# Verification report — KOU-20.52 (MacHale: is every positive multiple 16m equal to |G| − k(G) for a non-abelian group G of odd order?)

Verification date: 2026-10-27.

**Verdict.** The note gives a **partial answer**; the question itself is **not decided**.
- **Proved by hand.** The reduction to m = 2^j (r(A × G) = |A| r(G) for abelian A); explicit groups G_j with
  |G_j| − k(G_j) = 2^(j+4) for every 0 ≤ j ≤ 13, so every m < 16384 and every m with 2-adic valuation at most 13
  occurs (Theorem 1.2); the class-number formula for semidirect products of abelian groups by groups acting through
  cyclic quotients (Theorem 1.4); Lemma 2.7 on groups with an abelian normal subgroup of prime index.
- **Computer-assisted, with exact certificates.** For every 15 ≤ j ≤ 120 except j ∈ {34, 64} an explicit group G of odd
  order with r(G) = 2^(j+4) (Theorem 1.3; 104 certificates, plus 14 for j ≤ 13, in `certificates.json`).
- **Negative evidence for m = 2^14** (Proposition 1.6): no group of odd order with r(G) = 2^18 lies in the named
  families (a)–(e); the question for m = 2^14 = 16384 (and for 2^34, 2^64 among j ≤ 120) stays open.

## Statement checked

- Corpus record KOU-20.52 (UnsolvedMath, dataset release v1.8.0): the statement agrees with Problem 20.52 of the
  Kourovka Notebook (D. MacHale, 20th issue, 2022), read in version 49 of arXiv:1401.0300 (20 October 2026), where the
  problem has no mark and no comment.
- The question is Problem 5 of S. M. Buckley and D. MacHale, Conjugate deficiency in finite groups, Irish Math. Soc.
  Bull. 71 (2013) 13–19 (doi:10.33232/BIMS.0071.13.19), with k = j + 4; they settled 0 ≤ j ≤ 8 by groups of orders 21,
  39, 75, 147, 291, 579, 1161, 2307, 4221 and proved |G| ≤ 27 r(G)/16 for non-abelian G of odd order. This paper was
  read in full; the note credits it for the reduction, the bound and the cases j ≤ 8.
- Both directions of the reduction to m = 2^j were re-derived: a group with r(G) = 2^(j+4) gives r(G × C_c) = 16m for
  m = 2^j c (c odd), and every m has this form.

## Results in the paper

- Section 2: Lemma 2.1 (Burnside, Hirsch: 16 | r(G), and 48 | r(G) if 3 ∤ |G|); Lemma 2.2 (k(A ⋊ H) by commuting
  pairs, Clifford and Gallagher); the proof of Theorem 1.4; Lemma 2.3 (fixed-point-free modules of every o-size, and
  the converse); Lemmas 2.4–2.6 (Frobenius groups A ⋊ C_r, the type I factor F_I(X) with k = X + 8, coprime actions
  of prime order via Brauer's permutation lemma and Glauberman's lemma); Lemma 2.7.
- Section 3: Table 1 (G_0, …, G_13).
- Section 4: the four kinds of certificates and their formulas; Appendix A lists all certificates for 15 ≤ j ≤ 120.
- Section 5: Proposition 1.6 and its proof.

## Computations (sanity checks; programs and outputs in reproducibility/)

- `certificates.json` (one certificate for each of the 118 exponents j ≤ 120 outside {14, 34, 64}).
- `check_certificates.py`: from the JSON alone, the family formula, the identity |G| − k(G) = 2^(j+4), the
  factorizations, the norm and size conditions, and a primality proof for each of the 242 distinct prime factors
  (trial division below 10^6, otherwise Pocklington's criterion with p − 1 completely factored, recursively; 247 primes
  above 10^6 are proved in the recursion, the largest with 33 digits). Result: 118/118 OK.
- `check_certificates.g` (data `certificates_gap.g`, made by `gen_certificates_gap.py`): the class numbers of the named
  groups, the automorphisms of K with the stated k(C_K(s)), and the kernels with the stated indices, class numbers and
  intersections, plus small instances of the type I factor and of A ⋊ C_5. Result: 31/31 true.
- Direct class counts in GAP: the fourteen groups of Table 1 (`verify_j_le_13.g`); the group of order 1064583 for
  j = 16 (both extraspecial groups of order 243; k = 16007, `k_verify_j16.g`); small instances of every group H of type
  (iv) with the same kernels (`k_verify_bilin*.g`, `k_verify_271.g`).

## Independent verification runs

Two independent verification runs (AI-assisted, each with its own programs, neither reading the other's report)
examined the paper and its files on 27 October 2026 (their reports use an earlier numbering: their Theorems 1.1,
1.2, 1.3, Proposition 1.4 and Lemma 2.8 are Theorems 1.2, 1.3, 1.4, Proposition 1.6 and Lemma 2.7 of the note).

