# Verification report — KOU-21.23 (Bubboloni–Fumagalli–Praeger: chordal enhanced power graphs of finite simple groups)

Verification date: 2026-10-27 (runs A and B on the first write-up; the second verification run on the final text).

**Verdict.** The note gives a **partial answer**; every statement it makes is proved (some by computer; for n = 4
some assuming published lists of maximal subgroups), and the scope is as follows.
- **Part (b), conditional.** E(G) is chordal for PSL2(q) (q ≥ 4), Sz(q), PSL3(4), A7 (Theorem 1.1, known:
  Bubboloni, Fumagalli, Praeger), and has an induced cycle of length ≥ 4 (a hole) for M11, J1, 2F4(2)′, PSL3(q)
  with q odd, PSU3(q) with q ≥ 3, and 2G2(q) with q ≥ 27 (Theorem 1.2). Hence, **assuming the upper bound of
  [BFP, Theorem B]** (every simple group outside an explicit list has an induced 4-cycle), E(G) is chordal if and
  only if G is PSL2(q) (q ≥ 4), Sz(q), PSL3(4) or A7 (Corollary 1.3).
- **Part (a) for n = 4, conditional.** E(2F4(2)′) is C4-free (computer-assisted); E(PSL3(p)) and E(PSU3(p)) are
  C4-free for the primes p with p − 1, resp. p + 1, equal to 2^a or 3·2^a, by a proof that uses the lists of
  maximal subgroups of Bray, Holt and Roney-Dougal (verified independently of these lists by exhaustive computation
  for small p); induced 4-cycles exist for PSU3(q) with q even ≥ 4 or q + 1 not of that form, and for 2G2(q),
  q ≥ 27 (Theorem 1.4). Assuming the upper bound of [BFP, Theorem B] and the BHRD lists, this gives the set S4
  (Corollary 1.5).
- **Part (a) for n ≥ 5: open.** Only the least hole lengths of some small groups are determined (Proposition 1.6:
  10 for M11; 6 for J1, 2F4(2)′, PSL3(q), PSU3(q) with q = 3, 5, 7).
- **Not new.** Theorem 1.1 (for PSL2(q) through Cameron's result that E(PSL2(q)) is a cograph, as stated in [BFP]),
  the tools of Section 2 ([BFP] Lemmas 3.2, 4.9(ii), 4.12, 4.13, Proposition 4.14, Corollaries 4.15, 4.16), the
  C4-freeness of E(M11), E(J1), and the C4's in PSL3(q) for q ∉ {2, 4} ∪ P_L ([BFP] Lemma 6.5(b)); the elements of
  Section 3.4 are those of the proof of [BFP] Lemma 6.6.

The note is unrefereed; the upper bound of [BFP, Theorem B] that it assumes is proved in an unrefereed preprint.

## Statement checked
- **Primary source.** Kourovka Notebook No. 21, arXiv:1401.0300v49 (20 October 2026), Problem 21.23, proposed by
  D. Bubboloni, F. Fumagalli, C. E. Praeger; no star. (a) For a given n ≥ 4, determine the finite nonabelian simple
  groups G such that E(G) has no induced cycle with n vertices. (b) Determine those with E(G) chordal.
  E(G): vertex set G, edges {x, y} for x ≠ y with ⟨x, y⟩ cyclic.
- **Source of the question and of the known results.** D. Bubboloni, F. Fumagalli, C. E. Praeger, "Enhanced power
  graphs of finite groups with cograph structure", arXiv:2510.18073v1 (20 October 2025; the only arXiv version on
  2026-10-27), Open Problem 1, Theorems A and B. Read in full. All statements used, with page and line, are listed
  in audit/source_check.md (not part of the release package).
- **Corpus record.** ulamai/UnsolvedMath, KOU-21.23 (dataset version 1.8.0; status partially_solved); its statement
  is the v49 text.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (b): which simple G have E(G) chordal | PSL2(q) (q ≥ 4), Sz(q), PSL3(4), A7, assuming the upper bound of [BFP, Theorem B] | Corollary 1.3 |
| (b) without that assumption | chordal for these groups; a hole in every other group of the upper list of [BFP] | Theorems 1.1, 1.2 |
| (a) for n = 4 | S4 = L1 ∪ {A7, M11, J1, 2F4(2)′} ∪ {PSL3(p) : p ∈ P_L} ∪ {PSU3(p) : p ∈ P_U}, assuming [BFP, Theorem B] and the BHRD lists | Corollary 1.5 |
| (a) for n ≥ 5 | open; least hole lengths for M11 (10) and J1, 2F4(2)′, PSL3(q), PSU3(q) (q = 3, 5, 7) (6) | Proposition 1.6 |
| "Hole" | induced cycle of length ≥ 4; cycles with chords do not count | Section 1.1 |
| Groups up to isomorphism; 2G2(3), 2F4(2) not simple; 2F4(2)′ is the Tits group | as in [BFP] | Section 1.1 |

