# Verification report — KOU-19.82 (Kourovka Notebook, Problem 19.82, P. Shumyatsky: soluble subgroups and the generalized Fitting height) Verification dates: 2026-10-21 (four runs on the first written version of the results) and 2026-10-22 (one run on the final text of the note). **Verdict.** The note gives a **partial answer: yes for every finite group whose nonabelian chief factors are of type PSL(2,q) (q >= 4) or Sz(q), provided that no PSL(2,q)-factor with q odd and q = +-1 (mod 8) induces a group containing PGL(2,q); the general question stays open**. Every statement it makes is proved, modulo standard facts on quasinilpotent groups, PSL(2,q) and Sz(q) that are listed with their sources. - **Proved.** Theorem 1.2: for such G and every prime p with O_p(G) = 1 there is a soluble K <= G with h(K/O_p(K)) >= h*(G); with p not dividing |G| this gives a soluble K with h(K) = h*(G). It follows from the reduction theorem (Theorem 2.10: relative witness systems for the pairs (S, A) induced on the nonabelian chief factors, with the closure condition (Cl')) and the witness systems of Propositions 3.7 (PSL(2,q)) and 4.4 (Sz(q)). Special cases: all groups whose nonabelian composition factors are PSL(2,2^f), PSL(2,q) with q = +-3 (mod 8), or Sz(q); all groups with only A_5 as nonabelian composition factor; S_6 wr H, M_10 wr H. - **Taken from the literature.** The strategy, the lift of soluble subgroups of quotients (Lemma 2.4: the argument of the proof of Lemma 3.1 of Khukhro-Shumyatsky 2015) and the treatment of a minimal counterexample through the kernel of the action on the simple factors and Sylow normalizers (Lemma 3.2 of the same paper); standard facts (see "Readings"). - **Limits.** (i) Not covered: PSL(2,q)-factors with q odd, q = +-1 (mod 8) and induced group containing PGL(2,q); at p = l the method fails there (exhaustive searches for all eleven such q <= 73 find no witness), while in all such groups tested other soluble subgroups still reach h*; not a counterexample. (ii) The witness criterion is only sufficient and is not available for every simple group (it fails for A_6 with the PGL(2,9) class and for PSL(2,7), PSL(2,17), PSL(2,23), PSL(2,25) at the defining prime with the diagonal class). (iii) A_n (n >= 7), other groups of Lie type and the sporadic groups are not treated. (iv) O_p(G) = 1 is needed in the p-form (s_2(S_4) = 2 < 3 = h*(S_4); also 2^6:SigmaL(2,8) and 2^6:PGL(2,7)). - **Trivial range.** For h*(G) <= 2 the answer is yes for every finite group (Remark 1.1, Schmidt's theorem). 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 19.82 (P. Shumyatsky, 19th issue, 2018), read in version 48 (6 October 2026; local text extraction) and in version 49 (20 October 2026; downloaded by two verification runs). The text is the same in both versions; there is no comment, star or archive mark. - Definitions: h*(G) is the least k with F*_k(G) = G, where F*_1(G) = F*(G) and F*_{i+1}(G) is the inverse image of F*(G/F*_i(G)); for soluble G, h*(G) = h(G). - Question: does every finite group G contain a soluble subgroup K with h*(G) = h(K)? - **Corpus record.** ulamai/UnsolvedMath, KOU-19.82 (dataset version 1.8.0, snapshot d371ee9, record id 701630, status `open`): the statement field reproduces the Kourovka text in full. - **Conventions.** The note starts the series with F*_0 = 1, so h*(1) = 0; the Notebook starts with F*_1. The two conventions differ only for G = 1, where K = 1 answers the question. - **Reformulation.** The Fitting heights of the soluble subgroups of G are exactly 0, 1, ..., s(G) (Lemma 2.1), so the question is equivalent to h*(G) <= s(G), s(G) the largest Fitting height of a soluble subgroup. ## Readings | Reading | Answer | Where | |---|---|---| | The question for all finite groups | open; yes for the class of Theorem 1.2 | Theorem 1.2, Section 5.1 | | Groups with h*(G) <= 2 | yes, trivially (Schmidt's theorem on minimal non-nilpotent groups) | Remark 1.1 | | The opposite inequality h(K) <= h*(G) for all soluble K | false (S_3 <= A_5), not the question | Section 1.1 | | The stronger p-form s_p(G) >= h*(G) | proved for the class of Theorem 1.2 when O_p(G) = 1; false without O_p(G) = 1 | Theorems 1.2, 2.10; Section 5.1(iv) | | The nonsoluble length lambda(G) <= s(G) | known (Khukhro-Shumyatsky 2015, Corollary 1.2); weaker, as lambda(G) <= h*(G) | Section 1.1 | ## Results in the paper - **Section 2 (reduction).