# Verification report — KOU-7.27 (Kourovka Notebook Problem 7.27, V. M. Levchuk: diagonal matrices of SL_n(q) and their irreducible overgroups)

Verification date: 2026-10-18.

**Verdict.** The note gives a **complete affirmative answer**, which is **a short corollary of published theorems**;
every statement it makes is proved.
- **Settled.** For every fixed n ≥ 2, the proportion of regular diagonal matrices d of SL_n(q) with property P(d)
  tends to 1 as q → ∞ (Theorem 1.1). P(d): every proper subgroup of SL_n(q) that contains d and is irreducible over
  F_q permutes the summands of a decomposition of F_q^n into coordinate subspaces. So for all q ≥ q_1(n) such d
  exist, in each reading of the question (irreducibility over F_q or absolute irreducibility; blocks arbitrary or
  coordinate subspaces).
- **Not new mathematics.** Theorem 1.1 follows from the density theorems of Fulman and Guralnick (Durham 2001
  proceedings, World Sci. 2003, Theorem 3.2 with the four-type description of Section 3; Trans. AMS 364 (2012),
  Theorem 7.3). What the note adds is the deduction written out in full, which was not found in print, and the
  complete answer for n = 2.
- **n = 2, complete.** A diagonal d ∈ SL_2(q) with P(d) exists if and only if q ∈ {4, 8} or q ≥ 13; for q ≥ 13,
  diag(w, w^{-1}) works for every generator w of F_q^* (Theorem 1.2). Proposition 6.3 gives all such d for q ≥ 4.
  With absolute irreducibility the only change is that q = 2 becomes admissible (SL_2(2) has no proper absolutely
  irreducible subgroup).
- **Limits.** q_1(n) is not explicit for n ≥ 3; a bound independent of n is not treated; Levchuk's 1973 paper on
  n = 2, 3 was not seen.
- **Verification.** Three independent AI-assisted verification runs: runs A and B examined the first write-up;
  run 2 examined the final text and the package (programs in `reproducibility/independent_run_2/`). No run found a
  mathematical error; all required corrections were made (listed below).

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, Novosibirsk 2026, arXiv:1401.0300v48 (6 October 2026), page 24 (7th Issue, 1980), Problem 7.27, signed
  V. M. Levchuk. Read in the arXiv text extraction and in the PDF (page image). The question asks whether SL_n(q)
  contains, for sufficiently large q, a diagonal matrix lying in no proper irreducible subgroup of SL_n(q) "with the
  exception of block-monomial ones"; the Notebook adds that the answer is affirmative for n = 2, 3 (V. M. Levchuk,
  Krasnoyarsk 1973) and that similar problems for other Chevalley groups are of interest. The problem is in the
  list of open problems, without a star or a comment; no other problem refers to it (the name index lists
  "Levchuk V. M. 7.27–28"). The wording of version 46 is the same.
- **Corpus record.** ulamai/UnsolvedMath, record KOU-7.27 (dataset version 1.8.0; status `open`), statement equal
  to the Notebook text.

## Readings
| Reading | Answer | Where |
|---|---|---|
| n fixed, "sufficiently large q" = all q ≥ q_1(n) | yes, with almost all regular diagonal matrices | Theorem 1.1 |
| "irreducible" over F_q | yes (this is the property P(d) proved) | Theorem 1.1 |
| "irreducible" = absolutely irreducible | yes (fewer subgroups to consider) | Section 1.1 |
| "block-monomial" with respect to coordinate subspaces of the standard basis | yes (P(d) uses coordinate blocks) | Theorem 1.1, Lemma 4.3 |
| a bound q_1 independent of n | not asked, not treated (for q ≤ n no regular diagonal matrix exists) | Section 1.1 |
| n = 2, every q | good d exists iff q ∈ {4, 8} or q ≥ 13 (over F_q); with absolute irreducibility also q = 2 | Theorem 1.2 |

