# Verification report — KOU-8.3 (Kourovka Notebook, Problem 8.3, B. A. F. Wehrfritz: regularity of the Sylow p-subgroups P_n(Z/p^m) of GL_n(Z/p^m)) Verification date: 2026-10-22; revised 2026-10-23 (Theorem A', Section 6 of the note, and its independent verification run; see "Theorem A' and its verification" below). **Verdict.** The note gives a **partial answer**: P_n(Z/p^m) is regular in the boundary cases nm - 1 = p for (p, n, m) = (11,4,3), (17,6,3), (23,8,3), (29,10,3), (19,5,4), (23,6,4), (29,6,5), and P_n(Z/p^3) is irregular for n = (p+4)/3 and p in {11, 17, 23, 29}; hence for every prime p <= 29, P_n(Z/p^3) is regular iff 3n - 1 <= p, and for p = 11 the answer is complete except for n = 4, m >= 4; the question stays open in general. - **Settled (new cases).** Theorem A (regular: the seven boundary cases, and with them all P_{n'}(Z/p^{m'}) with n' <= n, m' <= m); Theorem B (irregular: P_n(Z/p^3) for n = (p+4)/3, p = 11, 17, 23, 29, and with them all n' >= n, m' >= 3); Corollaries 1-3 (m = 3: regular iff 3n - 1 <= p for p = 11, 17, 23, 29, and hence, with known results, for every prime p <= 29; p = 11 except n = 4, m >= 4; the range (p+4)/3 <= n < (p+1)/2, m >= 3, for p = 17, 23, 29). - **Proved by hand (new).** Theorem A': for a prime p >= 5 with p = 2 (mod 3) and n = (p+1)/3, P_n(Z/p^3) is regular if (L1_n) holds over F_p, that is, if S = e21 + ... + e_{n,n-1} and any element Y of the first subdiagonal that is not a multiple of S generate a Lie subalgebra of gl_n(F_p) containing e_{n1}. Also: the two reductions of Lemma 6.5, and Remark 6.3 ((L1_n) fails for every odd n >= 5). - **Computer-assisted.** Theorems A and B and Proposition 6.6. Theorem B by an exact computation in the group (pcgs by sifting with an end-of-run certificate); Theorem A for the five triples of Table 2 by a reduction proved in the note (Hall-Petresco identity, Magnus embedding, a graded lower bound for mho_1('), torus symmetry) to finite linear algebra over F_p; Theorem A for (23,8,3), (29,10,3) (and again for (11,4,3), (17,6,3)) by Theorem A' and Proposition 6.6, the computer check of (L1_n) for (n, p) = (4,11), (6,17), (8,23), (10,29). The second sentence of Corollary 1 is computer-assisted on the rows p = 11, 17, 23, 29 and rests on the literature for the other primes p <= 29. - **Not new.** The witness pair of Theorem B is the pair of Kolesnikov and Leontiev (SEMR 19 (2022), Section 4); the tools are classical (P. Hall; Petresco; Magnus; Jacobson). Known results used: Merzlyakov (class bound), Yagzhev (m = 1), Kolesnikov 2001 and Egorychev-Kolesnikov-Leontiev 2023 (m = 2), Kolesnikov 2006 (n^2 < p), Kolesnikov-Leontiev 2022, Theorem 2 (p = 1 mod 3, m >= 3). - **Open.** In general; e.g. (13, 4, m >= 4), (19, 5, m >= 5), (19, 6, m >= 4), (11, 4, m >= 4), (17, 5 or 6, m >= 4), (23, 8, m >= 4), (29, 10, m >= 4), and m = 3 for p = 2 (mod 3), p >= 41 (the boundary n = (p+1)/3, which by Theorem A' follows from (L1_n), and the range (p+4)/3 <= n < (p+1)/2). Three conjectures (m = 3: regular iff 3n - 1 <= p, now proved for p <= 29; nm - 1 = p with m >= 3 implies regular; (L1_n) for every even n and every prime p > 2n) are stated as conjectures. The note is unrefereed. ## Statement checked - **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, No. 21, arXiv:1401.0300, Problem 8.3 (B. A. F. Wehrfritz, 8th issue, 1982). - Version 48 (6 October 2026): local text extraction, lines 885 ff. Version 49 (20 October 2026): the arXiv PDF (sha256 `24cd6e5031baa7b1b4bb58f363b980c2f6f1769f9a8431d4fef0065dc777e196`), text line 918 ff. The entry is word for word the same in both versions, with the editors' comments of 2001 (Merzlyakov mn < p; Yagzhev m = 1, "erroneous for m > 1"; Kolesnikov m = 2) and 2009 (Kolesnikov, n^2 < p). It carries neither the star of solved problems nor the mark of unverified AI solutions introduced in version 49. - The question: P_n(Z_{p^m}) = all n x n matrices (a_ij) over Z/p^m with a_ii = 1 (mod p) and a_ij = 0 (mod p) for i > j; for which m and n is it regular? It is the preimage of the upper unitriangular group of GL_n(F_p), a Sylow p-subgroup of GL_n(Z/p^m), of order p^((m-1)n^2 + n(n-1)/2). - **Corpus record.