# Verification report — KOU-15.98 (Kourovka Notebook Problem 15.98, L. A. Shemetkov: minimal non-F-groups with Sylow residual) Verification date: 2026-10-21. **Verdict.** The note gives a **complete answer** to the question as posed: **the answer is no**. In the words of the note: the answer is no: there is a saturated formation F and a finite soluble minimal non-F-group G of order 2304 whose F-residual is a Sylow 2-subgroup isomorphic to no factor group of a Sylow 2-subgroup of PSU(3, 2^{2n}); 2304 is the least order of such a counterexample. - **Settled.** The counterexample (Theorem 4.2), a second one of order 3^8 * 2^4 (Proposition 4.3), the least order 2304 (Theorem 5.4), and the answer no under the additional conditions: F subgroup-closed, F containing all supersoluble groups, f(p) abelian, F inside N_p N_r (Proposition 6.1), F = N^2 and F = NA (Proposition 6.3). Nothing changes in the universe of soluble groups. - **Known or not claimed as new.** The case F = N (yes) is the theorem of Mazurov–Syskin (p = 2) and Zhurtov–Syskin on Schmidt groups (Remark 6.4). The case F = U (yes, Theorem 6.5) is proved in the note but not claimed as new; it uses the Zhurtov–Syskin theorem, read only in its zbMATH review. - **Dependence on unread sources.** Proposition 5.3(f), Remark 6.4 and Theorem 6.5 use Theorem 1 of Zhurtov–Syskin (Sib. Mat. Zh. 28 (1987), no. 2, 74–78) in the form stated in its zbMATH review (Zbl 0622.20010). The answer no and the least order (Theorem 5.4) do not use it. - **Open.** Which saturated formations give the answer yes (beyond N and U) is not determined. The note is unrefereed. ## Statement checked - **Primary source.** The Kourovka Notebook, No. 21, edited by E. I. Khukhro and V. D. Mazurov, arXiv:1401.0300; Problem 15.98 (L. A. Shemetkov, 15th issue, 2002), printed page 99. - Read in version v48 (6 October 2026) and again in version v49 (20 October 2026; downloaded on 2026-10-21, 1,485,651 bytes, sha256 `24cd6e5031baa7b1b4bb58f363b980c2f6f1769f9a8431d4fef0065dc777e196`). The six lines of the problem are identical in the two text extractions. No comment follows the problem (the next entry is 15.99), and the index of names lists 15.98 without the archive mark "A:", in both versions. - Content (paraphrase): F a saturated formation, G a finite soluble minimal non-F-group whose F-residual is a Sylow p-subgroup; is the residual isomorphic to a factor group of the Sylow p-subgroup of PSU(3, p^{2n})? The problem adds that this holds for the formation of nilpotent groups and cites Mazurov–Syskin (Math. Notes 14 (1973) 683–686) and Zhurtov–Syskin (Siberian Math. J., printed there as volume 26, 235–239). - **Corpus record.** UnsolvedMath v1.8.0 (snapshot d371ee9), record KOU-15.98, status "open"; the statement field is the Notebook text with its two references; nothing truncated. ## Readings - PSU(3, p^{2n}) = U_3(p^n), over the field with p^{2n} elements; its Sylow p-subgroup U(p^n) has order p^{3n}. This is also the notation of the zbMATH review of Zhurtov–Syskin. The other conceivable reading (field with p^{4n} elements, Sylow subgroup U(p^{2n})) would make the statement false already for F = N (SL(2,3) with residual Q_8 = U(2)), contrary to the remark in the problem; the counterexamples of the note work under both readings. The least-order statement refers to the first reading. - n is not fixed (some n >= 1); factor groups may be trivial or the whole group. - The question is universal over saturated formations; the restricted readings examined are listed in the verdict. In the universe of soluble groups the answers are the same. ## Results in the paper | Item | Statement | Proof status | |---|---|---| | Lemma 3.1, Corollary 3.2 | every nonabelian factor group Q of U(q) has Z(Q) = Q' = Phi(Q), |Q:Q'| = q^2, |Q'|^2 <= |Q:Q'|, and [y,Q] = Q' for y outside Z(Q) | complete, elementary | | Lemma 4.1 | construction: H a p'-group on P with P/Phi(P) irreducible, F = LF(f), f(p) = X, f(l) = all; then P x| H is minimal non-F with residual P | complete; X need not be subgroup-closed | | Theorem 4.2 | witness A: P = F64 x F4, H = C9, |G| = 2304; [(1,0),P] has 2 elements while |P'| = 4 | complete; finite facts also checked by programs | | Proposition 4.3 | witness B: P = F81 x F81, beta = x y^3 - x^3 y, H = C16; |P'| = |P:P'| = 3^4 | complete | | Lemmas 5.1, 5.2 | invariant forms; groups determined by squaring/commutator