# Verification report — OWR-11129-006 (Lübeck–Niemeyer–Praeger: elements powering to strong involutions, Oberwolfach Report 37/2011)

Verification date: 2026-10-19.

**Verdict.** The note answers the question for the groups of Lübeck, Niemeyer and Praeger (2009), Table 1: the
order Θ(1/log n) for every group H with S ≤ H ≤ X in a line of the table, and the asymptotic value for the eight
groups GL_n(q), GU_n(q), Sp_2m(q), GSp_2m(q), SO_2m+1(q), GO_2m+1(q), SO±_2m(q), GO±_2m(q)^0 (all groups X, and the
groups S of the symplectic and orthogonal lines). Every statement it makes is proved, and the scope is as follows.
- **Settled.** |Q|/|X| ≤ C/log n with absolute constants, uniformly in odd q: C = 100 for GL_n(q), GU_n(q)
  (Theorem 1.1, with the lower bound 1/(300 log n) for n ≥ 100); C = 45 for Sp_2m(q), GSp_2m(q), SO_2m+1(q),
  GO_2m+1(q) and C = 80 for SO±_2m(q), GO±_2m(q)^0 (Theorem 1.2). With the lower bound of [LNP09, Theorem 1.1],
  |Q|/|H| = Θ(1/log n) for every H with S ≤ H ≤ X, the upper constant carrying a factor at most q + 1 for the
  other H, among them SL_n(q) and SU_n(q) (Corollary 1.3). The asymptotic value, for the eight groups above:
  (log n)|Q|/|X| = Φ(log_2 log n) + o(1) uniformly in q,
  where Φ(u) = (log 2/3) Σ_k 2^(u−k) exp(−2^(u−k)/3) is 1-periodic with mean 1, not constant, and |Φ − 1| < 9.9·10⁻⁶
  (Theorem 1.4); so (log n)|Q|/|X| has no limit, its limit points forming [min Φ, max Φ].
- **Not new.** The question (Praeger, jointly with Lübeck and Niemeyer); the lower bound, the list of groups and the
  description of the tori and their involutions [LNP09]; the torus formula for quokka sets (Niemeyer–Praeger 2010,
  Theorem 1.3, used as [LNP09, Lemma 2.3] and Niemeyer–Popiel–Praeger 2014, Theorem 2.2); standard facts on
  random permutations.
- **Not covered.** The full orthogonal groups O±_2m(q); Theorem 1.4 and q-uniform constants for SL_n(q), SU_n(q)
  and for S < H < X (only the order of magnitude); a lower bound for Ω (only the upper bound); explicit constants
  close to the truth; a rate of convergence; even q (outside the question).

The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 37/2011, *Computational Group Theory* (B. Eick, G. Hiß, D. F. Holt,
  E. A. O'Brien, eds.), Oberwolfach Reports 8 (2011), no. 3, pp. 2113–2161, doi:10.4171/OWR/2011/37; problem
  session of 4 August 2011, first problem (C. E. Praeger, joint with F. Lübeck and A. C. Niemeyer), pp. 2154–2155.
  Read in a text extraction of the report.
  - G = Class(n, q), a finite n-dimensional classical group over a field of odd order q, V the natural module;
    Q = {g : |g| even, dim Fix_V(g^{|g|/2}) ∈ [n/3, 2n/3)}.
  - Known: |Q|/|G| ≥ c/log n with c depending on the Lie type [LNP09]; statistical tests reported in [LNP09].
  - Question: the asymptotic value of |Q|/|G|; in particular, is |Q|/|G| = O(log(n))? As printed this is vacuous;
    it is read as O(1/log n), which is supported by the adjacent lower bound and by [LNP09], whose statistical
    tests point to the order 1/log ℓ.
- **The paper of the question.** F. Lübeck, A. C. Niemeyer, C. E. Praeger, J. Algebra 321 (2009) 3397–3417,
  doi:10.1016/j.jalgebra.2008.05.009; read in the authors' preprint of 20 May 2008 (the publisher's file refused
  automated access). Table 1 (the lines (S, X)), Theorem 1.1 (the lower bound 1/(5000 log_2 ℓ) for S ≤ H ≤ X,
  ℓ ≥ 2), Lemma 2.3 (the torus formula for P(G, I)), Sections 3 and 5 (Weyl groups, Frobenius actions, tori and
  involutions), the paragraph "Upper bounds and other groups" (the method gives no upper bound; sampled proportions
  of about 15% in dimension 1000).
- **Corpus record.** ulamai/UnsolvedMath, OWR-11129-006, status `open`. In the current snapshot (commit d371ee9) the
  clean statement asks "Determine the asymptotic behavior of |Q|/|G|. In particular, is it of order 1/log n?", with
  the statement status `corrected_verified` and a note on the omitted reciprocal; the original statement keeps the
  printed O(log n). The local copy of the older release 1.6.0 has only the printed form.