** Standard facts (Q1)-(Q3); Lemma 2.1 (interval of heights); Lemmas 2.2-2.3 (quasinilpotent groups form a formation; residual series; properties of h*); Lemma 2.4 (the argument of Khukhro-Shumyatsky's Lemma 3.1, with Feit-Thompson); Lemma 2.5 (coprime stability, self-contained); Lemma 2.6 (q'-elements of a normal quasinilpotent subgroup centralize normal q-subgroups); Lemma 2.7 (F*(X) is the intersection of the centralizers of the chief factors below it); Definitions 2.8 (induced pairs, the class C(Sigma)) and 2.9 (witness systems, condition (Cl')); Theorem 2.10 (reduction), with the minimal counterexample over all pairs (G, p) and the change of prime in the abelian step; Lemmas 2.11 and 2.12. - **Section 3 (PSL(2,q)).** Facts (P1)-(P4); Lemma 3.1 (Lang-Steinberg for field automorphisms of PGL(2,q), proved from Lang's theorem; also for r = l; no involution in the coset delta*sigma^(f/2)); Lemma 3.2 (C(U) = U); Lemma 3.3 (field-type cosets, p != l: X = U); Lemma 3.4 (field-type cosets, p = l: a Sylow t-subgroup for a Zsigmondy prime t); Lemma 3.5 (the diagonal coset: U for p != l, a Klein four-group for p = l and q = +-3 mod 8); Remark 3.6 (the excluded pairs: check of candidates and computations, not a proof); Proposition 3.7; Example 3.8 (A_5). - **Section 4 (Sz(q)).** Facts (Z1)-(Z3); Lemma 4.1 (Lang-Steinberg for Sz(q)); Lemma 4.2 (p odd: X = U, with C_B(U) <= U proved from the fixed-point-free action on Z(U)); Lemma 4.3 (p = 2: Zsigmondy primes for 2^n - 1 or 2^(4n) - 1); Proposition 4.4. - **Section 5.** Proof of Theorem 1.2 and Corollary 1.3; the limits (i)-(iv); conclusion. ## Computations (sanity checks; programs and outputs in reproducibility/) No proof depends on a computation. Computations describe the limits of the method and serve as checks. - **Program written with the note** (`check_note.py`, standard library, 6 s, 2667 checks, all passed): the inequality of Lemma 3.3 for q <= 2^20; the choice of t in Lemma 3.4 for 731 field classes (l < 60, f <= 30) and in Lemma 4.3 for 150 classes (odd n <= 201), via the primitive parts of cyclotomic values; Zsigmondy's theorem for l < 60, n <= 60; Lemmas 3.1-3.5, (P3) and (Cl') by brute force in PGammaL(2,q) for the 16 prime powers 4 <= q <= 32; S_4 (h* = 3, s_2 = 2) and S_5 (h* = 2). - **Programs with which the results were first obtained** (`first_version/`): `check_A5_witness.py`, `check_psl2_witness.py` (16 values q <= 32), `check_zsigmondy_choices.py` (65 classes); all print "ALL CHECKS PASSED" and were re-run with identical output. - **Programs of the verification runs** (`independent_runs/`, see its README): exact h*, s, s_p over all soluble subgroups of 14 groups with composition factors A_5 and of 29 groups with socle PSL(2,q), Sz(8) or abelian socle; lower bounds for 12 larger groups; Lemma 2.7 on 13 groups; Theorem 2.10 on 169 classes of nonsoluble subgroups of 10 groups; witness tables for 14 simple groups; the witness lemmas for all 42 prime powers 4 <= q <= 128; Sz(8) and its automorphism group built in two independent ways; the exhaustive searches of Remark 3.6 (all subgroups for q = 7, 9, 17, 23, 25; all soluble subgroups for the eleven odd q = +-1 mod 8 up to 73). No violation of the inequality s_p(G) >= h*(G) (for O_p(G) = 1) or of the question was found. - **Programs of the run on the final text** (`independent_run_2/`, see its README): Definition 2.9 and Lemmas 3.1-3.5, (P2), (P3), (Cl') for the 16 prime powers 4 <= q <= 32 (295 checks); all subgroups of PSL(2,q) for the 12 odd prime powers 5 <= q <= 31 (Remark 3.6: no witness for (l, delta) when q = +-1 mod 8, exactly V and A_4 otherwise); Sz(8) built from the Suzuki-Tits ovoid with (Z1)-(Z3) and Lemmas 4.1-4.3 (34 checks); the arithmetic of Lemmas 3.3, 3.4, 4.3 by complete factorization of cyclotomic values (3674 checks); S_4 and the group G = of Section 5.1(i) (h*(G) = h_7(N_G(Q)) = 4). All passed. - **Re-runs on 21 October 2026** from a copy of the package: see `reproducibility/README.md`; all re-run outputs were reproduced (up to fields that record running times and, in one program, the choice of a class representative). ## Independent verification runs The results were first obtained in two stages: the reduction with the case of A_5, then the relative witness systems with PSL(2,q) and