# Verification report — KOU-16.60 (Kourovka Notebook Problem 16.60, D. MacHale: elements inverted by an automorphism and the sum of the character degrees)

Verification date: 2026-10-18.

**Verdict.** The note gives a **complete answer: yes**. For every finite group G and every automorphism α of G
one has |S_α| ≤ T(G), where S_α = {g ∈ G : α(g) = g⁻¹} and T(G) is the sum of the degrees of the irreducible
complex characters of G.
- **Settled.** Theorem 1.2: (a) |S_α| ≤ T(C_G(α²)) ≤ T(G), where C_G(α²) = {g : α²(g) = g}; (b) |S_α| = T(G) if and
  only if α² = 1 and every twisted indicator ι_α(χ) = |G|⁻¹ Σ_g χ(g α(g)) equals 1; (c) |S_α| < T(G) whenever
  α² ≠ 1, in particular whenever α has order at least 3.
- **Not new.** The case α = id is the theorem of Frobenius and Schur. The involutory case follows from the twisted
  Frobenius–Schur count of Kawanaka and Matsuyama (1990; see also Bump and Ginzburg 2004), printed in this form in
  Vinroot, Math. Z. 294 (2020), Section 2. Inner automorphisms of all finite groups and all automorphisms of
  complete groups: Bors, J. Algebra Appl. 18 (2019) 1950055, Section 2.2. Abelian groups, dihedral groups and
  groups of order p, 2p, p²: Yerrapati, Dixit and Shukla, arXiv:2605.22127 (2026). The full statement is
  Theorem 3.1 of Yerrapati, Dixit and Shukla, arXiv:2605.23195 (2026); the proof given there for its Lemma 2.19
  (the bound on the twisted indicators) is incomplete as written.
- **New, and modest.** A proof of |ι_α(χ)| ≤ 1 for an arbitrary automorphism (Proposition 3.1(c)), which completes
  the strategy of arXiv:2605.23195; a second proof of the inequality by reduction to the involutory automorphism
  α|C_G(α²) (Section 5); and the characterisation of equality.

The note is unrefereed.

## Statement checked
- **Primary source.** Kourovka Notebook, Problem 16.60, proposed by D. MacHale in the 16th issue (2006). The 2006
  printing was not consulted. The wording was compared in:
  - arXiv:1401.0300v1 (No. 18, 1 January 2014), printed page 104;
  - arXiv:1401.0300v48 (No. 21, 6 October 2026), printed page 106 (also the local text extraction);
  - arXiv:1401.0300v46 (No. 21, 1 September 2026), by verification run A.

  The wording is the same in all three: T(G) = Σ_{i=1}^{k(G)} d_i, the sum of the degrees of the irreducible
  complex representations of G, where G has k(G) conjugacy classes; for α ∈ Aut G, S_α = {g ∈ G | α(g) = g⁻¹};
  "Is it true that T(G) ⩾ |S_α| for all α ∈ Aut G?". Version 48 prints no comment on the problem; the only other
  occurrence of "16.60" is the index of proposers (MacHale: 16.60, 16.63, 17.76).
- **Corpus record.** ulamai/UnsolvedMath, KOU-16.60 (dataset release v1.8.0; status `open`). Its statement agrees
  with the Notebook.
- **Edition cited.** The note cites the current version (No. 21, arXiv version 48) and the 2014 printing (No. 18,
  arXiv version 1), and says that the problem dates from the 16th issue (2006). (arXiv:2605.22127 cites the 1995
  edition for this problem; the note does not copy that reference.)