maps | complete | | Proposition 5.3 | structure of all candidates, (a)–(f) | (a)–(e) complete; (f) uses Zhurtov–Syskin (review) | | Theorem 5.4 | no counterexample of order < 2304 | complete; does not use 5.3(f) | | Proposition 6.1 | the witnesses' formations are subgroup-closed, contain U, have abelian f(p); a variant lies in N_p N_r | complete | | Lemma 6.2, Proposition 6.3 | F = N^2 (SL(2,3), p = 5) and F = NA (M16, p = 3): answer no | complete | | Remark 6.4, Theorem 6.5 | F = N and F = U: answer yes | via Zhurtov–Syskin (review); not claimed new | ## Computations (sanity checks; programs and outputs in reproducibility/) - `check_note.py` (standard library; 62 checks, 3.6 s): witness A completely (field facts; associativity on all 256^3 triples; Z(P) = P' = Phi(P) of order 4; [(1,0),P] of order 2, distribution {2: 36, 4: 216} over the non-central elements; 111 involutions; h of order 9, h^3 trivial on Phi(P); irreducibility; the 105 h^3-invariant and 3 h-invariant subgroups; membership in F from the definition for the 108 subgroups representing all subgroups of order divisible by 3 up to conjugacy, with G the only one outside F; the normal subgroups of G and G^F = P); U(8) as the set of all unitriangular matrices preserving the form (64^3 candidates), Lemma 3.1 for it, and its 7 quotients of order 2^8 (|[y,Q]| = 4 for all non-central y, 39 involutions); witness B; Lemma 5.1 in the cases used. Deliberate errors in modified copies make it fail. - The five programs with which the results were first obtained (`certificates/`), run again from the package: outputs identical to the recorded ones, byte for byte (cert_lemma1 2.0 s, cert_2304 5.4 s, cert_104976 5.2 s, cert_small_cases 2.7 s, cert_restricted 0.2 s). In the package their docstrings refer to the note, and the statement that wrote the output file was replaced by a comment. - The five programs of the third verification run (`independent_run_2/`: r2_unitary, r2_witnessA, r2_witnessB, r2_least, r2_restricted; 250 checks in about 30 s; Python 3.13.5, NumPy 2.5.2): see the section on the third run below and the README of that folder. ## Independent verification runs The results were first obtained with proofs and with the five programs. Two independent verification runs, both AI-assisted, followed (2026-10-21); each re-derived the proofs line by line, wrote its own programs, and searched the literature. | Item | Run A | Run B | |---|---|---| | Statement and reading against the Notebook (v48) and the record | CONFIRMED | CONFIRMED | | Lemma 3.1 (factor groups of U(q)) | CONFIRMED (re-derived; own programs, q = 2..9, all quotients by central subgroups) | CONFIRMED (re-derived; all unitriangular unitary matrices for q = 2..9, and |SU(3,4)| by brute force) | | Lemma 4.1 (construction) | CONFIRMED | CONFIRMED; noted that subgroup-closure of X is not used | | Witness A | CONFIRMED (own field model x^6+x^5+1; involution counts 111 and 39) | CONFIRMED (own model; full table of G; membership in F tested from the definition for all subgroups up to conjugacy, with wrong local definitions as a test) | | Witness B | CONFIRMED (own model x^4+x^3+2) | CONFIRMED | | Proposition 5.3 | CONFIRMED; (f) checked against the zbMATH review | CONFIRMED; (f) checked against the review | | Least order 2304 | CONFIRMED_WITH_FIXES (direct argument for p = 2, m = 4, H~ = C5) | CONFIRMED (all 35 tuples (p,m,d) of small order covered; odd-order irreducible subgroups of GL3(2), GL4(2) enumerated) | | F = U (yes) | CONFIRMED_WITH_FIXES (sub-case r = 2, b = 0) | CONFIRMED_WITH_FIXES (wording in case 2a) | | F = N^2, F = NA | CONFIRMED (own programs) | CONFIRMED (own programs) | | Novelty | no published answer found | no published answer found | Neither run found a mathematical error. **Corrections required by runs A and B**, all applied in the note: 1. (Run A) Least order, case p = 2, m = 4, H~ = C5: the appeal to Proposition 5.3(f) is replaced by the direct argument (the C5-invariant quadratic forms on F16 are 0 and the forms pi o N; Lemma 5.1(b) and Lemma 5.2), so Theorem 5.4 does not use 5.3(f): Theorem 5.4, case (2,4). 2. (Run A) Theorem 6.5, Case 1: the sub-case r = 2, b = 0 (impossible, as it forces p = 2 and H~ = C2). 3. (Run A) [x,U], [y,Q], [x,P] denote sets of commutators, which are subgroups here: Section 2. 