Sz(q). Each stage was examined by two independent verification runs, all AI-assisted, each with its own programs: run A (statement, proofs, sources) and run B (independent computation, literature). | Item | Stage 1, run A | Stage 1, run B | Stage 2, run A | Stage 2, run B | |---|---|---|---|---| | Statement, record, Kourovka v48/v49, conventions, reformulation | CONFIRMED | CONFIRMED | CONFIRMED | CONFIRMED | | Lemmas 2.1-2.7 | CONFIRMED (re-derived); textbook numbers not checked | CONFIRMED (re-derived); Lemma 2.7 tested on 13 groups | used | used | | Theorem 2.10 (absolute version), Lemmas 2.11, 2.12 | CONFIRMED (re-derived, boundary cases) | CONFIRMED (re-derived); tested on 18 groups and 169 subgroup classes | — | — | | Theorem 2.10 (relative version: induced pairs, (Cl')) | — | — | CONFIRMED (re-derived) | CONFIRMED (re-derived) | | A_5 witness system (Example 3.8) | CONFIRMED (own program) | CONFIRMED (all subgroups) | recovered as q = 4, 5 | recovered | | Lemma 3.1 / Lemma 4.1 (Lang-Steinberg) | — | — | PROVED from Lang's theorem; checked for q <= 32 and Sz(8) | checked for q <= 128 (also r = l) and Sz(8) | | Lemmas 3.2-3.5, Proposition 3.7 | — | — | CONFIRMED (re-derived; definition-level check, 165 cases) | CONFIRMED (brute force, 42 values of q) | | Remark 3.6 (excluded pairs) | — | failure of the criterion found for A_6, PSL(2,7), PSL(2,17), PSL(2,23), PSL(2,25) | CONFIRMED for q = 7, 9 over all subgroups | CONFIRMED for 11 values q <= 73 over soluble subgroups | | Lemmas 4.2, 4.3, Proposition 4.4 | — | witness system for Sz(8) | CONFIRMED (re-derived; Sz(8) from scratch) | CONFIRMED (Sz(8) and Sz(8):3 by brute force) | | Theorem 1.2 | — | — | CONFIRMED_WITH_FIXES (hypothesis) | CONFIRMED_WITH_FIXES (hypothesis); tested on 29 + 8 groups | | Literature | no prior answer found | no prior answer found | no new paper | no prior answer found | No run found a gap in a proof. **Corrections required by the runs, all applied in the note:** 1. (Stage 2, both runs) The hypothesis of Theorem 1.2 is that the induced group does not contain PGL(2,q) (delta not in A/Inn(S)); an earlier formulation presented it as equivalent to "the induced group lies in PSigmaL(2,q)", which differs for f even (e.g. M_10). The cosets delta*sigma^(f/2) are field-type and covered by Lemmas 3.3-3.4. 2. (Stage 2, both runs) Remark 3.6 is labelled a check of candidates; only the (b1) part is proved; the exhaustive searches for q = 7, 9 and up to 73 are cited. 3. (Stage 2, run B) The Sylow 2-normalizers in (P3) (A_4, or self-normalizing; never S_4). 4. (Stage 2) The Lang-Steinberg statements are proved in Lemmas 3.1 and 4.1 (including r = l) instead of being cited by theorem number; Lemma 3.4 states that the r-part lies in c (e = 1 mod r) and uses t odd, t != l; Lemma 2.12 states that the transported induced groups equal A; Lemma 4.2 proves C_B(U) <= U. 5. (Stage 1, run B) The general question is not presented as depending on witness systems only: the criterion fails for some simple groups (Section 5.1(ii)); one term, "witness system", is used throughout. 6. (Stage 1, run A) The trivial range h* <= 2 is stated (Remark 1.1); the induction over pairs (G, p) and the change of prime are explicit; Lemma 2.7 says that the nonabelian chief factors below F* are X-isomorphic to the sections E_jZ/Z; the step "y = zz'N" in Lemma 2.11 is written out. 7. (Both stages) References: textbook numbering is marked as not checked; P. Hall 1958 and Gorenstein's coprime-action theorem are replaced by the self-contained Lemma 2.5; the remarks on A_n and Lie type are labelled unproved; "complete proof" reads "complete modulo the standard facts listed". The note was written after these runs. ### Verification run on the final text (2026-10-22) A further verification run, also AI-assisted, examined the final text as if nothing had been checked (earlier runs had examined the first written version of the results, not this text). - **Statement.** Problem 19.82 read again in Kourovka v48 (local text) and v49 (downloaded): same text, no comment; corpus record unchanged (status open); conventions and reformulation confirmed. Khukhro-Shumyatsky 2015 (arXiv v3) read again: Corollary 1.2 is lambda(G) <= max Fitting height of soluble subgroups; Lemma 3.1 is stated for the (2 x 2')-length and its proof contains the lift used in Lemma 2.4; Lemma 3.2 is the model of Lemmas 2.11-2.12; the paper uses the classification through Schreier's conjecture. - **Proofs.