## Readings
| Reading | Answer | Where |
|---|---|---|
| "Classical group" = the groups X and S of [LNP09, Table 1] (the source cites [LNP09] for the bound) | order Θ(1/log n) for all; q-uniform constants and the asymptotic value for all X and for Sp_2m, SO_2m+1, SO±_2m | Theorems 1.1, 1.2, 1.4, Corollary 1.3 |
| SL_n(q), SU_n(q), and groups H strictly between S and X | Θ(1/log n), upper constant ≤ 100(q + 1); asymptotic value not claimed | Corollary 1.3 |
| Ω_2m+1(q), Ω±_2m(q) (below S) | upper bound 2C_SO/log n only | Corollary 1.3 |
| Full orthogonal groups O±_2m(q) | not covered (disconnected group; torus formula not available) | Section 5 |
| "is |Q|/|G| = O(log n)?" as printed | vacuous; read as O(1/log n): yes | Section 1.1, Theorems 1.1, 1.2 |
| "asymptotic value" as a limit of (log n)|Q|/|G| | no limit; the limit points form [min Φ, max Φ], an interval of length < 2·10⁻⁵ around 1 | Theorem 1.4 |
| Constants "depending on the Lie type" | the upper constants are absolute and uniform in q | Theorems 1.1, 1.2 |
| Endpoints of [n/3, 2n/3) | kept as in the source; equivalently n/3 < dim V_{−1} ≤ 2n/3 | Section 1.1 |