## Results in the paper
- **Lemmas 2.1–2.5** (tools): Lemma 2.2 = [BFP] Lemma 4.13 (shortest hole with prime-power vertices, coprime
  consecutive orders), Lemma 2.3 = [BFP] Prop. 4.14 (C4 criterion; the sufficiency does not need prime powers),
  Lemma 2.4 = [BFP] Lemma 3.2 and Cor. 4.15, Lemma 2.5 (scalars and unipotent elements). All re-proved.
- **Theorem 1.1** via Lemma 3.1 (a partition into cyclic subgroups and p-groups gives chordality) and the
  partitions of PSL2(q), Sz(q); PSL3(4) (all elements of prime-power order); A7 ([BFP] Prop. 6.1).
- **Theorem 1.2**: explicit induced 6-cycles in ASL(2,q) ≤ PSL3(q) (point reflections and shears) and in PSU3(q),
  q odd (involutions t_P and unitary transvections); an induced C4 in PSU3(q), q even ≥ 4; Proposition 3.2: an
  induced C4 (e1, J, e2, J′) in 2G2(q), q ≥ 27, from C(J) = ⟨J⟩ × PSL2(q) (Ward 1966); certificates for M11
  (10-cycle), J1 (6-cycle with orders 3,5,2,5,3,2) and 2F4(2)′ (6-cycle in PSL3(3)).
- **Theorem 1.4**: Lemma 4.1 (tori: C2^2 × C_r^2), the non-prime PSL3(q) by [BFP] Lemma 6.5(b) (including q = 9, 25,
  49), Lemmas 4.2–4.6 (cyclic normal subgroup of prime index; Borel type; two-dimensional type; monomial type;
  affine type), Table 1 of maximal subgroups, Theorem 4.7.
- **Section 5**: the proof of the upper bound of [BFP, Theorem B] re-checked family by family (Table 2); four
  neutral remarks on [BFP] (the 3D4(2) step needs q ≥ 4, and the steps for F4(2), E6(2), 2E6(2), E7(2), E8(2) and
  the Table 1 rows of Th and M pass through 3D4(2); all closed by S3 × L2(8) ≤ 3D4(2); the subgroup A4 × C2^2
  listed for J1 in Table 1 fits J3, where an explicit C4 was found; the sentence on PSL3(3) in the 2F4(2)′ paragraph;
  "not C4-free" for M11, J1 on p. 29 should read "C4-free").
- **Lemma 6.1** (parity): if every edge of the prime graph contains 2, shortest holes are even; Proposition 1.6.

## Computations (programs and outputs in reproducibility/)
No theorem on an infinite family depends on a computation.
- **Programs of the first computation** (programs/, outputs in recorded/): the reduced graph Γ(G) on cyclic
  subgroups of prime-power order (epg_build.g, epg_analyze.py: C4 test, exact least hole length, chordality, every
  hole re-checked in GAP); the uniform constructions for 16 + 11 + 4 values of q; A7, A8, M11, J1, PΓL(2,8),
  PSL3(3), PSU3(3); the maximal-subgroup method (Lemma 2.4(b)) for PSL3(p), p = 3, 5, 7, 13, PSU3(p), p = 3, 5, 7,
  11, 23, M11, J1, 2F4(2)′; J3, Ree(27), twelve sporadic groups, 3D4(2); the CTblLib containments; the 56
  nonabelian simple groups of order < 10^6 (chordal exactly for the 39 PSL2(q), A7, PSL3(4), Sz(8)).
- **Programs written with the note**: check_note.g (re-checks every stored certificate pair by pair, the small
  cases of the constructions, Proposition 3.2 for q = 27 with membership of the stored 7 × 7 matrices in
  ReeGroup(27), the C4's of 3D4(2) and M12, the prime graphs; about 13 s; CHECK_NOTE ALL OK: true) and
  check_c4_direct.g (exhaustive whole-group search of Lemma 2.3(b) for 2F4(2)′, M11, J1, PSL3(p) for p = 3, 5, 7,
  13, 17 and PSU3(p) for p = 3, 5, 7, 11, 23, with controls; no C4 found; 15.5 minutes, of which 883 s for
  PSU3(23); CHECK_C4_DIRECT ALL AS EXPECTED: true).
- **Programs of the second verification run** (independent_run_2/, with a README): a GAP-independent computation
  of Γ(G) and of the least hole lengths (r2_gamma.py), a whole-group C4 search organised differently from
  check_c4_direct.g (r2_direct_c4.g), the constructions of the note rebuilt from its text, the stored certificates
  in rebuilt groups, the prime graphs, the remarks of Section 5 and CTblLib containments (r2_checks.g,
  r2_addendum.g); all outputs agree with the note.
- **Re-runs.** On 2026-10-27, from a copy of the package, all programs of the first computation were run again;
  the summary results (vertex counts, chordality, C4, least hole lengths, "ALL ... true" lines) are identical to
  the recorded ones; cert_c4free3.g (PSU3(23) through the ClassicalMaximals list) stopped with a GAP error inside
  the randomised SmallerDegreePermutationRepresentation with GAP's default seed, and reproduced the recorded results
  with another seed; PSU3(23) is also covered without maximal subgroups by check_c4_direct.g. A re-run may print a different but equally valid hole (observed for J1 and PSU3(3)),
  because GAP's class representatives can differ between runs. Details in reproducibility/README.md and
  reproducibility/RERUN_LOG.txt.

