (abstract in Report No. 12/2005, Groups and
  Geometries), doi:10.4171/OWR/2005/12.
  - Read in the publisher's open file of the report (692,596 bytes, sha256
    `8d874afa3364daae6ac781a57e3473eb38faad72fe4efae882d641aa67f4b8bf`), pages 690–693 completely: by the first
    stage, by both verification runs (each with its own text extraction), and again as page images when the
    note was written.
  - Definitions (p. 691): suborbits, subdegrees, non-trivial subdegrees, MinSubDeg(G); the embedding
    G ≤ H ≀ S_k in product action on Δ^k with H primitive on Δ and induced by G; H is called the primitive
    component of G relative to the Cartesian decomposition Δ^k; the socle.
  - Theorem 1 (quoted there from Theorem 1.1 of the full paper): MinSubDeg(G) = k · MinSubDeg(H) if
    Soc(G) = Soc(H ≀ S_k) is non-abelian.
  - Type TW (p. 692): G ≤ H ≀ S_k on Δ^k with primitive component H, k ≥ 2; Soc(G) = T^k with T non-abelian
    simple, Soc(H) ≅ T × T; the socle is regular on Δ^k; G = Soc(G) ⋊ P for a transitive subgroup P of S_k.
    MinDeg(P) is the least degree of a faithful transitive action of P.
  - Theorem 3: max{MinSubDeg(H), MinDeg(P)} ≤ MinSubDeg(G) ≤ k · MinSubDeg(H), with examples at the lower end.
  - The sentence before the question says that the authors wonder whether a group of type TW can ever achieve
    the bound of Theorem 1. Question 4 asks whether the strict inequality holds for all finite primitive groups
    G = Soc(G) ⋊ P ≤ H ≀ S_k of type TW, with P, H as in Theorem 3.
- **Corpus record.** ulamai/UnsolvedMath, OWR-793-006 (record id 30000203; dataset version 1.6.0; upstream
  status `open`). The field `statement.original` consists of the last two sentences of the abstract before its
  references, word for word (with "Sk" for S_k); the fields `statement.clean` and `statement.verification_note`
  are empty, and `statement.status` is `not_audited`.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Question 4 as posed: is MinSubDeg(G) < k · MinSubDeg(H) for all primitive groups of type TW? | no: equality for G = A_5 twr PGL(3,4), 252 = 21 · 12 | Theorem 1.1 |
| k = number of simple factors of the socle (= degree of P = exponent in Δ^k; Δ and T have the same size) | used; k = 21 | Section 1.1 |
| H = the group induced on one coordinate by the stabiliser of this coordinate (the source: H is induced by G) | H = T ⋊ φ(Q) = A_5 × A_5, MinSubDeg(H) = 12 | Lemma 2.1(b), Theorem 1.1 |
| H replaced by the full holomorph of T (not the definition of the source) | for the first example the product would be 315 > 252; for the top groups PΓL(3,4) and PSL(3,5) the induced group is the holomorph itself and equality holds (315, 465) | Remark 5.1, Theorem 1.2 |
| The action | the action with regular socle: the twisted wreath product on its base group | Section 2.1 |
| The non-strict inequality | Theorem 3 of the abstract; re-proved as Lemma 2.1(c) | Section 2.2 |
| A coarser Cartesian decomposition (blocks of simple factors) | none exists for the example: P is primitive on the 21 factors; and the source fixes Soc(H) ≅ T × T | checked by run A |
| Restricted classes: P primitive on the factors, φ(Q) = Inn(T), P almost simple | the example satisfies all of these | Theorem 1.1 |
| Additional hypotheses such as ker φ = 1 | not decided; in all three examples with equality ker φ ≠ 1 | Section 1.3 |
| "Non-trivial" subdegree = suborbit different from {α} (not "length > 1") | the same thing for these primitive groups | Section 1.1 |