4. (Run A) References: Zhurtov–Syskin cited by the Russian original (Sib. Mat. Zh. 28 (1987), no. 2, 74–78; Zbl 0622.20010) with the translation Siberian Math. J. 28 (1987) 235–239, doi:10.1007/BF00970869 (Crossref); the Notebook's volume 26 is noted; Mazurov–Syskin cited as Mat. Zametki 14 (1973) 217–222, translation Math. Notes 14 (1973) 683–686 (doi:10.1007/BF01147114), Zbl 0284.20025. Textbook theorem numbers are marked as not checked ("Scope and priority"). 5. (Run A) The answer is unchanged in the universe of soluble groups: Section 1.2. 6. (Run A) The involution counts (111 in P, 39 in each U(8)/N) as a second invariant: after Theorem 4.2. 7. (Run B) The reference "Shemetkov, Yi (2008)" (not found in Crossref) was dropped; it is not needed. 8. (Run B) Theorem 6.5, case 2a: "the prime s of Q is odd". 9. (Run B) Lemma 4.1: the hypothesis that X is subgroup-closed is dropped, with a remark that it is not needed. 10. (Run B) A test of the minimal non-F property in the package: `check_note.py`, Part D. **Changes made when the note was written, after runs A and B.** - The proof of Theorem 5.4 was shortened. The cases p >= 5, (p,m) = (3,3) and (2,5) now use that d >= 2 when m is odd (Proposition 5.3(c)), instead of the Singer-cycle and invariant-form arguments; the case (3,4) uses Lemma 5.2 (groups of exponent p are determined by their commutator map) instead of the classification of extraspecial groups or 5.3(f); the cases (2,4) and (2,6) use Lemma 5.1(b) and Lemma 5.2, with a proof of 5.1(b) by eigenvalues. Proposition 5.3(e) for odd p also uses Lemma 5.2. - Theorem 6.5, Case 2: the bounds m <= |H~:A| for abelian A and the scalar action of a central M are proved directly (diagonalizability over F_p), without absolute irreducibility; the fact that [Q,y] = Q in a Schmidt group with abelian Q is derived from coprime action. - These rearrangements were not examined by runs A and B. Their finite inputs are checked by `check_note.py` (Part G and Part E), and every step is written out in the note. They were examined by the third run (below). ## Third verification run (second stage; examined the final text) A third AI-assisted run (2026-10-21) examined the final text of the note and the package, the shortened proof of Theorem 5.4 first. Its programs and outputs are in `reproducibility/independent_run_2/` (with a README); they were written from the definitions printed in the note, with an own finite-field toolkit and a fourth model of F64 (x^6+x^5+x^4+x+1), before the programs of the package were run. | Item | Third run | |---|---| | Statement and reading | CONFIRMED: v48 (local copy) and v49 (arXiv, 20 Oct 2026), printed page 99, identical text, no comment, no star; record KOU-15.98 (open) as described; the Notebook's notation for unitary groups is not uniform (PSU(3,5^2) in 18.98), so the reading is settled, as in the note, by the known case F = N; the counterexamples work under both readings, the least order refers to the first | | Lemma 3.1, Corollary 3.2 | CONFIRMED_WITH_FIXES (abelian case of (b) written out); own program for q = 2..16, all unitriangular matrices tested for q <= 9, all factor groups by central subgroups | | Lemma 4.1, witness A | CONFIRMED; own program: associativity on all 256^3 triples, the 108 groups K x\| , K x\| tested from the definition of LF(f) (only G outside F), the Sylow reduction checked, normal subgroups of G, G^F = P, tests with f(2) = A(9) and A(1), involution counts 111 and 39, P not semi-extraspecial | | Witness B | CONFIRMED; own program (bilinearity on all triples, radical, span, exponent, orders, irreducibility) | | Lemmas 5.1, 5.2, Proposition 5.3 | CONFIRMED_WITH_FIXES (5.3(e): why H~ is nontrivial); Lemma 5.1 for the groups used and n = 1..4; Lemma 5.2 checked in the instances U(4)/N_0, U(8)/N_0 (all lambda), U(4) (d = 2) and U(9)/N_0 | | Theorem 5.4 (shortened proof) | CONFIRMED in every case; (2,4) with H~ = C5 by Lemma 5.1(b) and Lemma 5.2, without Proposition 5.3(f); odd-order irreducible subgroups of GL3(2) and (order < 72) of GL4(2) enumerated (C7, C7 x\| C3; C5, C15); facts on GL3(3) for the case (3,3); one slip corrected ({1,3,9}) | | Proposition 6.1, Lemma 6.2, Proposition 6.3 | CONFIRMED_WITH_FIXES (Lemma 6.2: members of LF(f) are soluble); own program for the SL(2,3) and M16 witnesses | | Remark 6.4, Theorem 6.5 | CONFIRMED (modulo Proposition 5.3(f), as stated); Case 1 arithmetic checked for p < 60 | | Package | re-run from an extracted copy