** ulamai/UnsolvedMath, KOU-8.3 (dataset v1.8.0, snapshot d371ee9, line 16502), status "open"; statement and comments as in the Notebook, up to TeX. - **Regularity.** P. Hall's definition: (ab)^p = a^p b^p s_1^p ... s_r^p with s_i in '; equivalently Z(a,b) = (a^p b^p)^-1 (ab)^p lies in mho_1(') = '>. (Kolesnikov-Leontiev use the equivalent form with all powers p^k.) ## Readings | Reading | Answer | Where | |---|---|---| | The group of the question (upper part free, lower part divisible by p) | isomorphic, by g -> (g^T)^-1, to G = {g : g_ii = 1 (mod p), g_ij = 0 (mod p) for i < j}, the convention of Kolesnikov-Leontiev; all computations are in G | Section 2.1 | | "a_ii = 1" read literally (diagonal exactly 1) | not the Sylow subgroup and not the reading of the Notebook's comments or of the literature; not treated | — | | Regularity of the whole group (all pairs) | Theorem A proves Hall's condition for all pairs; Theorem B gives one explicit pair violating it | Sections 4, 5 | | "For which m and n" (for every p) | partial: new triples decided, the general question open | Section 7.3 | ## Results in the paper - **Lemma 2.1** (subgroups and quotients: regularity passes to smaller n, m; irregularity to larger). - **Lemma 2.2** (the weight filtration w(p^k e_ij) = kn + i - j: normal subgroups G_w, [G_v, G_w] <= G_{v+w}, G_{mn} = 1, the layers A_w with the basis of positions, at most one per row); the graded algebra A, slots and walks; the top layer G_{mn-1} = , t = E + p^{m-1} e_{n1}. - **Lemma 2.3** (p-th powers: g^p = E + pX mod M_{w+n+1} when (p-1)w >= n+1, which holds for all w if n <= p - 2; the map pi). - **Lemma 3.1** (Hall-Petresco: regular iff c_p(a,b) in mho_1(') for all a, b). **Lemma 3.2** (mho_1(K) = mho_1(K') when gamma_p(K) = 1). **Lemma 3.3** (mho_1(K) = for a generating set adapted to the filtration, when p w_0 >= mn). **Lemma 3.4** (exactness of sifting under the closure hypothesis). - **Theorem B** with Table 1 (certificates). - **Lemma 5.1** (z^p = f(z) t-bar, the forced walk). **Proposition 5.2** (c_p(a,b) = t^{-sigma(x,y)}, via the Magnus map; Jacobson's formula as a remark). **Proposition 5.3** (o(V) <= gr(mho_1(')) for every pair with leading terms spanning V). **Lemma 5.4** (three equivalent forms of "sigma = 0 on V x V"). **Proposition 5.5** (finite criterion). **Lemma 5.6** (torus symmetry; normal forms and their number). **Theorem A** for the five triples of Table 2; validation; **Remark 5.7** (mechanism, an observation; for m = 3 now explained by the proof of Theorem A'). - **Section 6 (new, 23 October 2026).** **Lemma 6.1** (the level-zero map rho: A -> n, a homomorphism onto the strictly lower triangular matrices; pi^2 on A_{n-1} and pi on A_{2n-1} for m = 3). **Lemma 6.2** (the coordinate (-1)^(n-1) C(2n-2, n-1) of ad(omega)^(2n-2)(u_n) at eps(n,1,1)). **Proof of Theorem A'** (Step 1: no vector with all slots nonzero gives sigma = 0; Step 2: torus to omega + u_n in V; Case 1 by (L1_n) and pi^2; Case 2, V = V_KL, by Lemma 6.2 and pi). **Remark 6.3** ((L1_n) fails for every odd n >= 5: the automorphism X -> -J X^T J and palindromic D; (L1_3) holds). **Lemma 6.4** (bracket formulas for P S^d; L_{d+1} = [S, L_d] + [DS, L_d]). **Lemma 6.5** (two reductions proved by hand: l(D) != 0 suffices; l(D) = 0, c(D) != 0 and D(n) != D(2) suffice; the remaining sets H, A, B and their sizes). **Proposition 6.6** (computer-assisted: (L1_4) over F_11, (L1_6) over F_17, (L1_8) over F_23, (L1_10) over F_29) with Table 3. **Theorem A** for (23,8,3), (29,10,3), and a second proof for (11,4,3), (17,6,3). - **Corollaries 1-3** (proofs in Section 7, with the list of the primes p <= 29 and their sources); **Conjectures 1-3** (labelled as conjectures; Conjecture 1 proved for p <= 29, its remaining content p = 2 (mod 3), p >= 41; Conjecture 3: (L1_n) for even n and p > 2n, with evidence); evidence only (Section 7.2); the open region (Section 7.3). ## Computations (programs and outputs in reproducibility/) - **The check program of the note** (`check_note.py`, standard library; `TOTAL failures: 0`): - Part A (7 s): Theorem B for the four cases with end-of-run certificates for the pcgs of H, H' and mho_1(H'); orders 35/32/8, 63/60/12, 99/96/16, 143/140/20; Z of weight 7, 9, 11, 13; residues at weight p: (0,3,10,2), (0,11,8,2), (0,10,15,2), (0,24,7,2), with gr_p(mho_1(H')) spanned by (1,9,2,10), (1,7,10,16), (1,14,9,22), (1,6,23,28); mho_1(H') computed both by Lemma 3.3 and along the derived series (Lemma 3.2), with the same result; controls (7,3,3), (13,5,3), (5,3,2) (witness) and (11,4,3), (5,2,3), (11,3,4) (no witness). - Part B (20 s): (11,4,3) over all 16226 subspaces (15830 / 396 / 0) and the 44 normal forms (26 / 18 / 0, also with the p^4-pair criterion of the original program); (19,5,4), 682 normal forms (511 / 171 / 0); Lemma 5.1 on random elements; the torus symmetry on random samples. - Part C (1 s): validation (5,3,2), (7,4,2) not certified (the Kolesnikov-Leontiev subspace); the wrong rule o := d would certify them; (5,2,3), (11,3,4) certified. - Part D (10 s; added on 23 October 2026): Theorem A' in the graded algebra for (p, n) = (5,2), (11,4), (17,6), (23,8), (29,10): the coordinate of ad(omega)^(2n-2)(u_n) at eps(n,1,1) is 3, 2, 3, 18, 13 = (-1)^(n-1) C(2n-2, n-1) mod p and pi maps it to that multiple of t-bar (Case 2); 60 random Lie monomials of degree n-1 per case: the eps(n,1,0)-coordinate of w(omega + u_n, y) equals the e_n1-entry of w(S, Y) and pi^2 gives that multiple of t-bar (Case 1); the top lies in o(V) for V_KL and random V containing omega + u_n in (23,8,3) and (29,10,3). Lemmas 6.4 and 6.5 against matrix arithmetic. (L1_4) over F_11 (12 classes) and (L1_6) over F_17 (5220) on all classes, (L1_8) over F_23 on the hyperplane (292561); a second method with matrices on 220 classes; the failing classes for (5,11) and (7,13) are exactly the palindromic ones (1 and 14); the evidence for Conjecture 3 and the failures for small p. (L1_10) over F_29 is not run there (about 40 minutes with `L1fast2.py`); it is documented by saved outputs. - Part L: (17,6,3) 9891 forms, 7376 / 2515 / 0 (55 s); (23,6,4) 20391 forms, 16436 / 3955 / 0 (294 s); (29,6,5) 36723 forms, 30968 / 5755 / 0 (1039 s). - **The programs with which the results were first obtained** (`preg.py`, `liebound.py`, `allV.py`, `allV_gen.py`, `family.py`, `sanity.py`, `mech.py`, `targeted.py`, `rand_pairs.py`; outputs in `runs/`): the same counts for (11,4,3) (all subspaces and normal forms), (17,6,3), (19,5,4), (23,6,4), (29,6,5) (this run ended only after the note had first been completed, after 10671 s; output `runs/allV_gen_29_6_5.out`); Theorem B for the four cases; validation; Remark 5.7; evidence for P_4(Z/11^4), P_4(Z/11^5). For Proposition 6.6: `L1check.py` (all classes: (4,11), (6,17); controls (5,11), (5,19), (7,13)), `L1fast.py` (the hyperplane l(D) = 0: (8,23), 292561 classes, 19 s; also (6,17), (7,13), (8,11)), `L1fast2.py` (the sets A and B: (8,23), 12720 + 12720; (10,29), 21243690 + 21243690 classes in 2576 s; the printed times are cumulative); outputs `runs/L1_checks.out`, `runs/L1fast2_29_10.out`; no failure in the cases of Proposition 6.6. - **Writing stage** (`writing_stage_runs/`): the original `preg.py` on the Kolesnikov-Leontiev pair in P_4(Z/13^4), P_6(Z/19^4), P_5(Z/19^5) (Hall's condition holds; evidence only). - **Re-runs.