# Verification report — OWR-14299292-015 (Maglione: immediate descendants of the groups G_{E,P}(F_p))

Verification dates: 2026-10-27 (runs A and B on the first written version; the third run on the final text, UTC).
Report of 2026-10-28.

**Verdict.**
- **(Q1) answered.** For every prime p >= 5 and every pair (E, P), the group G_{E,P}(F_p) of Maglione and
  Stanojkovski has an immediate descendant of order p^10 (a central extension by Z/pZ of class 3 and order p^10)
  if and only if P is a point of order 2; otherwise it is terminal. For P of order 2 the nucleus has rank 2, there
  are descendants of orders p^10 and p^11 only, and both exponents p and p^2 occur at order p^10 (Theorem 1.1).
- **Counts for every group.** For P of order 2, the numbers d10 (all descendants of order p^10) and d10e (those of
  exponent p) are explicit functions of p, e3 = |E[3](F_p)| and s = |Stab(P)| (Theorem 1.2, Table 1); d10e =
  1 + |P^2(F_p)/Gamma|. This contains Theorem 1 of du Sautoy and Vaughan-Lee (J. Algebra 361 (2012)) as the case
  y^2 = x^3 - x, P = (0,0), and gives the number of all descendants of order p^10 of their group, which they proved
  not to be PORC but did not compute.
- **(Q2) answered in the source's sense.** The source defines an immediate descendant as a group of order p^10.
  The sum S10(p) over the isomorphism classes of G_p is PORC with modulus 72, and the exponent-p sum with modulus
  12, with explicit polynomials (Theorem 1.3). PORC is meant with rational coefficients (the usage of du Sautoy and
  Vaughan-Lee; the source's footnote does not specify; Maglione-Stanojkovski write Z[x]).
- **Open.** If all immediate descendants in O'Brien's sense are counted (orders p^10 and p^11), (Q2) remains open;
  the note records computed order-p^11 values only (Remark 6.6), reproduced by the third run's own programs, with
  nothing proved about them.

The note is unrefereed.

## Statement checked
- **Source.** Oberwolfach Report 27/2025, *Computational Group Theory*, doi:10.4171/OWR/2025/27: problem of
  J. Maglione in the problem session, pp. 1443-1444 (the question, with the footnote defining PORC), and Maglione's
  abstract (joint with M. Stanojkovski), pp. 1421-1422 (definitions of C_{E,P}, B_{E,P}, G_{E,P} and the set G_q).
  Read in the publisher's open file.
- **Definitions.** J. Maglione, M. Stanojkovski, *Smooth cuboids in group theory*, Algebra Number Theory 19 (2025)
  967-1006, doi:10.2140/ant.2025.19.967, read in the arXiv version v4 (arXiv:2212.03941; numbering used): Sections
  1, 2.4, 4, 5 (Theorems A, B, D, 2.13, 2.15, 5.6, 5.8 with proofs, Lemmas 5.4, 5.5, Corollaries 5.9, 5.10),
  Remark 2.10, Definition 3.2, Corollary 3.6, Conjecture 6.9. The journal version was not compared.
- **The question.** (Q1) For which pairs (E, P) does a central extension 1 -> Z/pZ -> G -> G_{E,P}(F_p) -> 1 with
  G of class 3 and order p^10 exist (the source calls such a G an immediate descendant)? (Q2) The source says it
  would be remarkable if the sum, over the groups of G_p up to isomorphism, of the numbers of immediate descendants
  were PORC.
- **Corpus record.** ulamai/UnsolvedMath v1.8.0, OWR-14299292-015; its statement is faithful to the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Q1 as posed (order p^10, class 3, central extension by Z/pZ) | exactly the pairs with P of order 2 | Theorem 1.1 |
| Q1 with O'Brien's immediate descendants (any order) | the same pairs; the others are terminal; descendants of orders p^10 and p^11 only | Theorem 1.1 |
| Q2 with the source's definition (order p^10, any exponent) | PORC, modulus 72, explicit polynomials | Theorem 1.3 |
| Q2 for descendants of order p^10 and exponent p | PORC, modulus 12 | Theorem 1.3 |
| Q2 with all immediate descendants in O'Brien's sense (orders p^10 and p^11) | open; order-p^11 data only | Remark 6.6 |
| PORC with integer coefficients (as written in Maglione-Stanojkovski, Section 1.4) | then neither the count of du Sautoy and Vaughan-Lee nor S10 would be PORC; the note uses rational coefficients | Section 1.1 |
| Groups over F_q with q = p^e, e > 1 | not treated (not asked) | Section 7 |