## Results in the paper
- **Theorem 1.1.** For fixed n ≥ 2, \|G_q\|/\|T^reg\| → 1 and \|G_q\|/\|T\| → 1, where G_q is the set of regular
  diagonal d with P(d).
- **Theorem 3.1** (quoted exactly): Fulman–Guralnick 2012, Theorem 7.3 — almost simple G with socle a Chevalley
  group S of fixed rank, G inside the inner-diagonal automorphisms of S; the union of the maximal subgroups of G not
  containing a maximal torus of S has proportion tending to 0. The statement gives only the limit; the rates
  O(log log q/q) and O(1/q) are in the proof.
- **Theorem 3.2** (quoted): Fulman–Guralnick 2003, Section 3 and Theorem 3.2 — for a fixed type of simple algebraic
  group (the characteristic may vary with q), the four types of maximal subgroups of X(q) (bounded order; subfield;
  Y_σ with Y semisimple of smaller rank; Y_σ with Y of maximal rank) and the vanishing proportion of the union of
  types 1–3 as q → ∞ through all prime powers.
- **Lemma 2.3** (passage SL_n(q) → PSL_n(q)): maximal subgroups contain the centre; correspondence and equal
  proportions.
- **Lemma 3.3** (step (a)): \|T \ R_N\| ≤ n(n−1)N²\|T\|/(q−1) for R_N = {d : d^e regular for e ≤ N}; for regular d,
  C(d) = T and the class of d meets T in the n! permutations of d; the classes of R_N cover the proportion
  \|R_N\|/(n!\|T\|) ≥ (1 − n(n−1)N²/(q−1))/n!.
- **Remark 3.4.** For large q, a maximal subgroup of SL_n(q) contains a maximal torus if and only if it is of maximal
  rank; so Theorem 3.1 (for PSL_n(q), through Lemma 2.3) and Theorem 3.2 describe the same subgroups, and the proof
  of Theorem 3.1 gives the rate O(log log q/q).
- **Lemma 4.1** (step (b)): a maximal subgroup Y^F of maximal rank is reducible (C1), permutes a decomposition (C2),
  or normalizes an extension field F_{q^r}, r | n prime (C3). Proof: unipotent radical (fixed vectors, Galois
  descent), structure of reductive maximal-rank subgroups of SL_n (root subgroups, equivalence relation, blocks
  W_s), action of the Frobenius on the blocks.
- **Lemma 4.2** (C3 excluded): no diagonal d with d^r regular normalizes such a field (eigenvalue argument).
- **Lemma 4.3** (coordinate blocks): if d^e is regular for e ≤ n!, every decomposition permuted by d consists of
  coordinate subspaces fixed by d.
- **Section 5.** Proof of Theorem 1.1: \|T \ G_q\| ≤ (n(n−1)(n!)²/(q−1) + n! ε_n(q))\|T\|.
- **Section 6 (n = 2).** Lemma 6.1 (irreducible and not block-monomial ⇔ all orbits on P¹ have ≥ 3 points);
  Lemma 6.2 (cyclic subgroups of order m ≥ 3 are conjugate to ⟨d̄⟩); Proposition 6.3 (criterion from Dickson's list:
  P(d) holds iff m = ord(l²) = 3 and q = 4, or m ≥ 6 and m divides no element order of a subfield subgroup
  PSL_2(q_0) or PGL_2(q_0) with q_0 ≥ 3); proof of Theorem 1.2; Lemma 6.4 (reduction for direct generation);
  Table 1 (witnesses for q = 2, 3, 5, 7, 9, 11); Table 2 (all good orders of l for q ≤ 64).
- **Section 7.** One paragraph on the second, independent proof through all Aschbacher classes, with the genericity
  conditions (G) and (F); not reproduced.

