# Verification report — KOU-2.78 (Trofimov: soluble and absolutely simple group-theoretic numbers)

Verification date: 2026-10-21.

**Verdict.** The note gives a **partial answer**: both sets are finite (the soluble numbers are exactly
1, 2, 3, 4, 5, 6, 8, and the absolutely simple numbers are 7 and possibly 11); the second question stays
open (a composite non-soluble number would exceed 323).
- **Settled.** Every k >= 9 with k != 11 is the number of IE-systems of a non-soluble non-simple finite
  group, and k(PSL(2,13)) = k(PSL(2,23)) = 11 (Theorem 1.1). Hence the soluble numbers lie in {1, ..., 8}
  and the absolutely simple numbers in {1, ..., 8, 11}: both sets are finite (Corollary 1.3(c)).
- **In print, used for the exact lists.** A non-soluble group has k(G) >= 7, with equality only for A5,
  and k(G) != 8 (Theorem 1.2): X. Zhong (2005) and J. Shi, C. Zhang, D. Liang (2015), known only from
  their reviews and abstracts; the note includes a proof based on Thompson's classification of the
  minimal simple groups. With it: soluble numbers = {1, 2, 3, 4, 5, 6, 8}; 7 is absolutely simple; 11 is
  absolutely simple if and only if no non-soluble non-simple group has exactly 11 IE-systems.
- **Open.** The value 11; the second question (data only: every k in {7} u [9,323] equals k(PSL(2,q))
  for an explicit q). Problem 3.57 is not treated.

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 2.78 (P. I. Trofimov, 2nd issue, 1967).
  - Read in v48 (6 October 2026; section "2nd Issue (1967)", page 8 of the PDF) and in v49
    (20 October 2026; the arXiv API lists v49 as the
    current version, updated 2026-10-20T09:15:18Z; PDF header "arXiv:1401.0300v49 [math.GR] 20 Oct
    2026"). The text of Problem 2.78 is the same in both, with no mark and no comment (listed as open).
  - Definitions: an IE-system of G is the set of all subgroups of one order that contains at least one
    non-normal subgroup; k is soluble (non-soluble; simple; composite; absolutely simple) if every
    finite group with exactly k IE-systems is soluble (some non-soluble group has k; some simple group
    has k; no simple group has k; some simple group and no non-soluble non-simple group has k).
  - Questions: are the sets of soluble and of absolutely simple numbers finite or infinite? Are there
    composite but not soluble numbers?
  - Companion Problem 3.57 (same proposer): the "laws of distribution" of the non-soluble and simple
    numbers. Not part of the record and not treated.
- **Corpus record.** ulamai/UnsolvedMath, KOU-2.78 (dataset version 1.8.0; status `open`). Its statement
  is the Notebook text (with LaTeX markup); no difference from the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| IE-system = the whole set of subgroups of one order (the source's wording "any set of all subgroups of the same given order") | k(G) = number of orders of non-normal subgroups = o(G) - u(G) | Section 1.1, 2 |
| IE-system = a conjugacy class of non-normal subgroups | excluded by the wording; not treated | Section 1.1 |
| "soluble" vacuous for some k? | no: every k >= 1 is k(S3 x C_{5^{k-1}}) | Section 2 |
| groups of prime order (simple, k = 0) | irrelevant for positive k | Section 1.1 |
| Question 1 | both sets finite; soluble numbers exactly 1-6, 8; absolutely simple numbers 7 and possibly 11 | Corollary 1.3 |
| Question 2 | open; a composite non-soluble number would exceed 323 (data) | Section 7 |

## Results in the paper
- **Lemma 2.1, Lemma 2.2.** Monotonicity of the sets of orders of non-normal subgroups under subgroups and
  quotients; o and u are multiplicative over coprime direct factors, k(G x H) = o(G)o(H) - u(G)u(H).
- **Lemma 3.1, Propositions 3.2, 3.3.** For C_{q^c} x| C_m (action of order d, d | q-1): (o,u) =
  ((c+1)tau(m), tau(m) + c tau(m/d)); for F_q^2 x| C_m (multiplication by an element of order d of
  F_{q^2}, d | q+1, gcd(d, q-1) = 1): (o,u) = (2 tau(m) + tau(m/d), tau(m) + tau(m/d)). The proofs use the
  Schur-Zassenhaus theorem explicitly (subgroups of order prime to q lie in complements; complements are
  conjugate). Both families are metabelian; the first is metacyclic, the second is not (its derived
  subgroup is F_q^2).
- **Theorem 4.1.** k(A5 x C_q x| C_{7^a}) = 16A - 2x and k(A5 x F_q^2 x| C_{7^a}) = 16A + 7x (A = a+1,
  x = a-b+1, kernel of order 7^{a-b}, q = +-1 mod 7^b from Dirichlet's theorem); every even k >= 126 and
  every odd k >= 361 is of this form.
- **Lemma 4.2.** Subgroup orders of PSL(2,p) (Dickson's theorem, Huppert, Endliche Gruppen I, Hauptsatz
  II.8.27) and of SL(2,p); u(PSL(2,p)) = 2, u(SL(2,p)) = 3.
- **Proposition 4.3.** An explicit list of 397 non-soluble non-simple groups G_k, one for each k in
  [9,406] minus {11}, with k(G_k) = k (Table 1: 17 base groups computed by exhaustive enumeration; 12
  lines with SL(2,p) or PSL(2,p) from Lemma 4.2; six lines with two blocks separated by `;`). Six values
  (25, 37, 41, 47, 53, 61) rest on Dickson's theorem beyond the range of exhaustive enumeration (the base
  groups PSL(2,13), PSL(2,29) and SL(2,29) of lines 173, 157 and 23 were also enumerated).
- **Theorem 1.1** (from 4.1 and 4.3) and **Theorem 1.2** (Lemmas 5.1, 5.2 and a minimal-counterexample
  argument; in print). **Corollary 1.3** as in the verdict; its finiteness part needs Theorem 1.1 only.
- **Remark 6.1.** The value 11 is open; among PSL(2,p), p <= 23, only p = 13, 23 give k = 11.
- **Section 7.** Data for the second question.