## Results in the paper
- **Lemma 2.1.** For H = G_{E,P}(F_p) the three notions of immediate descendant of order p^10 (the source's,
  O'Brien's, du Sautoy-Vaughan-Lee's) give the same groups; both exponents are allowed.
- **Lemmas 2.2-2.4.** The Lie p-covering ring L* = F_Z/([R,F_Z] + pR): graded pieces, multiplicator
  M = V + I + N, nucleus P_2(L*) = N = L_3(V)/[I,V]; under the Lazard correspondence L* is O'Brien's p-covering
  group and N its nucleus (Lemma 2.3); descendants of order p^10 = Aut(g)-orbits on hyperplanes of M not containing
  N, with the lifted automorphisms bijective (Lemma 2.4).
- **Lemmas 3.1-3.2, Remark 3.3.** N = (W (x) V)/phi(Lambda^3 V); the two 9 x 9 linear systems for N^* are written
  out: determinant 2 mu, and at mu = 0 the solutions are the multiples of an explicit matrix K_0, over every field
  of characteristic different from 2. The computer certificate (det phi1 = -2 mu, det phi2 = 2 mu, unit 8 x 8
  minors at mu = 0) is the same statement.
- **Lemmas 4.1-4.5.** Action of Aut(g) on M; tensor structure t = S (x) omega at mu = 0, the pairing kappa, N^* =
  {tau_xi}, the character chi; the structure of Psi from Maglione-Stanojkovski (Theorems 5.6, 5.8 and their proofs)
  with an explicit dictionary of conventions; lifts of the elements of Gamma = E[3](F_p) x| Stab(P).
- **Proposition 5.1, Lemma 5.2, Proposition 5.3.** The fibre over one nucleus functional and the fixed-point
  formula; a closed formula for the inner sum over t' (Lemma 5.2); the four contributions F_1, F_refl, F_transl,
  F_ord4 and their exponent-p analogues by written case analyses.
- **Theorem 1.2, Remark 5.4, Corollary 5.5.** The counts; d10e = 1 + |P^2(F_p)/Gamma|; the group of du Sautoy and
  Vaughan-Lee is G_{E,(0,0)}(F_p) for y^2 = x^3 - x (explicit isomorphism), its type for each residue class, V_p > 0
  iff e3 = 9, their Theorem 1 recovered, and d10 for their group (6780, 87328, 365098, 1451506 for p = 5, 7, 11, 13).
- **Lemmas 6.1-6.4, Theorem 1.3, Remark 6.5.** Classes of pairs with P of order 2 (2(p-1) + 2 delta_4 classes);
  Kubert's Tate normal form of level 6 (with proof); a cube criterion for full rational 3-torsion; the counts
  sum (e3 - 1) = (p-1)(p-3), #{e3 = 9} = delta_3 (p-1)(p-7)/24 and sum 1/e3 = (p-1) alpha(p); the totals S10 and
  S10e; the cancellation of the e3-dependence explained by genus-0 curves (X_1(6) and w^3 = 9c + 1; the Hesse
  pencil as the original route).
- **Remark 6.6.** Order-p^11 data for p = 5, 7, 11 (all values reproduced by the third run; nothing proved).