## Results in the paper
- **Lemma 2.1.** For G = T twr_φ P with T non-abelian simple and Inn(T) ≤ φ(Q): the non-trivial subdegrees are
  the indices |P : C_P(b)|, so MinSubDeg(G) = |P| / M_G; G ≤ H ≀ S_k with H = T ⋊ φ(Q) the group induced on a
  coordinate, H primitive with socle T × T, and k · MinSubDeg(H) = |P| / M; M_G ≥ M.
- **Lemma 2.3 (criterion).** M_G is the largest order of a subgroup S of P such that φ(S ∩ Q) fixes a
  non-identity element of T ("good" subgroups). Equality holds iff no good subgroup has order greater than M.
- **Lemma 2.4 (primitivity).** If Q is maximal and core-free in P, Inn(T) ≤ φ(Q), and φ does not extend to P,
  then G is primitive of type TW with socle the base group and point stabiliser P.
- **Remark 2.5.** Sources: these lemmas are known in substance.
- **Lemma 3.1.** The kernel of the action of a point stabiliser on the five lines through the point is the group
  of all 48 perspectivities with this centre: 16 elations (identity included) and 32 homologies, two for each
  of the 16 lines not through the point.
- **Theorem 1.1** and **Proposition 3.3**: see the verdict. M = 48 · 5 = 240.
- **Section 4.** Written proof of Proposition 3.3 from three facts (F1)–(F3) read off from the table of
  Connor and Leemans: the subgroups of order > 240 are PGL(3,4), PSL(3,4), the subgroups A_6, and the groups
  E ⋊ R in the two parabolic subgroups (Lemmas 4.1, 4.2); at every point the induced group has order 60, 12,
  10 or 6. Table 2 lists the 14 classes of order ≥ 240 with their orbit data.
- **Theorem 1.2, Table 1.** (M, M_G) = (80, 160), (128, 320), (240, 240), (384, 384), (800, 800) for
  PSL(3,4), PΣL(3,4), PGL(3,4), PΓL(3,4), PSL(3,5).
- **Table 3.** (M, M_G) for (A_6, A_5), (S_6, S_5), (A_7, A_6), (PSL(2,11), A_5), (M_11, M_10): (5, 24),
  (8, 48), (9, 36), (5, 55), (16, 144); minimal subdegrees 15, 15, 70, 12, 55.
- **Proposition 5.2.** Three sufficient conditions for strict inequality; the example satisfies none.