**Run A (statement and proofs).** It re-derived the reduction, all results of Section 2 including Theorem 1.4, the two
examples (3^(1+2), 3^(1+4)) and Table 1, re-ran the fourteen groups of Table 1 in GAP, re-checked all 118 certificates
with its own checker (its own recursive Lucas/Pocklington proofs; type counts 67/14/2/21 for 15 ≤ j ≤ 120), re-checked
the 31 group-data items in GAP with its own code, confirmed Theorem 1.4 by direct class counts in 25 built groups
(among them the group for j = 16), and checked every sentence about the sources. Verdict: theorems correct; the
written proof and the coverage of Proposition 1.6 had five gaps, each closed by its own computation or argument, none
giving a solution.

**Run B (independent computation and literature).** It wrote its own certificate checker (118/118), its own
recursive Pocklington prover (all 242 distinct prime factors), its own GAP checks (31/31), counted classes of 7
certificate groups (j = 15..21) with a formula-free orbit counter (j = 15, 16, 19 also in GAP), checked small instances
with the exact kernel data of all type (iii) and (iv) certificates, rejected 13 deliberately wrong certificates, re-ran
the paper's checkers from copies (identical output), re-ran Proposition 1.6(a) completely and spot-checked (b)–(e).
Verdict: Theorems 1.2–1.4 and Lemma 2.7 proved or confirmed; Proposition 1.6 true as far as checked, with two
computational gaps in its proof, closed by its own runs.

**Gaps found and how they were corrected (no new case gives a solution):**
1. Proposition 1.6(a): the search windows for F_I(X) omitted the term 8 in k(F_I(X)) = X + 8; corrected windows
   2^18/(9X) < |P| ≤ 2^18/(9X − 11(X + 8)/27). The corrected search covers 4264 non-abelian library groups (1617 pairs
   (|P|, k(P))) and 56 orders outside the library, of which again only 10125, 13419, 13689 admit an integral k(P).
2. Proposition 1.6(a), order 13419 = 3^3·7·71: the case of a cyclic normal subgroup of order 497 was missing (then the
   Sylow 71-subgroup is central and 71 | k(P), while 3931 is required); added.
3. Proposition 1.6(c), t = 3: the list of size multisets was incomplete (11 is a 5-size, 23 an 11-size). The complete list
   is {7,7,7}, {7,7,11}, {7,7,13}, {7,7,19}, {7,7,23}, {7,11,11}, {7,11,13}; the four new multisets need a quotient of order
   5 or 11 and were searched (167 groups, 924 choices). Both runs also searched them with their own code.
4. Proposition 1.6(c), t ≤ 2: the data of pairs of kernels does not contain the groups whose H/H' has prime order (one
   kernel); the 179 such groups of order ≤ 2187 were searched for j = 14, 34, 64 (one candidate for j = 64, not an
   admissible size).
5. Proposition 1.6(b): the exhaustive searches with C_3^2 and C_3^3 for j = 14 had not been archived; re-run and
   archived (`cover_elemH2_k2_j14.out`, `cover_elemH2_k3_j14.out`).
6. Smaller corrections: the converse of Lemma 2.3 is now stated and proved; the wording of the module argument in
   (b); the counts of primes (242 distinct factors; 247 proved in the recursion) and of the cases (o_1, o_2, h)
   (17363, of which 176 leave room for admissible sizes); a remark that Theorem 1.4 does not use the cyclicity of H/K_i;
   the formulation of Theorem 1.3 as 15 ≤ j ≤ 120, j ∉ {34, 64}.

## Relation to the literature, novelty and scope

- Searches (arXiv export API, zbMATH Open, OpenAlex including the works citing Buckley–MacHale, Semantic Scholar where
  it answered, Crossref for the references) found no work on Problem 5 of Buckley–MacHale or on Problem 20.52 beyond
  that paper; the works citing it (Degree sum deficiency in finite groups, 2015; Counting commutativities in finite
  algebraic systems, 2016; the solution of Kourovka 21.88, 2026) treat other questions. A search that finds nothing is
  not a proof of novelty; no priority is claimed.
- Theorem 1.4 is, in substance, a direct consequence of the Clifford–Gallagher orbit count; the note does not claim
  it as deep, only that it was not found in this form.
- The result is partial: the cases j = 14, 34, 64 (j ≤ 120) and all j > 120 are open. Proposition 1.6 is negative
  evidence for named families only; a search that finds nothing outside its stated scope proves nothing.

## Public release and license

Unrefereed preprint, CC BY 4.0. The source archive contains the paper sources, this report, and in
`reproducibility/` the programs and outputs of the note (`original/`) and of the two verification runs
(`verification_run_A/`, `verification_run_B/`); a private copy of the Buckley–MacHale paper used for reading is not
included.