## Computations (programs and outputs in reproducibility/)
No proof depends on a computation, apart from expansions of polynomials, which the programs also check.
- **Program written with the note** (`check_note.py`, numpy and sympy, about 25 seconds; TOTAL failures: 0):
  (A) the certificate and the systems of Lemma 3.2 over Z[lambda, mu, a]; (B) dim N from the definition for all
  394 pairs with p = 5, 7; (C) Proposition 5.3 and Lemma 5.2 against the exact sums for all primes below 100;
  (D) Lemmas 6.1-6.4 by brute force for p < 60 and the cube criterion for all curves y^2 + a1 xy + a3 y = x^3 with
  p < 32; (E) Theorem 1.2 for all 22 classes with p = 5, 7 by a direct Burnside count on the 20-dimensional
  multiplicator, with an exhaustive search for Psi_S; (F) Theorem 1.3 and Table 1; (G) Corollary 5.5.
- **Programs with which the results were first obtained** (`original/`): dim N from the definition (tensor
  algebra) and GAP's p-quotient algorithm, without Lie theory, for all 3632 pairs (E, P) with p <= 13; direct
  counts on M for all 182 classes with P of order 2 and p <= 23; the closed forms for the 76 primes below 400;
  Lemma 6.4 for p <= 127; D_p of du Sautoy-Vaughan-Lee for p <= 13; the order-p^11 data; and a GAP pipeline without
  Lie theory (p-covering group, AutPGrp, lifts, Burnside) which confirmed d10 and d10e for 9 of the 10 classes at
  p = 5 (for y^2 = x^3 + 2x, P = (0,0), AutPGrp did not finish within 11 minutes; at p = 7 it exceeded 5 GB).
- **Re-runs.** When the note was written, the quick programs of all three parts were run again from a copy of the
  package; all outputs were reproduced, byte for byte or up to running times (`reproducibility/RERUN_LOG.txt`).
  The third run repeated this from the archive `source.zip`, with the same result
  (`reproducibility/independent_run_2/outputs/rerun_package/`).
- **Programs of the third run** (`independent_run_2/`): exhaustive Psi for p = 5, 7; d10, d10e for all 68 classes
  with p <= 13 and one class per type at p = 17, 19; S10, S10e for p <= 13; the order-p^11 data for p = 5, 7, 11;
  Lemma 5.2, Proposition 5.3 and Section 6 for p < 100; dim N for all 3632 pairs with p <= 13; symbolic checks.
  `outputs/summary.out`: TOTAL failures: 0.

## Independent verification runs
Three independent verification runs, all AI-assisted, followed; each wrote its own programs. Runs A and B examined
the first written version of the results: run A examined the statement and all proofs; run B recomputed everything
with its own programs and examined the literature. The third run examined the final text of the note (below).