## Computations (programs and outputs in reproducibility/)
All computations are exact. Proposition 3.3, Theorem 1.2 and Table 3 are computer-assisted results; random
samples and small examples are tests.
- **Method (1), `original/`** (`maxstab.py`, 30 s): for each of the 192 subgroups R of the groups Q_t with
  |R| ≥ 12, all subgroups S with S ∩ Q = R and |S| > 240 by a depth-first search over the orbit of the base
  point: none exists. Control: with the bound 239 the search finds 102 good subgroups, all of order 240.
- **Method (2), `original/` and `verification_run_B/`** (`lattice.py`, 6 min; `b1_lattice.py`, 80 s): all
  subgroups up to conjugacy, by two programs: 100 classes, 81220 subgroups; 12 classes of order > 240, the
  smallest induced order is 6; two good classes of order 240. The two class lists correspond bijectively
  under an exact conjugacy test.
- **Method (3), `verification_run_A/`** (`t2_main.py`, 30 s): overgroups of a subgroup of order 7 (11), of
  order 5 (29), and of a Sylow 3-subgroup in each parabolic subgroup (14 each), with a reduction of the
  {2,3}-groups to these cases: the same 14 classes of order ≥ 240.
- **Method (4), `verification_run_B/`** (`b1_ext.py`, 10 s): all subgroups of order ≥ 240 organised by their
  intersection with Q: 252, 420, 168, 252, 105, 42, 42, 1, 1 subgroups of orders 240, 288, 360, 480, 576, 960,
  2880, 20160, 60480; the good ones are 102 subgroups of order 240.
- **Third run, `independent_run_2/`** (`r2_main.py`, 20 s; `r2_lattice.py`, 11 s): methods (3) and (2) again,
  with programs written from the statements of the note. Its form of method (3) uses no property of soluble
  groups: every divisor of 60480 which is at least 240 and prime to 35 is a multiple of 9, so every subgroup
  of order ≥ 240 is conjugate to an overgroup of a fixed subgroup of order 5, of a fixed subgroup of order 7,
  or of one of the four subgroups of order 9 of a fixed Sylow 3-subgroup. All overgroups of these six
  subgroups are listed (29, 11, 23, 41, 8, 8 groups): the same 14 classes of order ≥ 240, with the orbit data
  of Table 2. The lattice program finds 100 classes and 81220 subgroups.
- **Certificate without the reduction** (`b2_base.py fix`, `fixbig`; `r2_main.py`, part 3): for the 12 classes
  of order > 240, and for the 29 subgroups of order > 240 found by method (4), the only element of the base
  group A_5^21 fixed by the subgroup is the identity (from the equations b^s = b). Hence M_G ≤ 240 from the
  definition of the action and the list of subgroups alone. Each subgroup of order 240 fixes exactly four
  non-identity elements, and it is their stabiliser (no larger subgroup fixes a non-identity element). So
  252 · 4 elements have a stabiliser of order 240, and they lie in orbits of length 252: there are exactly
  four suborbits of length 252. The third run found them as four orbits of P on the base group: two consist
  of elements of one factor, two of elements supported on the 16 points off a line.
- **Base group, direct tests** (`b2_base.py`, `direct_check.py`, `t3_break.py`): the tuple form of the action
  agrees with the definition on functions on all 60480 elements; all 1239 elements supported on one
  coordinate have stabilisers of order 192, 144, 240; all 731,010 elements supported on two coordinates: at
  most 32; all 410,758 elements supported on a collinear triple or a triangle: at most 96; about 330,000
  further elements: at most 240.
- **Other rows** (`run_par.py`, `b1_lattice.py`, `b1_ext.py`, `b2_base.py fixbig`): PSL(3,4), PΣL(3,4),
  PΓL(3,4) by methods (1), (2), (4) (95, 152, 226 classes); PSL(3,5) by methods (1) and (4); for PΓL(3,4)
  and PSL(3,5) none of the 51 and 39 subgroups of order > M found by method (4) fixes a non-identity element
  of the base group. Table 3 by `classical.py` and `b4_classical.py`. The third run computed all rows of
  Tables 1 and 3 again from lists of all subgroups up to conjugacy (`r2_lattice.py`: 95, 152, 100, 226 and
  140 classes for the five projective groups; 22, 56, 40, 16 and 39 classes for the classical groups), and
  the rows of Table 1 also from the large subgroups alone (`r2_seeds.py`).
- **Validation.** Lemma 2.3 on 10 + 11 + 7 small twisted wreath products with completely listed base groups
  (three of the stages). The first two lattice programs reproduce the numbers of classes of subgroups and of
  subgroups of S_n (n ≤ 7) and A_n (n ≤ 8) of OEIS A000638, A005432, A029726, A029725; they agree with each
  other on M_11 (39, 8651) and PSL(3,2) (15, 179). The lattice program of the third run gives the same numbers
  for A_6, S_6, A_7 and M_11. For PGL(3,4) the pairs (order, number of conjugates) of all 100 classes coincide with
  the table of Connor and Leemans; for L_3(5) the numbers of subgroups of each order ≥ 800 coincide with those
  of method (4). The third run read both tables with a program of its own (`r2_cl_compare.py`): all 100 pairs
  for PGL_3(4) and all 140 pairs for L_3(5) coincide with its own lists of classes (81220 and 345809
  subgroups).
- **Re-runs.** On 2026-10-11 all programs of the package were run again from the package layout. The 93
  outputs and data files which have a counterpart in the first runs are identical to them up to fields which
  record running times, apart from a label now printed by one program and one reworded phrase
  (`reproducibility/RERUN_LOG.txt`). `reproducibility/run_main_certificate.sh` repeats the main computations
  in about three minutes; it was run from an extracted copy of the archive. The third run ran this script
  (12 of 12 comparisons identical) and all seven parts of `run_all.sh` again from an extracted copy of the
  archive: all 96 recorded outputs and data files were reproduced (part 3 of `RERUN_LOG.txt`).