## Independent verification runs A and B (on the first write-up)
The results were first obtained, with proofs, in a written-up form (not this text). Two independent verification
runs, A and B, both AI-assisted, followed; each re-derived the proofs and wrote its own programs. Their programs and
recorded outputs are in reproducibility/verification_run_A/ and verification_run_B/ (absolute local paths replaced
by placeholders). A second verification run examined the final text; see the next section.

| Item | Run A | Run B |
|---|---|---|
| Statement, definitions, quoted results of [BFP] | CONFIRMED (against the text) | CONFIRMED; Theorem B, Prop. 4.14, Cor. 4.15, 4.16, Lemma 6.7 against the arXiv HTML |
| Lemmas 2.1–2.5, Lemma 3.1, Theorem 1.1 | CONFIRMED (re-derived) | CONFIRMED (re-derived; Sz(32) and all 56 groups < 10^6 by computation) |
| Constructions for PSL3(q), PSU3(q) (q odd), PSU3(q) (q even) | CONFIRMED for all q (re-derived) | CONFIRMED; rebuilt in other coordinates for q ≤ 27, even q ≤ 64 |
| Proposition 3.2 (Ree) | CONFIRMED modulo Ward; q = 27 for all 26 cosets | CONFIRMED; own C4 in Ree(27) |
| M11, J1, 2F4(2)′ holes; least hole lengths | certificates re-verified | CONFIRMED with own Γ(G) programs; least hole 6 also for PSL3(7), PSU3(7), and 6 for 2F4(2)′ by parity |
| Lemma 4.1, [BFP] Lemma 6.5(b) | CONFIRMED (re-derived) | CONFIRMED; C4's found in 13 predicted groups PSL3(q), PSU3(q), q ≤ 29 |
| Lemmas 4.2–4.6, Theorem 4.7 | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| BHRD lists | compared with the ClassicalMaximals source; printed tables not available | ClassicalMaximals lists for p ≤ 23 and p = 17, 31 (partly) |
| C4-freeness by whole-group search | 2F4(2)′, A7, PSL3(3), PSU3(3), M11, PΓL(2,8), PSL3(5), PSU3(5), J1, PSL3(7), PSU3(7): no C4; A8, M12, PSU3(4): C4 | 2F4(2)′, PSL3(p) (p = 3, 5, 7, 13, 17), PSU3(p) (p = 3, 5, 7, 11): no C4; PSU3(23), PSL3(17) also by maximal subgroups |
| Re-check of the upper bound of [BFP, Theorem B] | CONFIRMED; 3D4(2) gap and J1/J3 slip confirmed | spot checks on 13 further groups; 3D4(2), J3 recomputed |
| Literature | Crossref data for Ward, Kleidman | arXiv, OpenAlex, zbMATH Open, Crossref, one web search: nothing found that answers the question |

Neither run found a mathematical error.

