# Verification report — OWR-793-006, second note (Giudici, Li, Praeger, Seress, Trofimov: minimal subdegrees of primitive groups of twisted wreath type; an infinite family attaining the upper bound)

Verification date: 2026-10-14.

**Verdict.** The note proves, by a written argument, that the upper bound MinSubDeg(G) <= k · MinSubDeg(H) of
Giudici, Li, Praeger, Seress and Trofimov (Theorem 1.5(a) of their 2006 paper) is attained by infinitely many
primitive groups of type TW: for every prime power q >= 4 the group G_q = PSL(2,q) twr PGL(3,q) (top group
PGL(3,q) on the points of PG(2,q), Q a point stabiliser acting on T = PSL(2,q) through its action on F_q^3/y) is
primitive of type TW and MinSubDeg(G_q) = k · MinSubDeg(H) = q(q^3 − 1)(q + 1)/c_q (Theorem B).
- **Settled.** Theorem A: no subgroup of PGL(3,q) of order greater than M = q^2(q − 1)c_q is good at any point,
  for every prime power q >= 4; hence Theorem B. This answers the question of M. Giudici, asked after the
  earlier note, whether infinitely many such groups exist; each G_q is a negative answer to Question 1.6 of the
  five authors (Question 4 of Praeger and Seress, Oberwolfach Report 12/2005).
- **Conditional on cited standard results, not re-proved:** the list of maximal subgroups of SL(3,q) (Mitchell
  1911, Hartley 1925; tabulated by Bray, Holt and Roney-Dougal 2013, read only in the transcription of Liu and
  Yang, arXiv:2209.02660, Lemma 2.2; the part that is used also follows from Alavi and Burness 2015); the
  automorphism groups of PSL(3,q0) and PSU(3,q0) (Schreier and van der Waerden 1928; Steinberg 1960); Dickson's
  list of the subgroups of PGL(2,q) for even q (Huppert II.8.27; for odd q it was read in Cameron, Omidi and
  Tayfeh-Rezaie 2006, Theorem 2). For q = 4, Theorem A is Proposition 3.3 of the earlier note.
- **Not new.** The bounds, the question and the criterion (Construction 4.1 and Lemmas 4.2, 4.3 of the 2006
  paper) are due to the five authors; the example q = 4, and the computed example q = 5, are those of the
  earlier note (doi:10.5281/zenodo.23342394).
- **Not claimed.** No classification of the groups of type TW with equality; nothing about other top groups.

The note is unrefereed. Three verification runs, all AI-assisted, examined it (two the first written version,
one the final text); none found a mathematical error.

## Statement checked
- **Primary source.** 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. 75–94, doi:10.1515/9783110199741.75. Read in the copy that M. Giudici sent for the earlier note (19 pages;
  text extracted locally; not redistributed): Theorem 1.5 and Question 1.6 (p. 4); Section 2 (product action,
  the primitive component as the group induced by the stabiliser of a coordinate, the O'Nan–Scott types,
  pp. 4–7; read by the third verification run); Section 4 up to Lemma 4.5
  (pp. 12–15): the twisted wreath product and its base group, the primitive component H = T ⋊ φ(Q),
  Construction 4.1 (the elements f_{R,t}), Lemmas 4.2 and 4.3 (MinSubDeg is attained on such an element, and
  the upper bound), Lemmas 4.4 and 4.5 (the lower bound).
  - Theorem 1.5(a): max{MinSubDeg(H), MinDeg(P)} <= MinSubDeg(G) <= k · MinSubDeg(H), with infinitely many
    examples of equality on the left.
  - Question 1.6 asks whether the strict inequality MinSubDeg(G) < k · MinSubDeg(H) holds for all finite
    primitive groups of type TW.
- **Abstract.** C. E. Praeger, Á. Seress, Oberwolfach Reports 2 (2005) 690–693, doi:10.4171/OWR/2005/12:
  Question 4 (p. 692) is the same question; read for the earlier note and again by the third verification run.
- **Corpus wording.** The record quotes the abstract, with its numbering (Theorem 1, Theorem 3, Question 4); in
  the 2006 paper these are Theorem 1.1, Theorem 1.5 and Question 1.6.
- **Earlier note.** A. Ferudun, "A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees
  of Primitive Groups of Twisted Wreath Type", Zenodo, version 1.1, doi:10.5281/zenodo.23342394 (all versions:
  doi:10.5281/zenodo.23299864): Lemmas 2.1, 2.3, 2.4 (used for the primitive component, the criterion and
  primitivity), Lemma 3.1, Lemma 3.2 and Proposition 3.3 with its proof (q = 4, including the subgroups A_6),
  Theorems 1.1, 1.2, Table 2 and Section 6 (q = 5 by complete enumeration).
- **Corpus record.** ulamai/UnsolvedMath, OWR-793-006 (dataset version 1.6.0, listed there as open).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Question 1.6 (Question 4): is MinSubDeg(G) < k · MinSubDeg(H) for all primitive groups of type TW? | No; G_q is a counterexample for every prime power q >= 4 | Theorem B |
| The follow-up question of M. Giudici: are there infinitely many primitive groups of type TW with MinSubDeg(G) = k · MinSubDeg(H)? | Yes | Theorem B |
| Primitive component | the group H = T ⋊ φ(Q) induced on one coordinate, as in Section 4 of the 2006 paper; here φ(Q) ≅ PGL(2,q) acting on T = PSL(2,q) by conjugation | Section 1.2, Lemma 1.1 |
| k | the number of simple factors of the socle, k = \|P : Q\| = q^2 + q + 1 | Section 1.3 |