## Independent verification runs
The results were first obtained with the lemmas of Section 2, methods (1) and (2), and a first form of the
written argument. Two independent verification runs, both AI-assisted, followed on 2026-10-11; neither saw the
other, and each wrote its own programs. Run A examined the statement and every proof; run B recomputed
everything, examined the first programs and the literature.

| Item | Run A | Run B |
|---|---|---|
| Statement: fidelity to the source, scope of the example | CONFIRMED_WITH_FIXES (wording) | CONFIRMED_WITH_FIXES (one sentence about the corpus record) |
| Lemmas 2.1 and 2.3 | CONFIRMED (derived before reading the proofs) | tested on eleven small examples; the upper bound M_G ≤ 240 obtained also without Lemma 2.3 |
| Lemma 2.4, primitivity and type TW of the example | CONFIRMED (own proof; compared with the criteria quoted in the literature) | CONFIRMED (own argument; hypotheses computed) |
| The datum (orders, kernel, Q_t, M = 240) | CONFIRMED (own program) | CONFIRMED (own program) |
| Proposition 3.3 by computation | CONFIRMED (method (3)) | CONFIRMED (methods (2) and (4)) |
| Proposition 3.3 by the written argument | CONFIRMED_WITH_FIXES (complete given the lists of maximal subgroups; a checked source required) | read, no gap seen |
| The first programs | not its part | CONFIRMED (code read; all outputs reproduced) |
| Theorem 1.1 | CONFIRMED | CONFIRMED |
| Other rows of Table 1, Table 3 | PSL(3,4), PΓL(3,4) and the row A_6 confirmed by its own computation; the other rows only in part | CONFIRMED (complete computations for all rows) |
| Proposition 5.2 | CONFIRMED | not its part |
| Novelty | no answer found (two searches) | CONFIRMED_WITH_FIXES: nothing found; the 2006 paper and its companion not readable; attributions required |

No run found a mathematical error or an error in a computation.

**Corrections required by the runs**, all applied in the note and the package:
1. (Run A, run B) The lists of maximal subgroups are taken from a source which was read (the tables of Connor
   and Leemans), with the row numbers of the table; no page numbers of other atlases are cited. The reference
   values of the self-tests are those of the OEIS, and the README says which values are not from a consulted
   source.
2. (Run A) The kernel K_d is described as the group of all 48 perspectivities with centre d (Lemma 3.1).
3. (Run A) In the written argument the points d ∈ l and d ≠ x are treated explicitly: for every such point, why
   an element with eigenvalue b ≠ 1 exists, and that the induced group lies in, and equals, the stabiliser A_4
   of a line of the pencil (Section 4).
4. (Run A) The statement on Cartesian decompositions is limited to what is proved; the attributions to the
   source are exact (the source does not say that P is a point stabiliser; the corpus fields are described as
   they are).
5. (Run A, run B) The note says in the abstract, in Section 1.3 and in "Scope and priority" that the full
   paper of 2006 and the companion paper were not read, and what is known about them.
6. (Run B) The reduction is presented as known in substance, with the sources (Remark 2.5).
7. (Run A) Wording: "a negative answer to Question 4"; the reading (k, H, the action) is stated; no claim of
   minimality or of a classification.
8. (Run B) Sample code which had not been executed is not included. A side remark of the first written
   version which is not needed for any result was dropped.
9. (Run B, recommended) The certificate without the reduction is part of the package; the note says that the
   two further examples have complete computations and that their primitive component is the full holomorph
   of A_5.

**Written after these two runs** (and examined by the third run, see the next section):
- the case S ≅ A_6 in the proof of Proposition 3.3 (orbit lengths and the order of the kernel; it replaces an
  argument with hyperovals which run A had checked);
- the arrangement of Section 4 around (F1)–(F3), Lemmas 4.1 and 4.2 (the same argument as checked by run A,
  with the parabolic subgroups written as E ⋊ L);
- the count of the four suborbits of length 252 (from the outputs of run B);
- the comparisons of complete lists with the tables of Connor and Leemans (`writing_stage/cl_compare.py`).