**Corrections required by run A**, all applied (details in audit/fixes_applied.md, not part of the package):
1. the exact external inputs of Corollaries 1.3 and 1.5 (CFSG, Liebeck–Saxl–Seitz 1992 Table 5.1, Craven 2023,
   Kleidman–Liebeck 1990 Lemma 4.1.1, ATLAS containments) are stated (Section 1.3, Section 5);
2. Lemma 2.3(a) without prime-power orders; PSL3(9), PSL3(25) (and PSL3(49)) are excluded by [BFP] Lemma 6.5(b), not
   by the torus argument (Section 4.1);
3. the Borel torus acts by w^(2p−1) for y^(−1)uy (Section 4.2);
4. the maximal-subgroup references (BHRD Propositions 3.2.1–3.2.6, Theorem 4.10.2, Tables 8.3–8.6; Mitchell 1911,
   Hartley 1925, Bloom 1967), the cross-checks with ClassicalMaximals and the ATLAS, and the plain statement that
   the printed tables were not consulted (Section 1.3, Section 4.2, Scope and priority);
5. Ward's theorem cited with its DOI (Section 3.5);
6. the least hole length of 2F4(2)′ is 6 (Proposition 1.6).

**Corrections required by run B**, all applied:
1. the stored J1 hole has orders 3,5,2,5,3,2 (Section 3.6; asserted by check_note.g);
2. least hole length exactly 6 for 2F4(2)′ (so 2F4(2)′ ∈ S5), PSL3(7) and PSU3(7) (Proposition 1.6, Lemma 6.1);
3. the dependence of Theorem 1.4(b) on the BHRD lists is stated in the theorem, and the machine-checked primes are
   named;
4. the passages of [BFP] behind the remarks are quoted (each under 15 words) with page numbers (Section 5); all
   remarks are neutral and in the body, none in the abstract;
5. the ambient representation of each certificate is stated (Section 7; check_note.g).

**Changes made when the note was written.** Two programs were added (check_note.g, check_c4_direct.g); the
whole-group search was extended to PSU3(23); the count of sporadic groups treated by containments was corrected from
nine to ten; Kleidman 1988 is not cited (not read), and the inclusion Ree(3) ≤ Ree(q), not used in any proof, is
justified by the fixed-point argument and checked for q = 27.

## Second verification run (on the final text)
A second verification run, AI-assisted and independent of runs A and B, examined the text of the note (27 October
2026). Its programs and outputs are in reproducibility/independent_run_2/ (with a README); none of them reads or
calls a program of the note or of runs A and B.

| Item | Result |
|---|---|
| Statement, definitions, scope | CONFIRMED against the Notebook (v49), the corpus record and [BFP] v1 (Theorems A, B, Open Problem 1, Definitions 2.1, 3.1, Lemmas 3.2, 4.9, 4.12, 4.13, Prop. 4.14, Cor. 4.15, 4.16, Prop. 6.1, Lemmas 6.3–6.7, the proof of Theorem B and Table 1); every theorem, hypothesis and list of groups and primes compared with the first write-up |
| Proofs | all re-derived (Sections 2–6, including the passage to the projective groups in Sections 3.2–3.4, Proposition 3.2, Lemmas 4.1–4.6, Theorem 4.7); no mathematical error |
| Least hole lengths, without GAP (r2_gamma.py) | A7 chordal; S6 6; M11 10; PΓL(2,8) 10; PSL3(3), PSU3(3), PSL3(5), J1 (from Janko's matrices over GF(11)) 6; A8 C4; vertex and edge counts as in the first computation; the M11 hole printed in the note is an induced 10-cycle |
| Whole-group C4 search of its own (r2_direct_c4.g) | none in M11, J1, 2F4(2)′ (primitive action on 1755 points), PSL3(p) for p = 3, 5, 7, 13, 17, PSU3(p) for p = 3, 5, 7, 11, 23, A7, S6, PΓL(2,8); a C4 in A8, M12, PSU3(4), PSL3(9), PSU3(9), PSL3(11) |
| Constructions rebuilt from the text (r2_checks.g) | Section 3.2 for 8, Section 3.3 for 6, Section 3.4 for 3 values of q, [BFP] Lemma 6.5(b) for q = 8, 9, 16, 25: all induced cycles in the projective groups |
| Other recomputations (r2_checks.g, r2_addendum.g) | prime graphs of Section 6; E(S3 × S3) C4-free; a C4 in S3 × L2(8) < 3D4(2); Ree(27): C(J) = ⟨J⟩ × L2(27), the stored 4-cycle, Ree(3) ≤ Ree(27); all stored certificates in rebuilt ambient groups; CTblLib containments of Section 5, including S6(2) < U6(2), S8 < S6(2), M22:2 < J4, 3D4(2):3 < F4(2) |
| Re-run of the package from the archive | all programs re-run from an extracted copy of source.zip: every recorded summary result obtained again (check_note.g identical; check_c4_direct.g with the same numbers of triples in all 14 exhaustive runs; cert_c4free3.g with the seed of RERUN_LOG.txt); only randomly chosen certificates differ, all re-checked (independent_run_2/RERUN_run2.txt) |
| Literature | arXiv API, Crossref, OpenAlex, zbMATH Open, one web search: nothing that answers the question; the AMS archive refused automated access, so Ward 1966, Mitchell 1911, Bloom 1967 were not read in this run either |