## Results in the paper
- **Lemma 1.1, Lemma 1.3** (from the earlier note and the 2006 paper): MinSubDeg(H) = \|Q\|/M,
  k · MinSubDeg(H) = \|P\|/M, MinSubDeg(G) = \|P\|/M_G with M_G the largest order of a good subgroup; equality
  if and only if no good subgroup has order greater than M. The proof of Lemma 1.3 is repeated.
- **Lemma 1.5.** S good iff good at y; g^{-1}Sg good iff S good at y^{g^{-1}}; φ(Q) ≅ PGL(2,q); M = q^2(q − 1)c_q.
- **Theorem A**, proved for q >= 5 in Sections 2–7: Section 2 (centralisers and c_q, Lemma 2.1, proved in
  full; the Borel lemma 2.2; Dickson's list 2.3, cited; large subgroups 2.4; lifting and subfield orbits 2.5);
  Section 3 (perspectivities, inverse-transpose duality); Section 4 (Inputs R1, R2; normalisers of PSL(3,q0)
  and PSU(3,q0) in PGL(3,q0^2), Lemma 4.1; the reduction, Proposition 4.2: a subgroup of order > M contains
  PSL(3,q) or fixes a point or a line; Remark 4.3: the part of R1 that is used follows also from Alavi–Burness;
  Lemma 4.4: overgroups of PSL(3,q); Remark 4.5: the reduction fails for q = 4 because of A_6 < PSL(3,4));
  Section 5 (line stabilisers, Proposition 5.2); Section 6 (point stabilisers via duality: all q^2 elations,
  then a translation argument; Proposition 6.1, Corollary 6.2). For q = 4: Proposition 3.3 of the earlier note.
- **Theorem B.** Primitivity (Lemma 1.1(b); φ does not extend, by the simplicity of PSL(3,q) and core-freeness
  of Q) and MinSubDeg(G_q) = k · MinSubDeg(H) = q(q^3 − 1)(q + 1)/c_q, with MinSubDeg(H) = q(q − 1),
  q(q + 1)/2, q(q − 1)/2 for q even, q ≡ 1, q ≡ 3 (mod 4). Table 1: q <= 25.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, 26 s, from the definitions; TOTAL failures: 0): for q = 5, 7,
  c_q and M by brute force; all 48 resp. 84 conjugacy classes of subgroups of GL(2,q) by a generic method; in
  both parabolic subgroups every subgroup of order > M contains the elation group E (all invariant subgroups of
  E listed), and none of the 10 (q = 5) resp. 9 (q = 7) subgroups is good at any of the 31 resp. 57 points; the
  control with |X| >= M finds the good subgroups of order M. Its lists coincide with those of verification run B
  (`compare_check_note.py`).
- **Programs of the first version** (`original/`): `parabolic_q.py` (q = 7, 8, 9: 9, 5, 19 subgroups of order
  > M in each parabolic, none good); `check_parabolic_prime.py` (q = 5, 7, 11, 13: 10, 9, 10, 34 subgroups,
  none good; q = 17 stopped after one subgroup, not covered); `lemma_checks.py` (Lemmas 2.1, 2.2, 2.4 for all
  twelve prime powers 4 <= q <= 25); `lemma17b_check.py` (Lemma 2.5(b), (16,4), (25,5)).
- **Verification run A** (`verification_run_A/`): `parab_check.py`, all subgroups of order > M of a line
  stabiliser including non-split ones (1-cocycles), q = 4, 5, 7, 8, 9, 11, 13, 16, 17 (5, 10, 9, 5, 19, 10, 34,
  20, 32 subgroups; none good), control at >= M for q <= 11; `values_check.py` (Table 1, q <= 32);
  `ineq_sweep.py` (all inequalities of Sections 2, 4–6 for all 217169 prime powers q <= 3·10^6; no failure;