## Computations (sanity checks; programs and outputs in reproducibility/)
- **The program of the note** (`check_note.py`, standard library, about 105 s; `--quick` about 25 s):
  644 checks, 0 failures. Its own enumeration of all subgroups up to conjugacy reproduces the known
  numbers of subgroups (59 for A5, 179 for PSL(2,7), 501 for A6, 942 for PSL(2,13)); it checks Lemma 2.2
  on nine coprime products (and its failure for A5 x C5), Propositions 3.2/3.3 in 22/15 instances, all 17
  rows of Table 1, Lemma 4.2 against enumeration, all 397 certificate lines, the arithmetic of Theorem
  4.1 for k <= 10^6, k(PSL(2,13)) = 11, the data of Section 7 and the class counts of Section 1.3.
- **The programs with which the results were first obtained** (`original/`): exhaustive enumeration
  (some 45 groups, among them those of Table 1), Dickson's list, the block checks (20 + 11 instances), the search for the
  certificate, the verifier and the scans. All were run again from the package: identical outputs up to
  running times, except one intermediate witness list recorded before a data file was completed (not
  used later; explained in the README).
- **Change to the verifier** (`original/verify_cert.py`): blocks with m < 2 are rejected; a line without
  blocks needs a non-trivial proper normal subgroup exhibited by a rule, instead of a list of names of
  simple groups. Tested on wrong inputs (a trivial block `B0:1` with a simple base is now rejected).
- **The programs of the third verification run** (`independent_run_2/`, Python 3 with numpy, sympy for
  one program; recorded outputs in `independent_run_2/outputs/`): own subgroup enumeration and group
  constructions, own coding of Dickson's list, own certificate verifier, the block and tail checks, the
  scan of Section 7, and the record of the rerun of this package (`outputs/package_rerun.txt`).

## Independent verification runs
The results were first obtained with proofs and the programs in `original/`. Three independent
verification runs, all AI-assisted, followed (2026-10-21); each re-derived the proofs and wrote its own
programs. Runs 1 and 2 examined the first write-up of the result; run 3 examined the text of the note and
this package (its programs and outputs are in `reproducibility/independent_run_2/`).