## Results in the paper
- **Lemma 2.1, Proposition 2.2.** Q depends only on the semisimple part; the torus formula (quoted).
- **Proposition 2.4.** GL, GU: P(G) is the probability that, for a uniform permutation of n points whose cycles carry
  independent 2-adic levels (a(λ) − X)^+, the cycles of maximal level (≥ 1) have total length in (n/3, 2n/3]. Sp,
  SO_2m+1: the same with independent fair signs (levels (a(λ) − X)^+ or (b(λ) − X)^+). SO±_2m: twice the expectation
  with the parity of the number of negative cycles.
- **Lemma 2.5.** The groups of similitudes GSp_2m(q), GO_2m+1(q), GO±_2m(q)^0: tori with the multiplier; at the
  levels v ≥ a_1 + 1 (resp. v ≥ a_1) the contributions equal those of the Sp (resp. SO_2m+1) model; the lower
  levels are bounded by E[(3/4)^{N_k0}].
- **Proposition 3.1, Lemmas 3.2, 3.3.** The witness formula P(M = v, D ∈ I) = Σ_{d∈I} E_d(v) A_{N−d}(v); the
  weights at the levels v = a_1 − 1 + k (only lengths divisible by 2^k active, independent of q); two closed forms.
- **Lemma 4.1.** A level-by-level upper bound with explicit constant K; Theorems 1.1 and 1.2 follow (constants
  K = 48.37, 37.50, 40.62; majorants M(e^45) = 39.25, 42.21, 42.90 and M(e^80) = 76.72, 77.97). Lower bound of
  Theorem 1.1 by the witnesses of [LNP09, Section 5] and a Jensen bound.
- **Corollary 1.3.** Indices [X : S] ≤ q + 1.
- **Theorem 1.4.** Φ: Fourier coefficients 3^{−iτ_j} Γ(1 − iτ_j); the levels with y_k = log n/2^k ≤ log log n and
  2^k ≤ √n contribute (log 2/3) 2^{−k} e^{−y_k/3} each, up to 2^{−k} O((log n)^{−2/3} (log log n)^3); the others
  o(1/log n); the families Sp, SO, SO±, GSp, GO, GO^0 by halving, a parity argument and Lemma 2.5.
- **Table 1.** (log n)P(X) for N = 10² … 10⁷ (evidence only), near 1.10 at 10⁶–10⁷.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation, except the finite evaluations of Σ_H, g(0), g'(0), K, M(e^C) and Σ|c_j|.
- **Program written with the note** (`check_note.py`, standard library, about 20 s; TOTAL failures: 0): the
  constants; Φ; the inequalities (9) on a finite range; in exact arithmetic the valuations (2), Lemma 2.3, (6), (7),
  P(no odd cycle), Proposition 2.4 against the witness formula level by level (130 cases), the level identities of
  Lemma 2.5 (40 cases); C_v ≤ T_k and the lower bound for GL_n(q), n ∈ {100, 1000}; brute force over 29 groups with
  the order checked; the model against all 35 recorded brute-force values.
- **Programs with which the results were first obtained** (`original/`): exact torus sums; brute force over 16
  groups; the witness formula in floating point (Table 1, N up to 10⁷); the constants; Φ in 40 digits. Of the
  brute-force runs, the second has an automatic order check; in the first, the printed orders agree with the order
  formula except for one line (a proper subgroup of SO+_4(3) of order 192), which is discarded; the GO_5(3) run was
  not finished.
- **Verification run B** (`verification_run_B/`): brute force over 35 groups with the order checked, including
  GO_5(3) = 21/64 (|GO_5(3)| = 103680); enumeration of semisimple classes (Steinberg's count of unipotent elements),
  independent of the torus formula, which agrees with its own torus model at 326 values.
- **Independent verification run 2** (`independent_run_2/`, the third AI-assisted run, on the written note): own
  constants (50 digits), Φ and (9); brute force over 26 groups with the order checked; an exact torus model built
  from the 2-parts of the torus elements (also for the groups of similitudes); the witness formula in floating point
  with weights computed from the valuations, up to N = 10⁷ for GL, q = 3; comparisons with Table 1, with the exact
  outputs of `original/` and with `fastcl_grid.out`. See the section below and `independent_run_2/README.md`.
- **Re-runs (2026-10-19).** The constant programs, the bound checks, the GL q = 3 grid of `fastcl.c`, the level
  listing of `fastgl.c` and samples of the exact sums were re-run from the package; all outputs are identical to the
  recorded ones (`reproducibility/README.md`). Verification run 2 re-ran `check_note.py`, `bound_check2.py`,
  `phi_check.py` and the whole grid of `run_fastcl.sh` from an extracted copy of the archive: identical up to the
  timing fields and time stamps.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two independent verification
runs, both AI-assisted, examined this first written version (2026-10-19); each re-derived the proofs line by line and
wrote its own programs, and neither read the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement and reading against the source and [LNP09] | CONFIRMED (source text and [LNP09] preprint read; the misprint reading checked) | CONFIRMED (also [LNP09]'s own "O(1/log l)") |
| Reduction to tori, Proposition 2.4 | CONFIRMED (re-derived; [LNP09, Lemma 2.3] and [NPP14, Theorem 2.2] compared with the texts) | CONFIRMED (re-derived; exact class enumeration independent of the torus formula agrees at 326 values) |
| Groups of similitudes (Lemma 2.5) | CONFIRMED (re-derived; own 2-part model and brute force) | CONFIRMED (brute force over q ∈ {3, 5, 7}, including GO_5(3)) |
| Witness formula, closed forms | CONFIRMED | CONFIRMED |
| Lemma 4.1, Theorems 1.1, 1.2 (constants) | CONFIRMED (constants recomputed) | CONFIRMED (constants recomputed) |
| Lower bound of Theorem 1.1 | CONFIRMED | CONFIRMED |
| Corollary 1.3 | CONFIRMED; remark on [LNP09, Lemma 4.4] required | CONFIRMED |
| Theorem 1.4 | CONFIRMED_WITH_FIXES (one bound in the I_2 case; cut-off of the series B) | CONFIRMED_WITH_FIXES (cut-off of the series B; constant 9/2) |
| Φ (mean, Fourier coefficients, range) | CONFIRMED (40 digits) | CONFIRMED (40 digits) |
| Table 1 | reproduced for N ≤ 10⁴ (own recursion) | reproduced for N ≤ 10⁴ (class enumeration), GL at 10⁵; Monte Carlo at 10⁵–10⁷ within 2.2 standard errors |
| Literature and novelty | no prior upper bound found; caveat on Dixon–Praeger–Seress (2018) required | no prior upper bound or asymptotic found (arXiv, OpenAlex, Semantic Scholar, zbMATH, Crossref, three web searches); bibliographic corrections required |