| Item | Run A | Run B |
|---|---|---|
| Statement and readings against the source | CONFIRMED; Q2 answered in the source's definition, open in O'Brien's | CONFIRMED (the source defines immediate descendants with order p^10) |
| Theorem 1.1 (criterion, nucleus of rank 2) | PROVED (re-derived; certificate re-run; dim N over F_p for all (lambda, mu, a), p = 5, 7, 11) | PROVED independently (own certificate: determinant -4 mu^2, unit 16 x 16 minor at mu = 0; group cohomology in GAP for all 3632 pairs with p <= 13; Lie cocycles for all pairs p <= 23) |
| Lemmas 2.2-2.4 (p-covering ring, orbits) | CONFIRMED_WITH_FIXES (surjectivity of lifts; nucleus identification) | own model of the extensions; group-level confirmation (below) |
| Lemma 4.2(c) (chi) | CONFIRMED_WITH_FIXES (uniqueness of kappa had been proved only over Q(lambda, a)) | not separately examined |
| Lemma 4.3 (structure of Psi, Maglione-Stanojkovski) | CONFIRMED (reading of Theorems 5.6, 5.8 checked against the text); WITH_FIXES (dictionary of conventions; image as a set) | order |Psi| = e3 s |GL_2(F_p)| confirmed by an exhaustive search for all 182 classes p <= 23; structure not examined |
| Proposition 5.1, Lemma 4.5 | CONFIRMED (re-derived) | values confirmed |
| Proposition 5.3 (four contributions) | CONFIRMED for F_1, F_refl, F_transl (case analyses redone); F_ord4 WITH_FIXES (interpolation needed a written argument) | values confirmed at all computed primes; F_ord4 WITH_FIXES (written case analysis required) |
| Theorem 1.2 | CONFIRMED (Prop. 5.1 formula = direct count for 22 classes p <= 7; GAP pipeline for 3 classes at p = 5) | CONFIRMED for all 182 classes with p <= 23 (orbit enumeration at p = 5 and on a fibre at p = 7; Burnside); group-level confirmation for all 68 classes with p <= 13, incl. y^2 = x^3 + 2x at p = 5 (6780 / 12), and one class per type at p = 17 |
| Lemmas 6.1, 6.4 | CONFIRMED (Tate normal form and Hesse pencil checked from standard facts; numerics p <= 127) | statements CONFIRMED by brute force for p <= 97 |
| Theorem 1.3 | CONFIRMED (assembly; moduli 72 and 12; polynomials re-derived) | values CONFIRMED at p = 5, ..., 23; polynomials consistent with the general formula for p < 5000 |
| du Sautoy-Vaughan-Lee | Theorem 1 recovered as polynomial identities | D_p = 12, 34, 30, 52 on their own presentation; V_p > 0 iff e3 = 9 for p = 1 mod 12, p < 3000 |
| Order-p^11 data | not checked | not checked (reproduced by the third run) |
| Novelty | no prior source found | no prior source found (searches listed below) |

No run found a wrong theorem, proposition or lemma, or a gap that could not be closed.

**Corrections required by run A**, all applied:
1. Lemma 4.2(c): chi is obtained from the 2-dimensionality of N^* and the fact that the tau_xi span it, which
   Lemma 3.2 proves over every F_p (the uniqueness of kappa itself is also proved there, over every field of
   characteristic different from 2).
2. Proposition 5.3, F_ord4: derived by a written case analysis (via Lemma 5.2), like the other three; the
   interpolation at primes remains only as a check.
3. Lemma 2.4(b): the lifted automorphism is surjective (Burnside-basis argument), hence bijective.
4. Lemma 4.3: explicit dictionary with the conventions of Maglione-Stanojkovski (Theorems 5.6, 5.8, Lemmas 5.4,
   5.5, Corollary 3.6); the image of phi is the set {sigma o tau_T}, not only of that order.
5. Lemma 6.2 (Kubert's Tate normal form, with a proof, for all c with Delta(c) != 0) and Remark 6.5 (Hesse pencil,
   Artebani-Dolgachev, Enseign. Math. 55 (2009)); references checked with Crossref.
6. Theorem 1.1(iii): O'Brien's nucleus is identified with N inside the Lie model (Lemma 2.3).
7. The two readings of (Q2) are stated in the abstract, the introduction and the scope paragraph.

**Corrections required by run B**, all applied:
1. The PORC convention (rational coefficients; the source's footnote does not specify; Maglione-Stanojkovski write
   Z[x]; the polynomials of Theorem 1.3 have denominators 3 and 27 and integer values).
2. F_ord4 derived by a written case analysis (as in correction A2).
3. The group-level confirmation for all classes with p <= 13 (including y^2 = x^3 + 2x at p = 5) and one class per
   type at p = 17 is reported as that of run B; the GAP pipeline of the original programs is reported as covering
   9 of the 10 classes at p = 5.
4. Remark 4.4: only the order of Psi was confirmed by exhaustive computation; the structural description rests on
   the proofs of Maglione-Stanojkovski's Theorems 5.6 and 5.8, whose reading run A checked against their text.