| Item | Run 1 | Run 2 | Run 3 (on the note) |
|---|---|---|---|
| Statement and reading against the Notebook (v48) and the record | CONFIRMED | CONFIRMED | CONFIRMED (v49 fetched again and compared with v48; record identical to the source) |
| Lemmas 2.1, 2.2 | CONFIRMED (re-derived; Lemma 2.2 also checked for products of two non-abelian groups) | CONFIRMED (re-derived; five products of order up to 12180 enumerated) | CONFIRMED (re-derived; three products of order up to 18060 enumerated, among them A5 x B1(43,1,7,7)) |
| Propositions 3.2, 3.3 | CONFIRMED_WITH_FIXES (cite Schur-Zassenhaus explicitly); 22 instances enumerated | CONFIRMED; 866 and 57 instances enumerated, more with composite d and tail-type members | CONFIRMED_WITH_FIXES (the groups of 3.3 are metabelian, not metacyclic; q odd and e \| m stated); 20 and 15 instances enumerated, three invalid ones rejected |
| Theorem 4.1 | CONFIRMED; recipes checked up to 20000 | CONFIRMED; arithmetic up to 2 * 10^6, explicit groups with probable primes for k <= 1200 | CONFIRMED; arithmetic up to 2 * 10^6, explicit groups for k <= 800 |
| Base groups (Table 1) | CONFIRMED by own enumeration (22 groups, same o, u and subgroup counts) | CONFIRMED by own enumeration (all 40 groups of the first table, 41 more) | CONFIRMED by own enumeration (all 17 rows, PSL(2,13), PSL(2,29), SL(2,29); four whole certificate groups, lines 26, 32, 34, 36) |
| Dickson's list | CONFIRMED against own enumeration (PSL(2,p), p = 7, 11, 13; SL(2,p), p = 5, 7, 13) | CONFIRMED against own enumeration (PSL(2,q), all prime powers q <= 31; SL, PGL(2,q), odd q <= 23) | CONFIRMED against own enumeration (PSL(2,q), 13 values q <= 29; SL(2,q), q = 5, 7, 9, 11, 13, 17, 29) |
| Certificate (397 lines) | CONFIRMED with own verifier and data | CONFIRMED with own data; independent second covering of [9,406] minus {11}; original programs rerun with identical outputs | CONFIRMED with own verifier and data; whole package rerun from the source archive with the recorded outputs |
| Theorem 1.2 (in print) | CONFIRMED (re-derived; reviews checked) | CONFIRMED (re-derived; reviews and abstract checked) | CONFIRMED (re-derived; reviews and abstract read again; class counts of Section 1.3 recomputed) |
| Corollary 1.3 | CONFIRMED (finiteness from Theorem 1.1 alone) | CONFIRMED | CONFIRMED |
| Second question | data reproduced (to 323) | data extended: every k in {7} u [9,331] for q <= 6 * 10^6 | data reproduced (primes to 5 * 10^6, same least primes for all 422 values) |
| Presentation | five fixes (status of 11 in the examples; Schur-Zassenhaus; inference behind the reading of Shi-Zhang-Liang; separation of the results in print; two-block lines) | four fixes (reading of Shi-Zhang-Liang; a citation; hardening of the verifier; residual literature risk) | five fixes (metabelian, not metacyclic; credit for k(G): Trofimov's number, Zhong's r(G), Xiang-Li's notation J(G); two short justifications; record of this run; a heading) |

All required fixes were made in the note and in the package. While the note was written, the program
`check_note.py` was added, the proof of Theorem 1.2 was given computation-free lower bounds for
k(A5 x C_r) and k(S5), and a case of Theorem 4.1 that does not occur for k >= 126 was dropped; runs 1 and
2 have not seen the final text. Run 3 examined it: it found no error in the statements or proofs and
required the corrections listed in the table (the groups F_q^2 x| C_m of Proposition 3.3 had been called
metacyclic; Section 1.3 had said that Xiang and Li introduced k(G)), all of which were made.

## Relation to the literature, novelty and scope
- **Searches (21 October 2026).** zbMATH Open API (Trofimov's papers; orders of non-normal subgroups;
  non-normal subgroups and solubility; "group-theoretic number", "IE-system"; citing works), OpenAlex
  (topic searches, citing works), Crossref, the arXiv API, a few web searches. mathnet.ru unreachable;
  zbMATH web pages refused automated access (HTTP 403); Project Euclid answered with a bot check;
  Semantic Scholar with HTTP 429; none of this was worked around. Run 3 repeated the searches in part on
  the same day (arXiv API, zbMATH Open API, OpenAlex citing works and topic searches, Crossref, one web
  search) and found nothing new; the AMS file of Thompson 1968 answered HTTP 403 (not worked around).
- **Found.** Zhong 2005 (zbMATH 1070.20016): non-soluble G has at least seven orders of non-normal
  subgroups, seven only for A5. Shi-Zhang-Liang 2015 (Hokkaido Math. J. 44, 387-396,
  doi:10.14492/hokmj/1470053370): at most nine classes of non-normal non-trivial subgroups of the same
  order forces solubility except for A5, PSL(2,7), SL(2,5). The zbMATH review says "conjugacy classes";
  the reading as orders is inferred (PSL(2,7) has six conjugacy classes of non-normal subgroups of
  non-prime-power order but four such orders, and 13 conjugacy classes of non-normal subgroups but
  k = 9). Xiang-Li 2007 (zbMATH 1144.20304) introduced the notation J(G) for k(G), which is Trofimov's
  number of IE-systems and Zhong's r(G); He-Lv-Chen 2024 (Bull. Aust. Math. Soc. 110, 121-128) studies it
  for soluble groups and p-groups. None answers Problem 2.78.
- **Not read.** The full texts of Zhong 2005, Xiang-Li 2007 and Shi-Zhang-Liang 2015; Trofimov's papers
  in Sib. Mat. Zh. 3 (1962) 876-881 and 5 (1964) 192-200 and his notes Sib. Mat. Zh. 16 (1975) 416
  (translation Sib. Math. J. 16 (1975) 321), where the group-theoretic numbers may have been introduced
  or studied. These are the main residual risk.
- **Novelty.** No printed answer to the question was found in the sources searched; a search that finds
  nothing is not a proof of novelty, and no priority is claimed. To our knowledge not in print: Theorem
  1.1 with Theorem 4.1 and Proposition 4.3, hence the finiteness of both sets and the exact set of
  soluble numbers.
- **Scope.** The first question is answered (both sets finite; absolutely simple numbers up to the value
  11); the second question is open; Problem 3.57 is not treated.

## 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/