** On 2026-10-22 all programs with recorded outputs were run again from the package folder; the outputs agree with the recorded ones up to running times (`reproducibility/RERUN_LOG.txt`; `targeted.py` took 28 min). Not repeated with the original program: `allV_gen.py` for (23,6,4) (46 min) and (29,6,5); these computations are covered by `check_note.py L` and by the second run's program, both run from the package. The programs of the two verification runs were re-run as well (identical outputs), except a few long or memory-heavy ones listed in `independent_runs/README.md`. Run 3 repeated all these re-runs from a copy of the package extracted from `source.zip` (same results up to running times; `RERUN_LOG.txt`, section [7], and `independent_run_2/package_rerun.txt`). On 23 October 2026 the files added for Section 6 were re-run from the package folder: `L1check.py`, `L1fast.py`, `L1fast2.py` on all cases of `runs/L1_checks.out`, every program of the fourth run (including its two hyperplane pieces for (10,29), 623 s and 621 s), and `check_note.py`; all agree with the recorded outputs up to running times (`RERUN_LOG.txt`, section [8]). Not repeated: `L1fast2.py 29 10` (43 minutes in one process; its sets are covered by the fourth run's `l1words.py`, which was repeated). - **The programs of run 3** (`independent_run_2/`, written for that run without the package code): `ga.py` (the graded algebra; closures from the definitions and in a simplified form; all counts of Table 2 for the five cases, the validation and the wrong rule; torus orbits; Remark 5.7), `gb.py` (exact group computations with another pivot convention and end-of-run certificates: Table 1, controls, brute-force validation, the evidence of Sections 7.1-7.2), `gx.py` (Hall-Petresco words and exact tests of Propositions 5.2 and 5.3 on 4100 random pairs). - **The programs of run 4** (`independent_runs/fourth_run/`, written for that run before it read the package code): `l1matrix.py` (the generated subalgebra with honest n x n matrices), `l1words.py` (all brackets of degree n-1, vectorised, numpy), `l1rank.py` (the spaces L_d by elimination), `cases_check.py` (both cases of the proof of Theorem A' in M_n(Z/p^3)), `tiny_witness.py` (brute-force witnesses for p = 2, 3). ## Independent verification runs Four independent verification runs, all AI-assisted, were made, each with its own programs; none found a mathematical error. Runs 1 and 2 examined the first write-up of the result (which contained Theorem A for (11,4,3), (17,6,3) and Theorem B); run 3 examined the text of the note before Section 6 was added, including Lemma 3.3 and the cases (19,5,4), (23,6,4), (29,6,5) of Theorem A, which were added after runs 1 and 2; run 4 examined Theorem A', the check of (L1_n) and the second sentence of Corollary 1, which were added after run 3 (next section). Section numbers in the records of runs 1-3 refer to the earlier text, whose Sections 6 and 7 are Sections 7 and 8 of the present note. | Item | Run 1 | Run 2 | Run 3 (final text) | |---|---|---|---| | Statement, conventions, quantifiers (Notebook vs. record vs. Kolesnikov-Leontiev) | CONFIRMED | CONFIRMED (Kourovka v49 read: unchanged, not starred) | CONFIRMED (v48 and v49 compared word for word; arXiv: v49 still current on 22 Oct 2026; KL 2022, Kolesnikov 2013 and the zbMATH reviews