**Changes made when the note was written, after runs A and B.**
- Lemma 3.2: the certificate of Theorem 1.1 written as two explicit 9 x 9 linear systems that can be solved by
  reading (determinant 2 mu; solutions the multiples of K_0 at mu = 0).
- Lemma 5.2: a closed formula for the sum over t', from which all four contributions of Proposition 5.3 are
  derived by short case analyses (F_ord4 = 6p + 34 + 12 delta_8 + 18 delta_3, as found by interpolation before).
- Lemmas 6.2 and 6.3: Kubert's normal form with proof, and a cube criterion for full rational 3-torsion; together
  they give the count #{e3 = 9} = delta_3 (p-1)(p-7)/24 without the Hesse pencil (which run A had checked; it is kept
  as Remark 6.5).
- Lemma 2.3 and Lemma 2.4(b) written out.
`check_note.py` checks their statements (parts A, C, D); the third run examined them first (below). The count of
classes checked numerically for p <= 23 is 182 (the first written version had a wrong total).

## Third verification run (on the final text)
The third run (AI-assisted) examined the final text, the additions above first, without using the conclusions of
runs A and B; it wrote its own programs (`reproducibility/independent_run_2/`, with a README).

| Item | Third run |
|---|---|
| Statement and readings | CONFIRMED against the source pages (pp. 1421-1422, 1443-1444 with the PORC footnote); the abstract now uses the source's "would be remarkable" |
| Lemma 3.2 (elimination) | PROVED: re-derived; the 18 conditions generated from the definition of N^* are, up to sign, the printed equations (sympy) |
| Lemmas 2.2-2.4, 4.1-4.5 (incl. chi over every F_p, the dictionary with Maglione-Stanojkovski) | CONFIRMED; reading of MS Theorems 5.6, 5.8 checked against the text; citation of |T(t)| = p + 1 made precise |
| Psi | |Psi| = e3 s |GL_2(F_p)| by an exhaustive search assuming no structure (all totally isotropic 3-subspaces of F_p^6; the stabilizer of (V1, V2) over PGL_3(F_p)) for the 22 classes with p = 5, 7 |
| Proposition 5.1, eq. (5), Lemma 5.2, Proposition 5.3 (incl. F_ord4) | CONFIRMED_WITH_FIX: all four case analyses redone; the constancy of F_gamma on the translations (not all conjugate when e3 = 9) is now justified by Lemma 4.5(c); direct sums for all primes below 100 |
| Theorem 1.2, Table 1 | CONFIRMED: Burnside counts in its own model for all 68 classes with p <= 13 and one class per type at p = 17, 19 (delta_8 = 1, delta_9 = 1); all equal |
| Lemmas 6.1-6.4, Remark 6.5 | CONFIRMED (proofs re-derived; brute force for p < 100; Lemma 6.2, 6.3 also for p < 40 with the general group law); pinpoints to Silverman's book removed |
| Corollary 5.5 | CONFIRMED (isomorphism Q^T A(y1, y2, -y3) Q = C(y); V_p criterion; "up to sign" added) |
| Theorem 1.3 | CONFIRMED: own class sums at p = 5, 7, 11, 13; polynomial identities for every pattern of the deltas |
| Order-p^11 data (Remark 6.6) | REPRODUCED: own Burnside counts for all classes with p = 5, 7, 11; every printed value and S11 |
| Package | `check_note.py` and 19 quick programs of `original/` and runs A, B re-run from the archive: identical |
| Novelty | no prior source found (searches above) |

It found no wrong statement and no gap that could not be closed. **Corrections required by the third run**, all
applied:
1. The pinpoint references to Silverman's book (Section III.1, Proposition III.3.1(b), Exercise 3.7), which could not be
   checked in a readable copy, were replaced by "Chapter III", with the one-line reason for the 3-division
   polynomial (duplication formula).
2. The constancy of F_gamma on the translation type was justified (before Lemma 5.2).
3. Two citations of Maglione-Stanojkovski were made precise (|T(t)| = p + 1: Corollary 3.6 with the end of the proof of
   Theorem D; the non-PORC behaviour of e3: Corollary C with Remark 1.2).
4. Remark 6.6, the Verification and Scope paragraphs, this report and the package README were updated for its
   results; the abstract was shortened (scope statements kept); "first observed in the course of this work" was
   dropped; the date is now 28 October 2026.