  291 tight cases, all of the two shapes of Lemma 4.1); `lemma_pgl2_check.py` (Lemmas 2.2–2.4 for q <= 16,
  Lemma 2.5(b) for (16,4), (25,5), (49,7)).
- **Verification run B** (`verification_run_B/`): `vb_parab.py` (both parabolics via H^1, q = 5, 7, 8, 9, 11,
  13, 16, 17, 19, 23, 25: 10, 9, 5, 19, 10, 34, 20, 32, 28, 9, 84 subgroups, N = E in every case, none good;
  control with threshold M/2 for q = 5, 7, 8, 9 finds good subgroups; threshold 239 for q = 4 finds twelve
  subgroups, six in each parabolic subgroup, which give the eleven parabolic rows of Table 2 of the earlier
  note, the stabiliser of a flag lying in both); `vb_maximal.py` (C2, C3, C5, C6, C8 members for
  q = 5, 7, 8, 9, 11, 13, 16, 25, orders by Schreier–Sims, all < M; Lemma 4.4 for these q); `vb_c9_q11.py`
  (PSL(2,7) < SL(3,11), order 168); `vb_lemmas.py` (Lemmas 2.1, 2.2, 2.4, 2.5(a) on all subgroups of GL(2,q),
  q <= 19); comparisons with the programs of `original/` (identical lists) and re-runs of them (identical
  outputs).
- **Third verification run, on the final text** (`independent_run_2/`, own code): `thmA_dfs.py` tests
  **Theorem A itself, without the reduction and without any list of maximal subgroups**: for one t in every
  orbit of Q on T \ {1} and every class of subgroups R of Q_t with |R|k > M, an exhaustive search for all
  S <= P with S ∩ Q = R and |S| > M. q = 4, 5, 7, 8, 9: none (65, 41, 91, 487, 191 pairs (t, R)); with the
  threshold M − 1 the good subgroups of order M are found. `values.py` (Table 1; c_q, M, MinSubDeg(H) for
  4 <= q <= 49); `plane_lemmas.py` (Lemmas 2.1, 2.2 for q <= 32; Lemma 3.1, Corollary 3.2, Lemma 3.3,
  Lemma 5.1 for q = 4, 5, 7, 8, 9); `tight_cases.py` (the two cases of Lemma 4.1 used in Proposition 4.2,
  without Input R2: q = 25, 64, 121 and q = 16); `ineq_sweep2.py` (the sweep again, with the orders of
  Alavi–Burness, Lemmas 4.10, 4.11, 4.13); `lattice_selftest.py` (its subgroup program against brute force).
  No failure.
- **Coverage.** Theorem A as a whole, and so also the reduction, is tested for q = 4, 5, 7, 8, 9 (third run);
  for q = 4 and q = 5 the complete enumerations of the earlier note cover all subgroups. The parabolic part of
  Theorem A (Propositions 5.2, 6.1, Corollary 6.2) is tested by at least two independent programs for
  q = 4, 5, 7, 8, 9, 11, 13, 16, 17 and by one for q = 19, 23, 25. For q >= 11 the reduction (Proposition 4.2)
  is not tested by computation beyond its arithmetic, and the completeness of Input R1 is not tested.
- **Re-runs.** On 2026-10-14 `run_quick.sh` was run from a copy of the package and again from the extracted
  source archive: 31 recorded outputs reproduced each time, 128 s. The third verification run ran it again from
  the extracted archive of the version before its corrections (31 identical, 132 s), and ran
  `independent_run_2/run_all.sh` from the extracted archive of this version (`reproducibility/RERUN_LOG.txt`).