read again) | | Weight filtration, Lemma 2.3 and its hypothesis n <= p - 2 | CONFIRMED (re-derived) | CONFIRMED (re-derived) | CONFIRMED (re-derived; the model of the graded algebra checked against matrix arithmetic in 7 groups; the conclusion of Lemma 2.3 fails when its hypothesis fails) | | Lemmas 3.1, 3.2 (Hall-Petresco reduction, agemo recursion) | CONFIRMED | CONFIRMED (second proof of Lemma 3.2 via regularity of the quotient) | CONFIRMED (re-derived) | | Lemma 3.3 (agemo of a filtered generating set) | not part of the first write-up | not part of the first write-up | CONFIRMED (re-derived; hypotheses stated and sufficient) | | Sifting code and its exactness | CONFIRMED (own implementation with end-of-run certificates; brute force in 5 small groups) | CONFIRMED (code review; different method: Lazard/Howell) | CONFIRMED (own implementation with another pivot convention and end-of-run certificates; brute force for 40 pairs in 6 small groups) | | Theorem B, four cases | CONFIRMED (own exact program; all four certificates, numbers for p = 17, 23, 29 new) | CONFIRMED (different method; orders; canonical residue for (11,5,3)) | CONFIRMED (every entry of Table 1 reproduced) | | Proposition 5.2 (c_p = t^{-sigma}) | CONFIRMED (re-derived; exact tests on 489 pairs) | CONFIRMED (re-derived; 3200 pairs; Hall-Petresco words computed directly on 1410 pairs; mutations detected) | CONFIRMED (re-derived; Hall-Petresco words computed by Hall's recursion on 4100 pairs in 8 groups) | | Proposition 5.3 (graded lower bound) | CONFIRMED (re-derived; exact lift tests) | CONFIRMED (re-derived; exact cross-checks) | CONFIRMED (re-derived; exact tests on the same 4100 pairs) | | Torus symmetry and normal forms | CONFIRMED (re-derived; counts 44, 9891, 3927 reproduced) | CONFIRMED (two representative families; count formula; invariance on 81130 random pairs) | CONFIRMED (counts = formula in 12 cases; over all subspaces of (5,3,2), (7,4,2), (11,4,3) each torus orbit contains exactly one normal form and the verdict is constant on orbits) | | Theorem A for (11,4,3), (17,6,3) | CONFIRMED (all 16226 subspaces; 9891 forms) | CONFIRMED (all subspaces; two families) | CONFIRMED (all subspaces and normal forms; closures from the definitions and in a simplified form) | | Theorem A for (19,5,4), (23,6,4), (29,6,5) | not part of the first write-up | computed by the run's own program (suggestion; the cases were added afterwards) | CONFIRMED (normal forms; closures from the definitions and in a simplified form) | | Validation (known irregular cases not certified) | CONFIRMED ((5,3,2), (7,4,2)) | CONFIRMED (also (13,7,2): 79117 forms, 24 uncovered; mutation o := d) | CONFIRMED ((5,3,2), (7,4,2), (11,6,2), (13,7,2); 12 regular cases; the wrong rule o := d) | | Corollaries, table of known results, open region | CONFIRMED | CONFIRMED_WITH_FIXES (open region: p = 1 mod 3 with m >= 4 also open) | CONFIRMED_WITH_FIXES (see below) | | Novelty | supported "to our knowledge"; nothing found | supported; KL 2022 and KP 2025 read in full; nothing found | supported; searches repeated (arXiv, zbMATH Open, Crossref, OpenAlex, one web search); nothing found | | Package | — | — | run again from the archive: all recorded outputs reproduced up to running times | **Corrections required by run 1** (all applied in the note): g_ii = 1 (mod p) in the definition of G; the agemo written mho_1, not Omega; Yagzhev cited as Math. Notes 56 (1994) with the Notebook's year 1995 and the 1996 letter to the editor; the row "n >= (p+1)/2 irregular" stated for m >= 2 (false for m = 1); the hypotheses of Kolesnikov 2013 (m >= 2, p >= 5, 2 <= n <= (p-1)/2, read in the journal PDF); the sentence that o(V) and the sigma-check