## Relation to the literature, novelty and scope
- **Searches (27 October 2026).** arXiv API (authors Maglione, Stanojkovski, Vaughan-Lee, S. Lee; abstracts with
  "immediate descendants", "PORC", "descendants" and "elliptic", "p-groups" and "elliptic curve", "smooth cuboids",
  "E-groups"), OpenAlex (15 works citing du Sautoy-Vaughan-Lee, one citing Maglione-Stanojkovski, three citing
  Stanojkovski-Voll (Math. Ann. 2021), ten citing S. Lee (J. Algebra 2016)), Semantic Scholar, zbMATH Open, Crossref,
  and three web searches. Repeated in part on the same day by the third run: arXiv API (authors Maglione,
  Stanojkovski, du Sautoy; "immediate descendant(s)", PORC, descendants and elliptic, "smooth cuboids", "elliptic
  groups"/"E-groups", "p-group generation" and descendants, Higman and PORC), OpenAlex (works citing
  Maglione-Stanojkovski: 1; citing du Sautoy-Vaughan-Lee: 15; citing the Oberwolfach report: 0), zbMATH Open, Crossref
  (all 21 DOIs of the bibliography) and one web search. The only new hit, arXiv:2608.07275 ("A Finite E-Group of
  Nilpotency Class Three"), uses "E-group" in another sense.
  - No printed answer to (Q1) or (Q2) was found. Maglione-Stanojkovski mention immediate descendants only in passing
    (Section 1.2). The only work found citing them, O'Brien-Stanojkovski (J. Algebra 703 (2026)), concerns
    isomorphism invariants according to its abstract. Du Sautoy-Vaughan-Lee treat only y^2 = x^3 - x with P = (0,0).
    S. Lee (2016), Vaughan-Lee (2018; arXiv:1702.04898) and Evseev (2008) concern other families of groups.
- **What was read.** The source pages; Maglione-Stanojkovski as listed above; du Sautoy-Vaughan-Lee in the arXiv
  version v2 (Sections 1, 2 and the beginnings of 4, 5: Theorems 1, 2, Corollaries 1, 2, the Lie-ring method);
  Artebani-Dolgachev, Section 2 (arXiv:math/0611590); Vaughan-Lee, "Graham Higman's PORC theorem"
  (arXiv:1808.04145), Sections 1, 7 and 8. Kubert (1976) and Higman (1960) were not read; the note proves what it
  uses of Kubert's paper. O'Brien (1990), Lazard (1954), Khukhro (1998) and Silverman (2009) are cited for standard
  definitions and facts and were not consulted for the note (the pinpoint references to Silverman's book were
  reduced to Chapter III by the third run). Of S. Lee, Vaughan-Lee (2017, 2018), Stanojkovski-Voll and
  O'Brien-Stanojkovski only abstracts were read. All bibliographic data were checked with Crossref (DOIs), arXiv preprints with the arXiv API.
- **Credits.** The question is Maglione's; the groups, their isomorphism classification (Theorem A) and the structure
  of their automorphism groups (Theorems 2.15, 5.6, 5.8, Lemmas 5.4, 5.5) are Maglione and Stanojkovski's; the group
  of y^2 = x^3 - x goes back to du Sautoy (Israel J. Math. 126 (2001); J. reine angew. Math. 549 (2002)); its
  exponent-p count and non-PORC results are du Sautoy and Vaughan-Lee's; p-group generation is O'Brien's (J. Symbolic
  Comput. 9 (1990)); the correspondence is Lazard's (1954; see Khukhro's book); the PORC conjecture is Higman's
  (Proc. London Math. Soc. (3) 10 (1960)).
- **Caveats.** A search that finds nothing is not a proof of novelty; no priority is claimed. The order-p^11 data
  were computed twice, by independent programs, but nothing is proved about them.

## 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/