Neither run found an error in a statement.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) Section 6.3(a), case I_2: the extra integer changes Σβ_ℓ/ℓ by at most β_ℓ*/ℓ* ≤ 3/(8m) ≤
   2^{−k}(3/8)m^{−1/2}, which is 2^{−k}o(1) (the first version had a bound false for k ≥ 3).
2. (Runs A and B) Step 2 of Section 6.2: the series B is cut off at degree n (this does not change E_d for d ≤ n).
3. (Run A) Dixon–Praeger–Seress (J. Algebra 498 (2018)) had not been read and was the one listed paper that could
   contain an upper bound for SL_n(q): first a caveat was added; the paper has since been read in full (see
   "Relation to the literature" and verification run 2).
4. (Run A) The proof of [LNP09, Lemma 4.4] treats a larger X than GO±(q)^0, so [LNP09, Theorem 1.1] covers the
   connected group a fortiori: proof of Corollary 1.3.
5. (Run A) Which brute-force runs were order-checked: Verification paragraph, `reproducibility/README.md`.
6. (Run B) Guest–Praeger is J. Algebra 372 (2012) 637–660; the Carter reference completed (Compositio Math. 25
   (1972) 1–59); Crossref gives no pages for the DMTCS paper, which is cited without pages.
7. (Run B) GO_5(3) = 21/64 by brute force in verification run B: its program and output are in
   `reproducibility/verification_run_B/`.
8. (Run B) The scope limits are kept in the abstract and in "Scope and priority".
9. (Run B, optional) The constant of X_3 is 9/2.
10. (Run B) The statement of the first version that (log n)P had begun to fall after N = 10⁷ rested on one program
    only; it has been dropped.

**Changes made when the note was written.** Lemma 4.1 is stated once and Theorem 1.1 is derived from it (the
first version gave the GL case separately with the same constant). The bound (9) is stated with the constants 2 and
5/Y (proved in the note; the first version had 3 and 8/X). The bound on π in Step 2 is 4·2^{−k} log L (the first
version wrote 3·2^{−k} log L, which needs an additive constant; the conclusion is unchanged). Table 1 collects the
values up to N = 10⁷.

## Independent verification run 2 (on the written note, 2026-10-19)
A third AI-assisted verification run examined the written note (`main.tex` as of 19 October 2026), first the parts
written or changed when the note was written, then all other proofs, line by line. Its programs and outputs are in
`reproducibility/independent_run_2/`.

| Item | Verdict |
|---|---|
| Statement fidelity (OWR 37/2011 text; corpus record) | CONFIRMED; the record's "of order 1/log n" and the source's printed O(log(n)) are both answered |
| Lemma 4.1 and the derivation of Theorems 1.1 (K = 48.37) and 1.2 | CONFIRMED (re-derived); constants recomputed with 50 digits: Σ_H = 57.6286, 116.0403, 121.4112; K = 48.3714, 37.4955, 40.6163; M(e^45) = 39.2534, 42.2069, 42.9043; M(e^80) = 76.7182, 77.9683 |
| Section 6.3(a) (I_2) and the cut-off of B in Section 6.2; π ≤ 4·2^{−k} log L; (9) with 2 and 5/Y | CONFIRMED (re-derived; (9) also numerically: 0.128 for M ≤ 2·10⁶, Y·error ≤ 0.250 for 4 ≤ Y ≤ 2·10⁵) |
| \|Φ − 1\| ≤ 2Σ\|c_j\| < 9.9·10⁻⁶ | CONFIRMED (2Σ\|c_j\| = 9.88445·10⁻⁶); the tail justification replaced by an explicit bound |
| Lemma 2.5 (groups of similitudes), (U1), (U2) for each family | CONFIRMED (re-derived; own 2-part torus model = own brute force for GSp_2(3), GSp_2(5), GSp_4(3), GO_3(3), GO_3(5), GO_5(3), GO±_4(3)^0, GO±_2(5)^0, GO±_4(5)^0) |
| Corollary 1.3, remark on [LNP09, Lemma 4.4] | CONFIRMED |
| Theorem 1.4, Steps 1–3 and Section 6.3 | CONFIRMED (re-derived) |
| Scope statements | FIX REQUIRED and made: q-uniform constants and Theorem 1.4 are not proved for SL_n(q), SU_n(q) |
| [DPS18] | read in full (publisher's version): Theorem 1.1 is a lower bound κ/log n (n_0 = 700, κ = 0.0002) for a different event; no upper bound for \|Q\|/\|G\| |
| Computations | brute force over 26 groups (orders checked) = own exact model; 63 exact values of `original/` reproduced; Table 1 reproduced to all printed digits (N ≤ 10⁶, and GL q = 3 at 10⁷); max difference from `fastcl_grid.out` 4.5·10⁻¹¹; C_v ≤ T_k at every level in the cases computed |
| Literature | arXiv (5 queries), OpenAlex (citing works of [LNP09], [NP10], [DPS18]), zbMATH (2 queries), Crossref (all 12 DOIs), one web search: nothing new |