cover every pair whose leading terms span V; the full data of Table 1 for all four cases; documentation of `mech.py`, and the statement that the computer part of Theorem A is finite F_p-linear algebra; an end-of-run certificate for the sifting in the package (`check_note.py`, Part A). **Corrections required by run 2** (all applied): the hypotheses of Kolesnikov 2013; v49 unchanged and not starred, KL 2022 and KP 2025 read in full; EKL 2023 known only from its zbMATH review (said in the note); the description of the open region (now Section 7.3); the (13,7,2) count (79117 forms, 24 uncovered); DOIs and pages (EKL 2023, Kolesnikov 2006, KP 2025, Yagzhev, Levchuk 1992, KL 2022, all checked with Crossref); the description of the later additions ((19,5,4), (23,6,4), (29,6,5), `mech.py`). **Corrections required by run 3** (all applied): the sentence on the condition nm - 1 < p in the abstract and in Section 1.3 (that nm - 1 < p is not necessary for m >= 3 follows already from Kolesnikov's n^2 < p, e.g. P_2(Z/5^m); now stated as: for m = 3, 4, 5 the group can be regular although neither nm - 1 < p nor n^2 < p holds, whereas for m <= 2 the condition nm - 1 < p is necessary); the caption of Table 2 (the column "sigma = 0 on V x V" counts only the subspaces with the top not in o(V)); the sentence after (K6) on Chevalley groups of other types (only the type-A results of those papers are used); one sentence of Remark 5.7; the record of this run (paragraph "Verification", Sections 5 and 7, this report, the README, RERUN_LOG.txt, the folder `reproducibility/independent_run_2/`); the status of the original run for (29,6,5). Recommended by run 3 and made: a sentence after the proof of Theorem B saying that only the verified properties (i)-(iv) enter the conclusion; the inclusions that hold by construction are used only for the data of Table 1. **Changes made when the note was written.** The cases (19,5,4), (23,6,4), (29,6,5) were added to Theorem A ((19,5,4) and (23,6,4) computed by the original program, by the second run's program and by `check_note.py`; (29,6,5) by the second run's program and by `check_note.py`, the original program's run not having finished; that run ended later with the same counts, and run 3 computed the case again with its own program). The lemma on filtered generating sets (Lemma 3.3) was added; `check_note.py` uses it besides Lemma 3.2 and checks that both give the same subgroup. The proof of the formula for c_p was written with an explicit comparison of the Magnus map and the filtration (the claim in the proof of Proposition 5.2), and the test for "sigma = 0 on V x V" was given the equivalent forms of Lemma 5.4. EKL 2023 was not available in full when the note was completed; the note says so. ## Theorem A' and its verification (run 4; added on 23 October 2026) After run 3, a proof by hand was found that reduces the boundary case n = (p+1)/3, m = 3, p = 2 (mod 3) to the condition (L1_n) on the Lie algebra of strictly lower triangular matrices over F_p (Theorem A'), together with a computer check of (L1_n) for n = 4, 6, 8, 10 after two reductions proved by hand (Lemma 6.5, Proposition 6.6). This adds the regular cases P_8(Z/23^3) and P_10(Z/29^3) to Theorem A (until then evidence only, Section 7.2), gives (11,4,3) and (17,6,3) a second proof, and gives the second sentence of Corollary 1 (every prime p <= 29). Run 4 examined these additions; its programs and outputs are in `reproducibility/independent_runs/fourth_run/`. | Item | Run 4 | |---|---| | Proof of Theorem A' from (L1_n) and Propositions 5.2, 5.3, 5.5, Lemma 5.6(a) | CONFIRMED (re-derived by hand: a vector with all slots nonzero exists unless sigma = 0 on V x V; the torus moves it to