## Computations (sanity checks; program and output in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, standard library, about 82 seconds; `TOTAL failures: 0`):
  - (A) arithmetic of F_q for all 70 prime powers q ≤ 256; subgroup orders in PGL_2(q) by a stabilizer chain
    against brute-force closure on 700 pairs of generators;
  - (B) Lemma 3.3(a) by exact counts: n = 2, q ≤ 1024 (197 prime powers); n = 3, q ≤ 256 (69); n = 4, q ≤ 128 (43);
  - (C) Lemma 3.3(b) by brute force: n = 2, q ∈ {4,5,7,8,9,11,13}; n = 3 in SL_3(4) and SL_3(5);
  - (D) Table 2 by Proposition 6.3 and, independently, by direct generation (Lemma 6.4), with a checked witness for
    every failure; identical, and equal to Table 2 as printed; the same comparison for the 107 pairs (q, ord(l)),
    64 < q ≤ 128, ord(l) ∤ 4;
  - (E) Table 1 by closure in SL_2(q) (non-abelian witnesses for q ≥ 3; every proper subgroup of SL_2(2) abelian);
  - (F) diag(w, w^{-1}) by direct generation for the 62 prime powers 13 ≤ q ≤ 256; the inequalities of the proof of
    Theorem 1.2 for the 2320 prime powers 13 ≤ q ≤ 20000.
- **Programs with which the results were first obtained** (not in the package): field arithmetic, an exact test of
  P(d) for n = 2 and 4 ≤ q ≤ 64 (and for primitive l at q = 67, 71, 73) by closure with early stop, counting data for
  n = 2, 3, 4, and random tests of the elementary lemmas of the second proof.
- **Verification run A** (own programs, not in the package): P(d) for n = 2 from Dickson's list for all prime
  powers 4 ≤ q ≤ 256; by direct generation for q ≤ 32 (agreement with the Dickson route: 0 mismatches); explicit
  certificates for every diagonal matrix for q = 5, 7, 9, 11 (28 matrices); an exhaustive search of two-generated
  subgroups for q = 2, 3, 4, 8 (found that q = 2, 3 fail as well); exact agreement with the original data for all
  q ≤ 64 and q = 67, 71, 73.
- **Verification run B** (own programs, not in the package): P(d) for 4 ≤ q ≤ 64 by direct generation with exact
  group orders (stabilizer chain) and by Dickson's list (139 pairs (q, k), no differences; 44 good, 95 bad);
  Dickson's list for all prime powers q ≤ 20000 (the q without a good d are exactly 2, 3, 5, 7, 9, 11); counting
  data for n = 2, 3, 4; random overgroup searches for n = 3, q ≤ 13 (evidence only); random tests of the
  symplectic and unitary C8 eigenvalue arguments; deliberately wrong inputs caught by its self-tests; re-run of the
  original programs from a copy (identical outputs).
- **Verification run 2** (programs and recorded outputs in `reproducibility/independent_run_2/`, see its README):
  Lemma 3.3 by exact enumeration (n = 2, 3, 4; q ≤ 1024, 256, 128) and, parts (b), (c), over all of SL_3(7) and
  SL_3(8); Lemma 4.2 on all subfields of M_2(F_q) with q² elements (q ≤ 32), for n = 3 through ΓL_1(q³) and for
  n = 4, r = 2 by linear algebra; Lemma 4.3 for n = 3 on all decompositions permuted by d (8 ≤ q ≤ 32);
  Theorem 1.2 and Table 1 for q ≤ 17 from the definitions alone (whole overgroup lattice of every diagonal element,
  no Dickson list, no Lemma 6.4) in four readings; Proposition 6.3 and Table 2 against direct generation with
  sympy's Schreier–Sims for all 288 pairs (q, ord(l)), 4 ≤ q ≤ 128; P(diag(w, w^{-1})) for the 62 prime powers
  13 ≤ q ≤ 256 and q ∈ {289, 343, 361, 509, 512}; the arithmetic of Theorem 1.2 for all prime powers ≤ 10^6;
  re-run of `check_note.py` from the archived package (identical output apart from running times).

## Independent verification runs
The results were first obtained with the self-contained proof described in Section 7 of the note, together with
the observation that the answer follows from the Fulman–Guralnick theorems, and with programs for n = 2. Two
independent verification runs, both AI-assisted, followed (2026-10-18); each wrote its own programs, and neither
read the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, readings, source (Notebook v48 text and PDF, v46) | CONFIRMED (record = text = PDF) | CONFIRMED |
| Short route via Fulman–Guralnick | VALID; the two named steps written out (density 1/n!; maximal-rank subgroups lie in C1–C3) | VALID; re-derived as a sketch (order count only outlined) |
| FG 2003 Thm 3.2 and FG 2012 Thm 7.3: statements and hypotheses | CHECKED in the arXiv texts; two imprecisions in the first write-up (rate stated as part of Thm 7.3; almost simple hypothesis) | CHECKED in the arXiv texts; same wording issue |
| Self-contained proof through all Aschbacher classes | CORRECT AND COMPLETE at research level, modulo standard cited theorems | read; no error found |
| Counting lemma (of the first write-up) | CONFIRMED (re-derived) | CONFIRMED (own enumeration, identical numbers) |
| n = 2 data, q ≤ 64 | CONFIRMED by three methods; q = 2, 3 also fail | CONFIRMED by two methods; Dickson route to 20000 |
| Novelty | known by corollary; not new | short corollary of printed theorems; not in print |
| Bibliography | data checked; some theorem numbers of textbooks quoted from memory | data checked (Crossref, zbMATH); Vavilov years |