It found no error in a theorem, proposition, lemma or proof. **Corrections required by run 2**, all applied:
1. Scope (abstract, Section 1.2, "Scope and priority", this report, the Zenodo description): absolute constants
   and Theorem 1.4 for the eight groups of Theorems 1.1 and 1.2 only; SL_n(q), SU_n(q): order of magnitude only.
2. [DPS18]: the earlier caveat replaced by a precise statement of its Theorem 1.1 (a lower bound for a different
   event) and of the absence of an upper bound for \|Q\|/\|G\|.
3. Table 1 caption: GL(q) ↔ GU(q') needs v_2(q−1) = v_2(q'+1) and v_2(q+1) = v_2(q'−1).
4. Section 7: the eighth level ratio for GL_{10⁶}(3) is 1.01 (printed as 1.00, from a rounded level value).
5. Theorem 1.4(a): explicit tail bound Σ_{j≥2}\|c_j\| < 5·10⁻¹² in place of "decreases faster than e^{−14j}".
6. Proposition 2.4(a): with the Frobenius morphisms of [LNP09, Section 3.1], F acts trivially on W for GU as well.
7. Credit: [GP12, Theorem 1.2] (2-parts of order t·log n with probability bounded below), Sections 1.3 and 7.
8. Records: the Verification paragraph, this report, `independent_run_2/`.

## Relation to the literature, novelty and scope
- **Searches (18–19 October 2026).** arXiv (strong involutions; proportions of elements powering to involutions in
  classical groups; the authors of the question; quokka sets and maximal tori), the lists of citing works in OpenAlex
  (24 works citing [LNP09], 18 citing Niemeyer–Praeger 2010, 10 citing Niemeyer–Popiel–Praeger 2010), Semantic
  Scholar (in part refused), zbMATH, Crossref, and five web searches in all; on 19 October verification run 2
  repeated the queries to arXiv, OpenAlex (also the 2 works citing Dixon–Praeger–Seress 2018), zbMATH and Crossref.
  - No printed answer to the question was found. [LNP09] states that its method gives no upper bound; the citing
    works concern recognition algorithms, other proportions, or lower bounds for other events (Guest–Praeger 2012;
    Dixon–Praeger–Seress 2018; Glasby–Praeger–Roney-Dougal 2020).
- **What was read.** The problem session (text extraction); [LNP09] in the authors' preprint; Niemeyer–Popiel–Praeger
  (arXiv:1205.1454, Theorem 2.2 and the definition of quokka sets); Guest–Praeger (arXiv:1007.2983),
  Glasby–Praeger–Roney-Dougal (arXiv:1812.01362) and Niemeyer–Praeger, DMTCS 16 (arXiv:1405.2385) in their arXiv
  versions; Dixon–Praeger–Seress, *Strong involutions in finite special linear groups of odd characteristic*,
  J. Algebra 498 (2018) 413–447, completely, in the publisher's version. Seen only through metadata, abstracts or
  restatements: Niemeyer–Praeger 2010 (the quokka theorem; through [LNP09, Lemma 2.3] and [NPP14, Theorem 2.2]);
  Niemeyer–Popiel–Praeger 2010; Praeger–Seress 2012 and the survey Niemeyer–Praeger–Seress 2013 (abstracts); Carter
  1972 and 1985 (as cited in [LNP09]).
- **Dixon–Praeger–Seress (2018).** Its Theorem 1.1: for a strong involution t in GL_n(q) and a uniform
  g ∈ GL_n(q), the restriction of the involution z(g) of ⟨tt^g⟩ to an eigenspace of t is a strong involution with
  probability at least κ/log n for n ≥ n_0 (Remark 1.2: n_0 = 700, κ = 0.0002). This is a lower bound for a different
  event. The proportion |Q|/|G| appears only in Section 1.1, through [LNP09, Theorem 1.1]; the paper contains no upper
  bound and no asymptotic value for it. This negative search is not a proof of priority.
- **Guest–Praeger (2012), Theorem 1.2.** The 2-part of the order of a uniform element lies in an interval of the form
  [½ a t log n, 4 a t log n) with probability at least 0.22 (linear and unitary groups) or 0.11 (symplectic and
  orthogonal groups); this is the range of levels that carries the main term of Theorem 1.4, and the note now says so.
- **Scope.** As in the verdict. No priority is claimed.

## 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/