## Readings
| Reading | Answer | Where |
|---|---|---|
| All automorphisms (the literal reading) | yes | Theorem 1.2(a) |
| Involutory automorphisms only | yes; a consequence of known results | Section 1.3; Theorem 1.2(a) |
| Inner automorphisms only | yes; due to Bors (2019) | Remark 6.2 |
| α = id | yes; Frobenius–Schur | Section 1.3; Example 6.3(2) |
| T(G) read as Σ d_i² (the order of G) | not the source's definition (it would make the question trivial) | Section 1.1 |
| Can equality hold? | yes, exactly when α² = 1 and all twisted indicators are 1; never for α of order ≥ 3 | Theorem 1.2(b), (c) |

## Results in the paper
- **Lemma 2.1** (orthogonality of matrix coefficients), **Lemma 2.2** (regular character), **Lemma 2.3** (counting
  formula |S_α| = Σ_χ ι_α(χ) χ(1), valid for every map α; this is Lemma 2.20 of arXiv:2605.23195).
- **Proposition 3.1.** For α ∈ Aut G and χ ∈ Irr(G) of degree d afforded by a unitary representation ρ:
  (a) ι_α(χ) is real; (b) ι_α(χ) = 0 unless χ∘α = χ̄; (c) if χ∘α = χ̄, there is a unitary U with
  conj(ρ(α(g))) = U ρ(g) U⁻¹ and ι_α(χ) = tr(U Ū)/d, so |ι_α(χ)| ≤ 1 in all cases; (d) for α² = 1,
  U Ū = ι_α(χ) I and ι_α(χ) ∈ {0, 1, −1} (the Kawanaka–Matsuyama integrality, reproved).
- **Example 3.3.** Q_8 with the automorphism i → j → k → i of order 3: |S_α| = 2, T(Q_8) = 6, ι_α = 1, 0, 0, 0, 1/2;
  C_G(α²) = {±1} and T(C_G(α²)) = 2 = |S_α|.
- **Section 4 (first proof).** Lemma 2.3, Proposition 3.1(c) and the triangle inequality: the argument of
  arXiv:2605.23195, Theorem 3.1, with Proposition 3.1(c) in place of its Lemma 2.19.
- **Section 5 (second proof).** Lemma 5.1: S_α ⊆ H = C_G(α²), α(H) = H, β = α|_H is involutory and S_β = S_α.
  Lemma 5.2: T(H) ≤ T(G) for H ≤ G, strict for H ≠ G (Frobenius reciprocity; the constituent count of (1_H)^G).
  Then |S_α| = Σ_ψ ι_β(ψ) ψ(1) ≤ T(H) ≤ T(G).
- **Section 6.** Proof of Theorem 1.2(b), (c); Remark 6.1 (equality from Proposition 3.1 alone: U Ū = I for every
  irreducible ρ forces α² = 1); Remark 6.2 (inner automorphisms: x ↦ xt is a bijection from S_α onto the square
  roots of t², so |S_α| = Σ_χ ν(χ) χ(t²) ≤ T(G), Bors's argument); Example 6.3 (abelian groups with inversion;
  α = id and S_3, S_4, S_5; GL(3,2) ≅ PSL(2,7) with α(g) = (gᵀ)⁻¹: S_α = the 28 invertible symmetric matrices,
  T = 28, all twisted indicators 1, an instance of Vinroot 2020, Theorem 10.1, while |S_id| = 22; the strict
  examples C_7 ⋊ C_3 (max |S_α| = 7 < 9) and AGL(1,5) (max 6 < 8)).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`reproducibility/check_note.py`, standard library only, about 1 second; output
  `check_note_output.txt`, ending with `ALL CHECKS PASSED`). 42 groups: cyclic groups of orders 1–10 and 12;
  C_2², C_2³, C_3², C_4 × C_2, C_4²; dihedral groups of orders 6, 8, 10, 12, 14, 16, 18, 20, 24; Q_8, Q_16, the
  dicyclic group of order 12; A_4, S_4, A_5, S_5; C_2 × S_3, C_3 × S_3, S_3 × S_3, C_2 × Q_8, C_2 × D_8, C_2 × A_4;
  C_7 ⋊ C_3, AGL(1,5), SL(2,3), GL(3,2).