omega + u_n; Case 1 by a Lie monomial of degree n-1, the multiplicative level-zero projection and pi^2; Case 2, V = V_KL, by the forced walk of eps(n,1,1) and the unit (-1)^(n-1) C(2n-2, n-1); the memberships in d(V)) | | Side conditions | CONFIRMED_WITH_FIXES (n even and n <= p-2 hold for p >= 5; for p = 2, n = 1 and the proof does not apply, so the theorem is stated for p >= 5; n = 2 (p = 5): Case 1 is empty, Case 2 gives -2 != 0 mod 5) | | Points of Section 5 used by the proof (top layer, Lemma 2.3, Propositions 5.2 and 5.3, torus symmetry) | re-derived in outline; otherwise relied on runs 1-3 | | Lemma 6.5 (the reductions (a), (b)) and the sets H, A, B | CONFIRMED (re-derived; sizes (p^(n-3)-1)/(p-1) and (p^(n-4)-1)/(p-1)) | | Remark 6.3 (odd n) | CONFIRMED_WITH_FIXES (the automorphism X -> -J X^T J and the eigenvalue argument are correct; the statement must read "odd n >= 5", since (L1_3) holds) | | (L1_4) over F_11, (L1_6) over F_17 | CONFIRMED (all 12 and 5220 classes; three own programs) | | (L1_8) over F_23 | CONFIRMED (all 6 728 904 classes, without any reduction; also H, A and B) | | (L1_10) over F_29 | CONFIRMED (A and B, 2 x 21 243 690 classes, complete with Lemma 6.5(a), (b); and the whole hyperplane H, 616 067 011 classes, complete with Lemma 6.5(a) alone) | | Controls | CONFIRMED (n = 5 over F_11, F_19: one failing class, palindromic; n = 7 over F_13: the 14 palindromic classes; n = 10 over F_7: 8 failures; the three programs give identical failing sets) | | The two cases inside M_n(Z/p^3) for (p, n) = (5,2), (11,4), (17,6), (23,8), (29,10) | CONFIRMED (coefficients 3, 2, 3, 18, 13 mod p; random words consistent) | | The note's programs `L1check.py`, `L1fast.py`, `L1fast2.py` (re-run from a copy, small cases) | identical to the recorded outputs | | Corollary 1, second sentence (every prime p <= 29) | CONFIRMED_WITH_FIXES (true as stated, given Theorem B and the cited literature; the rows p = 2, 3 and the sources for p = 5 were missing, and the status, computer-assisted and unrefereed on the rows p = 11, 17, 23, 29, had to be stated; own brute-force witnesses of irregularity for P_2(Z/4), P_2(Z/8), P_2(Z/9), P_2(Z/27)) | | Theorem B (used by Corollary 1) | not re-checked by run 4 (confirmed by runs 1-3); run 4 re-ran the package's `preg.py` for P_7(Z/19^3) (a witness, consistent with KL 2022, Theorem 2) | | Literature | no external request by run 4; novelty taken from the searches of 22 October 2026 | **Corrections required by run 4** (all applied): (1) Theorem A' stated for p >= 5; (2) "odd n >= 5" in Remark 6.3; (3) the second sentence of Corollary 1 with the full list of primes p <= 29 and their sources (p = 2: the m = 2 theorem, or a regular 2-group is abelian; p = 3: the m = 2 theorem; p = 5: Kolesnikov 2006 for n = 2 and the m = 2 theorem for n = 3; p = 7, 13, 19: Merzlyakov and Kolesnikov-Leontiev 2022, Theorem 2; p = 11, 17, 23, 29: Theorems A and B, with monotonicity, Lemma 2.1, throughout), with the words computer-assisted and unrefereed; (4) the whole note made consistent with the new result (abstract, introduction, the list of theorems, Corollary 1 now for p in {11, 17, 23, 29}, the conjectures, the evidence section, the status wording, the notice for the corpus); (5) the times in `runs/L1fast2_29_10.out` named as cumulative; (6) the reductions (a), (b) named as lemmas proved by hand (Lemma 6.5), with the computation covering exactly the classes they leave; (7) the one-line reasons why the elements used in Cases 1 and 2 lie in d(V). **Changes made after run 4.