## Third verification run, on the final text
A third run, also AI-assisted, was made on 2026-10-11 on the final text of the note and on the package. It
wrote its own programs from the statements of the note (`reproducibility/independent_run_2/`), before it
read the programs of the earlier stages, and it examined first the four things listed above.

| Item | Third run |
|---|---|
| The question, Theorems 1 and 3 and the definitions in the source (fetched again, pages 690–693 read) | CONFIRMED: the note answers Question 4 as posed; the reading (k, H, the action) is stated exactly; the scope is stated correctly |
| Section 2.1, Lemmas 2.1, 2.3, 2.4, Remark 2.5 | CONFIRMED line by line (left cosets, the action on the base group, the conjugate which meets Q) |
| Lemmas 3.1, 3.2, the datum, Theorem 1.1(a), (b) | CONFIRMED (also by its program; primitivity also from Baddeley's criterion as stated by Fawcett, Theorem 4.6) |
| (F1)–(F3), Lemmas 4.1, 4.2, the arrangement of Section 4 | CONFIRMED; (F1)–(F3) read in the table |
| Proposition 3.3, the case S ≅ A_6 (the new argument) | CONFIRMED, every step; the conclusion checked on the stabiliser of a hyperoval: order 360, inside PSL(3,4), orbits (6, 60, 60) and (15, 24, 6), kernels of order 1 and 4 |
| Proposition 3.3, the other cases | CONFIRMED |
| Proposition 3.3 and Table 2 by computation | CONFIRMED by two complete enumerations of its own |
| The certificate without the reduction; exactly four suborbits of length 252 | CONFIRMED by its own program; the number four also derived from Lemma 2.3 and Table 2 |
| Theorem 1.2, Table 1, Table 3 | CONFIRMED: every row recomputed (Table 1 by two methods) |
| Proposition 5.2, Remarks 5.1 and 5.3, the witness for PSL(3,4) | CONFIRMED |
| The comparisons with the tables of Connor and Leemans | CONFIRMED and extended: all 100 classes of PGL_3(4), all 140 classes of L_3(5) |
| Attributions: Chua–Giudici–Morgan, Dolfi–Guralnick–Praeger–Spiga, Fawcett, the zbMATH review | CONFIRMED against the texts (arXiv versions fetched again) |
| Bibliography | the nine DOIs agree with Crossref; the four arXiv versions cited are the latest ones |
| Novelty | nothing found (arXiv, zbMATH, Crossref, OpenAlex, one web search); the 2006 paper and its companion are still not accessible |
| The package | all programs run again from the archive; all recorded outputs reproduced |

The third run found no error in a statement, a proof, a table or a computation. It required:
1. that the paragraph "Verification", Sections 1.2 and 6 and the sentence after the proof of Proposition 3.3
   describe the state after its checks (the numbers of programs, stages and enumerations; its two programs;
   the four things written after the first two runs are no longer unexamined);
2. a justification of the count of the four suborbits (Section 6: the stabiliser of each of the 252 · 4
   elements is the subgroup itself);
3. in "Scope and priority": its searches (nine web searches in all), the new attempt to reach the
   publishers' pages, and Section 1 of Fawcett's paper in the list of what was read (the note cites it for
   the definition of twisted wreath type);
4. its programs and outputs in the package, and the record of its re-run.

All were done. With these additions the note would have had 16 pages; to keep it at 15 pages the wording was
shortened in about twenty-five places (mostly paragraphs which ended with one or two words; the list of new
results in "Scope and priority" now refers to Section 1.3; one entry of the bibliography). No mathematical
statement, proof step or reference was removed; a few explanatory remarks in Section 6 and in the paragraph
"Verification" were dropped, and their content is in this report or in `reproducibility/README.md`. Title,
abstract, theorems, tables and all numbers are unchanged. These changes of wording were made by the third run
itself after its checks; they were not examined by a further run.