**Corrections required by the second run**, all applied:
1. remark (1) of Section 5: through the chain 3D4(q) < F4(q) < E6(q), 2E6(q) < E7(q), E8(q) of [BFP], the steps for
   F4(2), E6(2), 2E6(2), E7(2), E8(2) also rest on 3D4(2), and in [BFP, Table 1] the rows of Th and M (B also has
   the row A8); the subgroup S3 × L2(8) closes all of them (also Table 2);
2. the CTblLib confirmation of the containments over F2 (S6(2) < U6(2), S8 < S6(2)) is made by this run; J4 is
   covered through its maximal subgroup M22:2 (instead of 2^11:M24, whose splitness CTblLib does not show);
3. Proposition 1.6 states which values rest on computation besides those of M11 and J1;
4. the abstract no longer calls the cycles of all infinite families "explicit" (for 2G2(q) the 4-cycle is
   constructed from the involution centraliser);
5. credits: Cameron for the cograph property of E(PSL2(q)) (as stated in [BFP]); [BFP] Lemma 4.9(ii) for the last
   assertion of Lemma 2.2, [BFP] Lemma 4.12 = Aalipour et al. Lemma 32 for Lemma 2.1(c), [BFP] Lemma 6.6 for the
   elements of Section 3.4, [BFP] Theorem 5.3 and Lemmas 6.3, 6.4 as the model of Lemma 3.1;
6. the Verification paragraph (runs A and B, this run; no "hand proofs") and the Scope paragraph (this run's
   searches and sources);
7. three short additions: the simplicity of G in the proof of Lemma 2.4(b); the A7 argument in the proof of
   Theorem 1.1; the existence of β′ in Section 3.4; and 2G2(q) ∉ S10 at the end of Section 6.

## Relation to the literature, novelty and scope
- **Searches (27 October 2026).** arXiv API (enhanced power graphs with chordal, cograph, simple, perfect, hole,
  induced cycle, Ree, unitary, Tits, sporadic; the authors; "Kourovka" with 21.23), OpenAlex (no work citing
  arXiv:2510.18073), zbMATH Open, Crossref (no journal version of [BFP]), one web search (run B). Semantic Scholar
  refused automated access (HTTP 429) and was not used. The second verification run repeated the queries to the
  arXiv API, Crossref, OpenAlex and zbMATH Open and one web search on the same day, with the same outcome
  (reproducibility/independent_run_2/queries_run2.log).
  - No printed answer to the question was found in these sources; a search that finds nothing is not a proof of
    novelty.
  - Related but different: chordal *power* graphs of simple groups (Brachter–Kaja 2023; Manna–Cameron–Mehatari);
    van Doorn–Judin–Monticone–Morrison, arXiv:2607.17477, prove (according to the abstract) that a cograph power graph
    is chordal (Kourovka 21.24, not 21.23); Ma–Zahirović–Lv–She 2024 (forbidden subgraphs of enhanced power graphs,
    nilpotent groups).
- **What was read.** [BFP] in full (v1); the Notebook entry; the abstract of arXiv:2607.17477. Not read: the BHRD
  book, Mitchell, Hartley, Bloom, Liebeck–Saxl–Seitz, Craven, Kleidman–Liebeck, Ward, Levchuk–Nuzhin, the ATLAS;
  their bibliographic data were checked with Crossref; what the note uses from them is quoted in [BFP], implemented
  in ClassicalMaximals or CTblLib, or checked by computation for q = 27 (Ward). The second verification run could
  not read Ward, Mitchell or Bloom either (the publisher's archive refused automated access).
- **Scope.** Part (b) and part (a) for n = 4 are answered only conditionally (the upper bound of [BFP, Theorem B];
  for n = 4 also the BHRD lists); part (a) for n ≥ 5 is open. No priority is claimed.

## 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/