## Independent verification runs
The results were first found by computation for q <= 9 and then written up with complete proofs (the first
written version). Two independent verification runs, both AI-assisted, examined that version on 2026-10-14;
neither saw the other. Run A checked the proofs and the sources; run B the computations and the literature.

| Item | Run A | Run B |
|---|---|---|
| Setting, definition "good at d", criterion (Lemma 1.3 = Lemma 4.3 of the 2006 paper / Lemma 2.3 of the earlier note), M = q^2(q − 1)c_q, Table 1 | CORRECT (re-derived; compared with the 2006 paper and the earlier note) | Table 1 recomputed for q <= 19 |
| Lemmas on PGL(2,q) (Section 2) | PROVED (re-derived); Dickson's list for odd q compared with COT Theorem 2; for even q not checked against Huppert (standard) | CHECKED by computation for q <= 19; the source first cited for the centralisers covers only fixed points |
| Perspectivities and duality (Section 3) | PROVED | PROVED (read) |
| Input R1 | compared entry by entry with Liu–Yang Lemma 2.2; independently, the part used follows from Alavi–Burness (Theorem 4, Proposition 4.7, Table 7); BHRD not consulted | compared with Liu–Yang Lemma 2.2, Gill Theorem 19, and the online ATLAS for q = 5, 7, 8; BHRD not consulted |
| Lemma 4.1 (normalisers), Proposition 4.2 (reduction), Lemma 4.4 | CORRECT given R1, R2 (every case and every inequality re-done; sweep up to 3·10^6) | PROVED modulo R1, R2; orders of non-parabolic members computed for q <= 25 |
| Propositions 5.2, 6.1, Corollary 6.2 (parabolic part) | PROVED for all q >= 4; computation q <= 17 | PROVED (read); computation q <= 25 |
| Theorem B (primitivity, values) | CORRECT | CORRECT |
| Literature: is the bound known to be attained? | — | no prior construction found; a negative search |
| Overall | complete proof for all q >= 4, given the cited results; fixes editorial | CONFIRMED_WITH_FIXES |

No run found a mathematical error.

**Corrections required by runs A and B**, all applied:
1. (A, B) Input R1: cited as read in the transcription of Liu and Yang (Lemma 2.2, which states that it is taken
   from pp. 378–379 of BHRD); no BHRD table number is given, since the book was not consulted; Gill's Theorem 19
   and the online ATLAS (q = 5, 7, 8) as cross-checks; Alavi–Burness added as an independent published source
   for the part that is used (Remark 4.3).
2. (A) The alternative "F empty and two fixed lines" in the first case of the reduction cannot occur; it is
   removed (two fixed lines meet in a fixed point).
3. (A) Dickson's list: the types are said to overlap; only "at least one type" is used.
4. (A, B) The accounts of the computations and of the caveats are updated: independent checks exist; the
   reduction is not tested by computation for q >= 7 beyond its arithmetic.
5. (B) The centraliser lemma (Lemma 2.1) is proved in full; the source first cited (COT Theorem 1) is not used
   for it.
6. (B) A count "7 / 7" for q = 7 in a summary of the first computations is corrected to 9 (the note takes all
   counts from the recorded outputs).
7. (A, B; task) The automorphism groups of PSL(3,q0), PSU(3,q0) and Dickson's list for even q are cited as
   standard results, with the statements used, and marked as not re-proved (Input R2, Lemma 2.3).

**Changes made when the note was written, after the runs.** The first case of the proof of Proposition 4.2 was
shortened; Lemma 4.1 was restricted to the two cases that are used (q0 ≡ 2 resp. 1 (mod 3)), where the groups
have trivial centre; the non-extension argument in the proof of Theorem B uses core-freeness and the simplicity
of PSL(3,q) instead of the order of Aut(PSL(2,q)); the normaliser argument in the unitary case is phrased with
simultaneous eigenvectors of the diagonal torus; the program `check_note.py` was written and its lists were
compared with those of run B.

### Third verification run (on the final text, 2026-10-14)
A third verification run, also AI-assisted, examined the final text, and the four changes just listed first.

| Area | What it checked | Verdict |
|---|---|---|
| Statement fidelity | re-read the Oberwolfach abstract (pp. 690–693), Sections 1, 2 and 4 (to Lemma 4.5) of the 2006 paper and the cited parts of the earlier note (every lemma, proposition, theorem and table number cited); the question is answered as posed (each G_q is a negative answer to Question 1.6; infinitely many groups answer M. Giudici's question); definitions and scope as in the sources | PASS |
| Proofs | every lemma, proposition and theorem re-derived, including all cases of Proposition 4.2, Lemma 4.1 (trace and eigenvector arguments), Proposition 6.1 (all types of Lemma 2.3, the case q = 9) and the primitivity argument of Theorem B | PASS, no gap |
| Cited results | Input R1 against Liu–Yang Lemma 2.2 (row by row) and Gill Theorem 19; Remark 4.3 against Alavi–Burness (Theorem 4, Proposition 4.7, Lemmas 4.10, 4.11, 4.13, Table 7); Lemma 2.3 for odd q against Cameron–Omidi–Tayfeh-Rezaie Theorem 2 (q odd in that paper); Input R2 and Dickson for even q cited, not read | PASS |
| Computations | own programs (`independent_run_2/`, above): Theorem A for q = 4, 5, 7, 8, 9 without the reduction, with controls; Table 1; Lemmas 2.1, 2.2, Section 3, Lemma 5.1; the tight cases without Input R2; the sweep. The package from the extracted archive: 31 identical. The counts of Table 2 against the outputs of run B | PASS |
| Novelty and credit | arXiv API, OpenAlex (citing works of the 2006 paper: the same four), Crossref, zbMATH Open, one web search: nothing new; credits complete; the novelty claim is not stronger than the search | PASS |
| Presentation | house style; abstract; all DOIs (17 with Crossref, 2 with DataCite); build log (no overfull box); every page rendered and viewed | PASS with fixes |