## Relation to the literature, novelty and scope
- **Not read.** M. Giudici, C. H. Li, C. E. Praeger, Á. Seress, V. Trofimov, "On minimal subdegrees of finite
  primitive permutation groups", in: Finite Geometries, Groups, and Computation, de Gruyter 2006, pp. 75–93
  (doi:10.1515/9783110199741.75), and the companion paper of the same authors, "On limit graphs of finite
  vertex-primitive graphs", J. Combin. Theory Ser. A 114 (2007) 110–134. Both were not accessible to us, and no open copy was found; the third run tried once more on
  2026-10-11, without result. What is known about the 2006
  paper comes from the Oberwolfach abstract, from the zbMATH review (Zbl 1111.20003) and from the paper of
  Chua, Giudici and Morgan. The note makes no claim about the content of the two papers; in particular it is
  not known whether the question is repeated or restricted there, nor which examples are treated. A reader
  with access to them should compare. Dixon–Mortimer (1996) and Baddeley (1993) were not consulted either;
  they are cited through the papers which quote them.
- **Read.** The Oberwolfach abstract completely. Chua, Giudici, Morgan (arXiv:1801.02456v1): Sections 1, 2
  (Lemma 2.2, Theorem 2.3, Lemma 2.6). Fawcett (arXiv:2102.02190v2): the definition of twisted wreath type in
  Section 1, and Section 4 up to Lemma 4.7 (Theorem 4.6).
  Dolfi, Guralnick, Praeger, Spiga (arXiv:1109.6559v2): Example 4.4 (run B through an automatic reading of
  the HTML version; read in the PDF when the note was written). The journal versions were not compared. The
  table of the subgroup lattice of PGL_3(4) of Connor and Leemans (dated November 26, 2012) completely, and
  the table of L_3(5) for the orders and lengths of its classes; of their paper on the atlas only the
  abstract. The OEIS entries A000638, A005432, A029725, A029726. The third run fetched all of these again and
  read the same parts (the table of L_3(5) with its program, for all 140 classes).
- **Known in substance (not claimed as new).** Lemma 2.1(a): Chua–Giudici–Morgan, Lemma 2.2. Lemma 2.1(c): the
  upper bound of Theorem 3 of the abstract. Lemma 2.3(b): the function of Dolfi–Guralnick–Praeger–Spiga,
  Example 4.4, and of Chua–Giudici–Morgan, Lemma 2.6; the abstract says that the minimum is attained on a
  suborbit containing a special point constructed in the full paper. Lemma 2.4: contains the criterion which
  the two papers quote from Dixon–Mortimer (as Lemma 4.7A and as Lemma 4.7B); Baddeley's characterisation as
  stated by Fawcett, Theorem 4.6.
- **New, to our knowledge.** The example A_5 twr PGL(3,4) with MinSubDeg(G) = 252 = 21 · 12, and the two
  examples with top groups PΓL(3,4) and PSL(3,5), in which φ(Q) = Aut(T) and H is the full holomorph of A_5.
- **Searches (2026-10-11, at each stage).** Queries to the arXiv interface, zbMATH, Crossref, OpenAlex and
  Semantic Scholar (minimal subdegrees; smallest or shortest suborbits of primitive groups; twisted wreath
  products), the lists of works citing the 2006 paper (OpenAlex: 4, Semantic Scholar: 5), and nine web
  searches in all (the third run: arXiv, zbMATH, Crossref, OpenAlex and one web search). No answer to Question 4 was found in the literature accessible to us, and no mention of a
  twisted wreath product with top group PGL(3,4) in this context. Full texts were not searched; the authors of
  the question were not contacted. A search that finds nothing is not a proof of novelty.
- **Bibliographic data** were checked with Crossref on 2026-10-11 (the nine DOIs of the bibliography; again by
  the third run) and with the arXiv interface (four identifiers). For the 2006 paper Crossref gives the pages 75–94; zbMATH and the papers which cite it
  give 75–93, which the note uses.
- **Scope.** One explicit group answers the universal question in the negative; two further groups do so under
  either convention for the primitive component. Nothing is claimed about the smallest example, about a
  classification of the cases of equality, or about restricted classes of top groups.

## 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/