of the source archive: all six outputs identical to the recorded ones; the four deliberate-error tests of Section 8 reproduced on copies | | Sources and references | Lewis (arXiv:1709.03857) Sections 4 and 5 and the abstracts of the Ballester-Bolinches papers and of Brenner–Camina–Lewis read again; every DOI checked with Crossref, Zbl numbers and Russian originals with zbMATH | | Novelty | no published answer found (arXiv API, zbMATH API, Crossref, OpenAlex including the works citing Mazurov–Syskin, Zhurtov–Syskin and Semenchuk 1979, one web search) | The third run found no mathematical error. **Corrections required by the third run**, all applied: 1. Lemma 3.1(b): the case "Q abelian, N <= Z" is written out (then Z = U' <= N). 2. Proposition 5.3(e): H~ is nontrivial because V is irreducible of dimension 2. 3. Lemma 6.2: the members of LF(f) are soluble (a nonabelian chief factor embeds in its centralizer quotient), so LF(f) = N^2, respectively NA. 4. Section 2: the F-residual exists for nonempty formations; the converse of the theorem of Gaschütz, Lubeseder and Schmid is used only in Theorem 5.4 (Remark 6.4 and Theorem 6.5 use explicit local definitions). 5. Section 1.1: the least-order statement refers to the first reading (under the second, SL(2,3) of order 24 would already be a counterexample). 6. Conclusion: "yes" for N and U rests on the Zhurtov–Syskin theorem as stated in its review. 7. Remark 3.3 and Section 1.3: Beisiegel proved that the Sylow p-subgroups of SU3(p^{2a}) are semi-extraspecial (Lemmas 4 and 5 of his paper, as reported by Lewis); Lemma 3.1(a) is a direct proof. 8. Verification paragraph: the argument for the case p = 2, m = 4 is credited to both earlier runs; this run is recorded. 9. Section 8: the deliberate-error tests were made on modified copies, not included in the package. 10. Scope and priority: Sylow's theorems, Schur's lemma and the three subgroups lemma added to the standard tools; the description of the searches updated. 11. Theorem 5.4, case (2,4): |H~|/5 lies in {1, 3, 9}. ## Relation to the literature, novelty and scope - **Searches (2026-10-21).** zbMATH Open API, Crossref, arXiv API, OpenAlex (including the full lists of works citing Ballester-Bolinches–Esteban-Romero–Robinson 2005 and Ballester-Bolinches–Esteban-Romero 2007), and a few web searches, by the original work and the two runs; the search of Semantic Scholar refused automated access (HTTP 429). The third run repeated the searches (arXiv API, zbMATH API, Crossref, OpenAlex including the full lists of works citing Mazurov–Syskin, Zhurtov–Syskin and Semenchuk 1979, one web search). No printed answer to Problem 15.98 was found. A search that finds nothing is not a proof of novelty. - **Read.** The Notebook (v48, v49); the zbMATH review of Zhurtov–Syskin and the zbMATH record of Mazurov–Syskin; the abstracts of Ballester-Bolinches–Esteban-Romero–Robinson (PAMS 133 (2005)), Ballester-Bolinches–Esteban-Romero (Rev. Mat. Iberoam. 23 (2007)) and Brenner–Camina–Lewis (arXiv:2502.01598); Sections 1–5 of Lewis, "Semi-extraspecial groups" (arXiv:1709.03857), for the definition of semi-extraspecial groups, Verardi's characterization, Beisiegel's bound and Beisiegel's result that the Sylow p-subgroups of SU3(p^{2a}) are semi-extraspecial. The third run also compared the definitions of Section 2 (formation, saturated and local formations, minimal non-F-groups) with Section 1 of Murashka, arXiv:2405.20257 (which quotes Doerk–Hawkes IV, Definitions 3.1). - **Not read.** The papers of Zhurtov–Syskin and Mazurov–Syskin themselves; Semenchuk (Algebra and Logic 18 (1979)) and his later papers; the books of Doerk–Hawkes (1992), Ballester-Bolinches–Ezquerro (2006), Huppert (1967) and Shemetkov (1978); Beisiegel (1977). Chapter and theorem numbers of textbook results are not checked. - **Bibliographic data.** Checked with Crossref (DOIs of Mazurov–Syskin, Zhurtov–Syskin, Semenchuk, both Ballester-Bolinches papers, Doerk 1966, Beisiegel 1977, Doerk–Hawkes, Ballester-Bolinches–Ezquerro, Huppert), zbMATH (Zbl numbers; Shemetkov 1978) and the arXiv API. - **Scope.** The construction of Lemma 4.1 is elementary and may be known to specialists; no priority is claimed. The question as posed is answered completely; Theorem 6.5 (F = U) depends on the Zhurtov–Syskin theorem as stated in its review. ## 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/