It found no mathematical error. **Corrections required by the third run**, all applied:
1. The paragraph "Verification" and Section 8 describe its checks (a paragraph on `independent_run_2/`, the mark
   C in Table 2, and the coverage statement: Theorem A as a whole is tested for q = 4, 5, 7, 8, 9).
2. The description of run B said that the threshold-239 run for q = 4 "reproduces the twelve parabolic rows" of
   Table 2 of the earlier note; it finds twelve subgroups, six in each parabolic subgroup, which give the eleven
   parabolic rows of that table (the stabiliser of a flag lies in both). Corrected in the note, in this report
   and in the README.
3. Section 2 of the 2006 paper (cited in Section 1.1 of the note) and the Oberwolfach abstract are now listed
   among what was read, and the searches of the third run are recorded.
4. The package contains the programs and outputs of the third run (`independent_run_2/`), and the README and
   `RERUN_LOG.txt` describe them.

## Relation to the literature, novelty and scope
- **Credits.** Bounds, Question 1.6 and the method of Section 4 of the 2006 paper (Construction 4.1, Lemmas
  4.2–4.5): Giudici, Li, Praeger, Seress and Trofimov; Question 4: Praeger and Seress (2005). Example q = 4
  and computed example q = 5: the earlier note. Subgroup lists: Mitchell (1911), Hartley (1925), Bray–Holt–
  Roney-Dougal (2013, as transcribed), Dickson (1901). New, to our knowledge and relative to the literature
  found: the infinite family and its proof.
- **Searches (2026-10-14, run B; the OpenAlex list of citing works retrieved again when the note was
  written).** arXiv API, OpenAlex, zbMATH Open, Semantic Scholar (minimal subdegrees; twisted wreath products;
  works citing the 2006 paper: Giudici–Li–Praeger–Seress–Trofimov 2007 (not accessible), Bamberg–Giudici–
  Liebeck–Praeger–Saxl 2013 (abstract), Chua–Giudici–Morgan 2019 (full text searched), Burness–Shalev 2022
  (abstract), and the Oberwolfach abstract); three web searches. The third verification run repeated the
  search on the same day: arXiv API (no abstract with "minimal subdegree(s)"; the 23 abstracts with
  "subdegree(s)" and the 9 with "twisted wreath" contain nothing on the question), OpenAlex (the same four works
  citing the 2006 paper; searches for "minimal subdegree" and "twisted wreath" with "subdegree"), Crossref,
  zbMATH Open, and one web search. No construction of an infinite family attaining the bound, and no statement
  that the bound is attained, was found. A search that finds nothing is not a proof of novelty; no priority is
  claimed.
- **What was read.** The Oberwolfach abstract; the 2006 paper (Sections 1, 2 and 4 up to Lemma 4.5); Liu–Yang
  Section 2; Gill Section 3.1;
  Alavi–Burness (arXiv v2) Sections 1 and 4.2–4.4; Cameron–Omidi–Tayfeh-Rezaie Section 3; ATLAS pages of
  L3(5), L3(7), L3(8). Not read: Bray–Holt–Roney-Dougal, Mitchell, Hartley, Dickson, Huppert, Kleidman–Liebeck,
  Aschbacher, Schreier–van der Waerden, Steinberg. All DOIs were checked with Crossref (the Zenodo DOIs with
  DataCite) on 2026-10-14, and again by the third verification run.
- **Caveats.** The proof depends on the completeness of the list of maximal subgroups of SL(3,q) (Input R1),
  read through a transcription (the part used is also given by Alavi–Burness), and on Input R2 in the two tight
  cases q = q0^2 with q0 ≡ 2 or 1 (mod 3) (for example q = 25 and q = 16). For q = 4 the result rests on the
  earlier note. The journal version of Alavi–Burness was not compared with the arXiv version. These
  dependences concern the written proof; as a test, Theorem A was also confirmed for q = 4, 5, 7, 8, 9 by an
  exhaustive computation that uses neither input, and the tight cases for q = 16, 25, 64, 121 by computations
  that do not use Input R2.

## 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/