No run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (A1, B1) Present the result as an answer obtained as a short corollary of Fulman–Guralnick, not as new: title,
   abstract, Section 1.3, "Scope and priority".
2. (A2, B3) State FG 2012 Theorem 7.3 exactly (almost simple G, G ≤ InnDiag(S), limit 0; the rate only in the
   proof) and add the passage from SL_n(q) to PSL_n(q): Theorem 3.1, Lemma 2.3, Remark 3.4.
3. (A3, B2) Write out the two steps: the density 1/n! − O(1/q) of the classes of diagonal d with d^e regular for
   e ≤ n! (Lemma 3.3), and the description of the maximal subgroups of maximal rank (Lemma 4.1, with the reference
   to the end of FG 2003, Section 4); the exclusion of C3 (Lemma 4.2).
4. (A4) Theorem numbers of Kleidman–Liebeck, Borel and Humphreys quoted from memory: the note cites these books by
   content only, without theorem numbers.
5. (A5) "q ≤ n" instead of "q < n": Section 1.1 and Remark 5.1.
6. (A6) SL_2(3) (not only A_4) has no subgroup of index 2; the cases q = 2, 3 also fail: Theorem 1.2, its proof,
   Table 1.
7. (B4) Testerman–Zalesski (Transformation Groups 20 (2015)) is the algebraic-group analogue and does not settle
   7.27 (abstract read): Section 1.3.
8. (B5) Vavilov's series (1985–1993, parts I–V) and the earlier work on overgroups of the whole diagonal group
   (Borevich 1976, Vavilov, King 1990) answer a weaker question: Section 1.3, "Scope and priority", references.

**Changes made when the note was written, after runs A and B.**
- The deduction is organized around FG 2003 Theorem 3.2, whose conclusion ("of maximal rank") is the hypothesis of
  Lemma 4.1; FG 2012 Theorem 7.3 is stated exactly, and Remark 3.4 shows that for large q it concerns the same
  subgroups (with a rate in its proof).
- Lemma 4.1 received a proof that also covers the extension field case (Frobenius orbits on the blocks).
- Proposition 6.3 (all good d for every q ≥ 4, from Dickson's list) and Lemma 6.4 (reduction used for direct
  generation) were added, with Tables 1 and 2; the absolutely irreducible reading for n = 2 (q = 2) was settled.
- The program of the note was written; it recomputes Table 2 by two methods and Table 1 by closure.

## Verification run 2 (the final text and the package)
A further independent verification run, AI-assisted, examined the final text, the release package and the claims
made in it (2026-10-18). It read the sources again (the Notebook, v48 text and the PDF page; FG 2003, Sections 3
and 4, including the setting of Theorem 3.2; FG 2012, Sections 1 and 7; the abstract of Testerman–Zalesski),
re-derived every proof of Sections 2–6 line by line, wrote its own programs (`reproducibility/independent_run_2/`)
before re-running `check_note.py` from the archived package, repeated part of the literature search and checked
all DOIs of the reference list on Crossref.

| Item | Run 2 |
|---|---|
| Statement and readings (v48 text and PDF page 24, v46, corpus record) | CONFIRMED |
| FG 2003 Theorem 3.2 as quoted | the note said "X over F̄_p … for fixed X"; FG 2003 fix a *type* and let only the field size vary, so the characteristic may vary — required fix 1 (applied) |
| FG 2012 Theorem 7.3, FG 2003 Section 4 | CONFIRMED (exact statement; rate only in the proof; density 1/\|C_W(σw)\|) |
| Lemmas 2.1–2.3, 3.3, Remark 3.4, Lemmas 4.1–4.3, Section 5 | CORRECT (re-derived) |
| Section 6: Lemmas 6.1, 6.2, 6.4, Proposition 6.3, Theorem 1.2 | CORRECT (re-derived); wording "(in every reading)" after Lemma 6.1 — required fix 2 (applied) |
| Lemma 3.3 by computation | CONFIRMED (enumeration n = 2, 3, 4; all of SL_3(7), SL_3(8)) |
| Lemmas 4.2, 4.3 by computation | CONFIRMED (n = 2, 3, 4, resp. n = 3), with controls showing the hypotheses are needed |
| Theorem 1.2, Tables 1 and 2 | CONFIRMED: from the definitions for q ≤ 17 in four readings (only q = 2 depends on the reading); by direct generation (sympy) for all 288 pairs with q ≤ 128; Table 2 identical |
| `check_note.py` | re-run from the archive: identical output (timing lines aside); it tests what Section 8 says |
| Novelty, credit | no printed answer found; "corollary, no priority" wording appropriate |
| Presentation | 13 DOIs verified; house style; all pages inspected |