** The level-zero map and the coefficient of Case 2 were made separate statements (Lemmas 6.1 and 6.2), and so were the bracket formulas (Lemma 6.4); Table 3 was added; Conjecture 1 now records that it is proved for p <= 29 and that its remaining content is p = 2 (mod 3), p >= 41; Conjecture 3 ((L1_n) for every even n and every prime p > 2n) was added, with evidence computed by Part D of `check_note.py` ((L1_4) for 11 <= p <= 31, (L1_6) for 13 <= p <= 37, (L1_8) for p = 17, 19 on the hyperplane; random elements of the hyperplane for n = 10, 12, 14; failures for small p: (L1_8) over F_5, (L1_10) over F_7); Part D was added to `check_note.py`; the new files were re-run from the package folder (`RERUN_LOG.txt`, section [8]). ## Relation to the literature, novelty and scope - **Searches (22 October 2026).** zbMATH Open (the 32 works of S. G. Kolesnikov; works of V. M. Leontiev and of G. P. Egorychev; "Wehrfritz problem"; "regular Sylow subgroup"; titles with "regular" and "Sylow"; Wehrfritz and "regular" since 2010); arXiv API (queries on regular p-groups, Sylow subgroups, Wehrfritz, Hall-Petresco, the authors); OpenAlex and Semantic Scholar citing works (Kolesnikov 2006, KLE 2020, KL 2022, EKL 2023, KP 2025); Crossref; two web searches; the publishers' pages of KL 2022, Kolesnikov 2013 and KP 2025. mathnet.ru refused the connection to one run, and no workaround was tried. Run 3 repeated the search on the same day (arXiv API; zbMATH Open: works of S. G. Kolesnikov and of V. M. Leontiev since 2023, "Wehrfritz" and "regular" since 2022, the reviews and records cited; Crossref: all DOIs and a bibliographic query; OpenAlex: works citing Kolesnikov 2006, KL 2022, EKL 2023, KP 2025; one web search) and found nothing new. - No printed decision of the cases of Theorems A and B was found. KL 2022 says that its condition allows a partial study of n >= (p+1)/3 and proves irregularity only when (p+2)/3 is an integer; Kolesnikov 2013 proves necessary conditions only; KP 2025 treats F_4 and E_6; EKL 2023 (by its review) treats m <= 2. - OpenAlex and Semantic Scholar list no work citing KL 2022, EKL 2023 or KP 2025. - The cases (23,8,3) and (29,10,3) added with Theorem A' lie in the region covered by these searches (no decision of a boundary case n = (p+1)/3 with p = 2 (mod 3), p >= 11, was found). When Theorem A' was added, an arXiv API query for the newest abstracts containing "regular" and "Sylow" (anonymous; logged) found nothing new. The literature on Lie algebras of triangular matrices was not searched for (L1_n), which may be known in another form. - **What was read.** Entry 8.3 in Kourovka v48 and v49; KL 2022 in full (run 2) and, for the note, its introduction, statements, Section 4 and references; Kolesnikov 2013 in the journal's PDF (introduction, Theorem 1, Section 3, references); KP 2025 in full (run 2) and its introduction; zbMATH reviews of Yagzhev 1994, Kolesnikov 2006, EKL 2023, KP 2025; zbMATH records of Merzlyakov 1964, Kolesnikov-Mal'tsev 2011, Kolesnikov 2013. Not read: EKL 2023 (paywalled; only the review and the Crossref record), Kolesnikov 2001 (not accessible), Merzlyakov 1964 and Levchuk 1992 (the class bound is re-proved), Yagzhev's paper and its letter to the editor (only the review). - **Caveats.** A search that finds nothing is not a proof of novelty; conference abstracts were not searched. No priority is claimed. Yagzhev's printed list for m > 1 (as reviewed) contradicts known results and Theorem A; the Notebook calls the paper erroneous for m > 1, and the note uses it only for m = 1. - **Scope.** New cases for specific primes; for m = 3 the answer is now complete for every prime p <= 29 (computer-assisted and unrefereed on the rows p = 11, 17, 23, 29); the question stays open in general (Section 7.3 of the note); the conjectures are conjectures. ## 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/