  - Character tables computed from the central characters modulo a prime p ≡ 1 (mod exp G), lifted to Z[ζ_e] and
    certified exactly (class-algebra identities, χ(1) = d, the norm identity, distinct rows; orthogonality).
  - Aut(G) enumerated completely: 1,778 automorphisms, 492 of them involutory; every |Aut(G)| agrees with the known
    value; every automorphism found is re-checked against the full multiplication table.
  - For every automorphism: Lemma 2.3; Proposition 3.1(a), (b), (d) exactly; the bound of Proposition 3.1(c)
    exactly when ι is rational, otherwise in floating point with margin 10⁻⁹; Lemma 5.1; Theorem 1.2(a) with
    T(C_G(α²)) from the exact table of C_G(α²) (257 subgroup tables) and Lemma 5.2 for these subgroups;
    Theorem 1.2(b), (c). 209 automorphisms attain equality, all involutory.
  - For every t ∈ G: the count of Remark 6.2. Examples 3.3 and 6.3 as printed. T(G × H) = T(G) T(H) for the 8
    direct products.
  - Negative controls, run when the program was written (not part of the package): with T(S_4) replaced by
    T(S_4) − 1, with the inversion map of S_3 added as a false automorphism, with a wrong degree in the table of
    S_4, and with a perturbed character value of A_4, the corresponding checks fail.
- **Program with which the results were first tested** (floating-point character tables, not part of the package):
  38 groups, 1,315 automorphisms, 133 equality cases; re-run by verification run B with byte-identical output.
- **Verification run A** (own floating-point program; not part of the package): 44 groups of orders 4 to 720,
  7,104 automorphisms, all statements confirmed, 356 equality cases; among them S_6, A_6 and PSL(2,7) with all
  their automorphisms. Outer automorphisms of S_6: at most 36 < 76 inverted elements (the value of
  arXiv:2605.23195, Theorem 3.5); PSL(2,7): equality exactly for the 28 outer involutory automorphisms.
- **Verification run B** (own exact programs; not part of the package): the complete list of the 586 groups of order
  at most 64, rebuilt as cyclic extensions and compared with the known numbers of groups of each order; exact
  character tables (modular method, lifted to cyclotomic integers, with a floating-point cross-check of the degrees);
  every automorphism of every group of order ≤ 64 except three abelian groups (C_2⁵, C_4 × C_2⁴, C_2⁶; 200,000
  automorphisms each): 4,319,984 automorphisms in the complete enumerations (3,923,398 of them for the 469
  non-abelian groups), 120,818 involutory, 4,261 equality cases (4,263 with the capped runs), all involutory, no
  violation of Theorem 1.2,
  Lemma 2.3 or Proposition 3.1(a), (b), (d) and the bound of (c). Nine further groups of orders 72 to 720 with all
  their automorphisms. The identity ι_α(χ) = tr(U Ū)/d with explicit unitary matrices in 9,391 cases (numerical
  agreement to 1.3 · 10⁻¹⁴). Bors's count |S_inn(t)| = #{r : r² = t²} for every t in all 586 groups. Deliberately
  wrong inputs were detected.