**Corrections required by run 2**, all applied:
1. Theorem 3.2 quoted for a fixed type, with the characteristic allowed to vary and N depending only on the type;
   the application paragraph names the type A_{n−1} and every prime p; after (1): "q runs through all prime powers".
2. Section 6, after Lemma 6.1: "P(d) fails (in every reading)" replaced by "P(d) fails, whether the blocks are
   arbitrary or coordinate subspaces", with a pointer to the absolutely irreducible reading in Theorem 1.2.
3. Abstract (and Zenodo description): "a diagonal matrix with this property exists if and only if q ∈ {4, 8} or
   q ≥ 13 (if irreducibility is meant absolutely, also for q = 2)".
4. Section 1.3: the sentence on overgroups of the whole diagonal group now says that containing T is a much stronger
   condition than containing one d ∈ T (the earlier wording said the opposite).
5. Record of the earlier runs: run B tested the symplectic and unitary C8 arguments, not C5 (Verification paragraph
   and this report); Section 7 now says that the two runs on the first write-up re-derived the second proof and that
   nothing in the note depends on it.
6. This run recorded in the Verification paragraph, in Section 8 and here; its programs in
   `reproducibility/independent_run_2/`; six web searches in "Scope and priority".
Also: Section 8 running time "about 80 seconds" (recorded 81 s); Testerman–Zalesski's "regular torus" defined;
Q_8 for q = 3 noted to be monomial over F_9 (block-monomial is meant over F_q).

## Relation to the literature, novelty and scope
- **Searches (17–18 October 2026).** The Notebook (7.27 has no comment; no other problem refers to it); the arXiv
  API (Kourovka and Levchuk; "diagonal matrix" with "irreducible subgroup"; overgroups of regular semisimple
  elements and of tori; "block-monomial"; Fulman and Guralnick); zbMATH Open (the records of the authors Levchuk and
  Nuzhin; overgroups of semisimple elements; block-monomial groups); Crossref; OpenAlex (five searches); six web
  searches (one of them by run 2). Semantic Scholar was not available and was not used. Run 2
  repeated part of the search (arXiv API: 11 queries; zbMATH Open; OpenAlex; Crossref for all 13 DOIs of the
  reference list; one web search, which found only copies of the Notebook and arXiv 2607.17477, a paper on eight
  other Kourovka problems).
  - No printed answer to the question was found. Judging from titles and reviews, none of Levchuk's later papers
    treats the question for n ≥ 4.
  - The Fulman–Guralnick papers do not mention diagonal matrices, Levchuk or the Notebook.
  - Borevich (1976), Vavilov (1985–1993) and King (1990) describe the subgroups containing the whole diagonal group
    (nets); this is a weaker requirement than containing one diagonal matrix.
  - Testerman–Zalesski (2015): maximal closed connected subgroups of simple algebraic groups containing a regular
    torus (abstract read); an algebraic-group analogue, not the finite single-element question.
- **What was read.** The Notebook (v48 text and PDF; v46 text); FG 2003 in full and FG 2012 Sections 1 and 7 (arXiv
  version 1 of each; the journal versions were not compared); the abstract of Testerman–Zalesski; the zbMATH review
  of Vavilov's part I; bibliographic data of Borevich, King and the textbooks (Crossref and zbMATH). The proofs of
  the cited theorems were not checked. Levchuk's 1973 paper was not available.
- **Caveats.** The deduction rests on the four-type description of maximal subgroups in FG 2003, which that paper
  calls well known for classical groups (with a reference to Aschbacher's theorem), and on FG 2003 Theorem 3.2;
  q_1(n) is not explicit. This negative search is not a proof of priority.
- **Scope.** Specialists will regard Theorem 1.1 as a corollary of the Fulman–Guralnick theorems; no priority is
  claimed for it or for the complete answer for n = 2.

## 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/