** All proofs of Sections 2-5 re-derived line by line (Lemmas 2.1-2.7 including the coprime Lemma 2.5 in place of P. Hall's theorem; Theorem 2.10 with the induction over pairs (G, p), the change of prime, Step 3 and the transversal argument of Lemma 2.12; Lemmas 3.1-3.5 including r = l, the case f even and the coset delta*sigma^(f/2); Lemmas 4.1-4.3; Theorem 1.2 and Corollary 1.3): CONFIRMED, no error and no gap. - **Computations.** Own programs (`reproducibility/independent_run_2/`, listed under "Computations"): all passed. The package was re-run from the archive: all recorded outputs reproduced (up to running times and, in one program, the class representative attaining a maximum); one recorded output (`stage2_run_B/out/zsig_arith.txt`) has a last line written by a separate call, now documented in `independent_runs/README.md`. - **Literature.** arXiv API, zbMATH API, OpenAlex, Semantic Scholar, Crossref (all 15 DOIs) and one web search (21 October 2026 UTC): no answer to Problem 19.82 found; the related papers are described correctly. - **Corrections required by this run, all applied in the note:** (1) Lemma 2.4 is attributed to the argument of the proof of Lemma 3.1 of Khukhro-Shumyatsky, which is stated there for the (2 x 2')-length (Lemma 2.4, Section 1.3, Scope). (2) Remark 3.6 and Section 5.1(ii) state the same coverage of the exhaustive searches: no witness among all subgroups for q = 7, 9, 17, 23, 25, 31, among all soluble subgroups for the eleven odd q = +-1 mod 8 up to 73. (3) Section 5.1(i) gives h* = 3, 4, 3, 4 for the four groups listed. (4) The abstract says "the nonsoluble length lambda(G), which is at most h*(G)" and names unipotent subgroups among the ingredients. (5)-(8) The Verification paragraph, Section 6, the package READMEs and the Scope paragraph record this run. ## Relation to the literature, novelty and scope - **Searches (21 October 2026, by the verification runs and at the writing).** arXiv API (all nine abstracts containing "generalized Fitting height", re-queried at the writing; "nonsoluble length" and variants; "Fitting height" with soluble/solvable subgroups; the papers of P. Shumyatsky), OpenAlex (works matching "generalized Fitting height"; the 40 works citing Khukhro-Shumyatsky 2015, titles read), zbMATH API (16 documents matching "generalized Fitting height"; review 1570.20021), Semantic Scholar (51 works citing arXiv:1310.2434, titles read), Crossref (all bibliographic data with a DOI), five web searches in all (the last one by the run on the final text, which also repeated the arXiv, zbMATH, OpenAlex and Semantic Scholar queries). - No answer to Problem 19.82 was found in print or as a preprint; Kourovka v49 (2026-10-20) records none. - Related, none answering 19.82: Murashka-Vasil'ev 2025 (h*(G) - h*(M) <= 2 for maximal M); Khukhro-Shumyatsky 2026 (arXiv:2605.08596, Hall subgroups); Guralnick-Tracey 2020 (Engel-type conjectures); Guo-Vdovin 2018 (Carter subgroups; only the bibliographic data were seen at the writing, the publisher's page refused the automated request; a verification run read its abstract); Gong-Yan-Cosme Llopez 2026 (arXiv:2610.28082, soluble groups only). - **What was read.** Khukhro-Shumyatsky 2015 in the arXiv version 3 (Sections 1-3, with Theorem 1.1, Corollary 1.2 and the proofs of Lemmas 3.1 and 3.2); the Kourovka text; the arXiv abstracts of the related papers and the zbMATH review of Murashka-Vasil'ev; the introduction of Guralnick-Tracey (by two verification runs). The books (Huppert I, Huppert-Blackburn III, Aschbacher, Kurzweil-Stellmacher, Gorenstein-Lyons-Solomon 3) were not available; their section and theorem numbers were not checked (chapter titles of Huppert I and Huppert-Blackburn III checked with Crossref). - **Caveats.** A search that finds nothing is not a proof of novelty; no priority is claimed. The classification of finite simple groups is not used. - **Scope.** The note answers the question for the class of Theorem 1.2 only; the general question stays open. Suggested label for the corpus: open, unchanged (unrefereed partial answer; at most "partially solved" at the maintainers' discretion; never "solved"). ## 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/