- **Verification run C** (own numpy programs, in `reproducibility/independent_run_2/`; on the final text): 59 groups
  of orders 1 to 720 with all their automorphisms (7,832; among them the 42 groups of the program of the note, S_6,
  A_6 and PSL(2,11)), every statement of the note checked for every automorphism, every figure of the recorded
  output of `check_note.py` reproduced (42/42); the identity ι_α(χ) = tr(U Ū)/d with explicit unitary intertwiners
  in 6,326 cases (worst deviation 3.5 · 10⁻¹⁴); the recorded output reproduced byte for byte from the archived
  source; the totals of run B recomputed from its result files. 0 failures.

## Independent verification runs
The results were first obtained with written proofs and the floating-point program above. Three independent
verification runs, all AI-assisted, followed. Runs A and B (2026-10-17) examined the proofs before the note was
written; each re-derived the statement and both proofs step by step and wrote its own programs, and neither read
the other's report. Run C (2026-10-18) examined the final text; it is described after the table.

| Item | Run A | Run B |
|---|---|---|
| Statement against the source; readings | CONFIRMED (versions 1, 46, 48; 2006 printing not consulted) | CONFIRMED (version 48) |
| Lemmas 2.1–2.3 | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Proposition 3.1(a)–(c) for arbitrary α | CONFIRMED (re-derived; indices and conjugations checked) | CONFIRMED (re-derived; tr(UŪ)/d checked numerically in 9,391 cases) |
| Proposition 3.1(d) (involutory case) | CONFIRMED | CONFIRMED (exactly on 120,818 involutory automorphisms) |
| First proof | CONFIRMED | CONFIRMED |
| Second proof (Lemmas 5.1, 5.2) | CONFIRMED | CONFIRMED |
| Equality and strictness | CONFIRMED (also by a second route, now Remark 6.1) | CONFIRMED |
| Special cases and examples | CONFIRMED (Gow/Vinroot/Potter details not checked against the papers) | CONFIRMED |
| arXiv:2605.23195, Lemma 2.19 | proof incomplete as written: the Cauchy–Schwarz step from display (2) to (3) is not justified for non-linear χ; statement true | same assessment; witness at g = 1 (summands d and d²) |
| Literature | CONFIRMED_WITH_FIXES (crediting wording; printings; unverified counts) | CONFIRMED_WITH_FIXES (Bors 2019, Vinroot 2020 missing) |
| Computations | 44 groups, 7,104 automorphisms, 0 failures | 586 groups of order ≤ 64 and 9 larger groups, 4,319,984 automorphisms in complete enumerations, 0 failures |

Neither run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) Neutral crediting of arXiv:2605.23195: its Theorem 3.1 states the inequality; the proof given for its
   Lemma 2.19 is incomplete (the Cauchy–Schwarz step from display (2) to display (3) on p. 6 is not justified for
   non-linear characters; at g = 1 the two summands are d and d²); the first proof follows the preprint's strategy
   (Lemma 2.20, a bound on ι, the triangle inequality); the new content is the proof of the bound, the second
   proof and the equality characterisation; Lemma 2.19 is quoted exactly ("ι_α(χ) ≤ 1"). Section 1.3, Section 4,
   abstract, "Scope and priority".
2. (Run A) The printings of the Notebook that were compared are named, and the 2006 printing is said not to have
   been consulted: Section 1.1, "Scope and priority".
3. (Run A) Counts from rate-limited services are marked as unverified, and the Notebook edition is cited correctly
   (No. 21, version 48; No. 18, version 1; 16th issue 2006): "Scope and priority", references.
4. (Run B) Bors, J. Algebra Appl. 18 (2019) 1950055 (arXiv:1601.04311), Section 2.2, Proposition 2.2.2 and
   Corollary 2.2.5, credited as prior partial results (inner automorphisms; complete groups) and recovered as a
   special case: Section 1.3, Remark 6.2, abstract, "Scope and priority".
5. (Run B) Vinroot, Math. Z. 294 (2020) 1759–1785, Section 2 (and Lemma 2.1), cited as the printed source of the
   twisted count for involutory automorphisms, together with Kawanaka–Matsuyama (1990) and Bump–Ginzburg (2004);
   the note says that only the zbMATH reviews of the last two were read: Section 1.3, "Scope and priority".
6. (Run B) Taylor–Vinroot is cited with its journal year 2018.

**Changes made when the note was written, after runs A and B** (all four examined by run C, below).
- Remark 6.1 (equality from Proposition 3.1 alone) was written out from the second route given by run A.
- The proof in Remark 6.2 (Bors's argument) was written out; run B had checked the count computationally.
- Example 6.3(3) describes the equality case of PSL(2,7) through GL(3,2) and symmetric matrices and relates it to
  Vinroot 2020, Theorem 10.1.
- The program `check_note.py` was written; it has not been examined by the runs.

### Verification run C (on the final text; 2026-10-18; AI-assisted)
Run C examined the paper as written, line by line, before any further edit; its programs and outputs are in
`reproducibility/independent_run_2/` (named after its position among the program packages on this record: runs A and B examined
the proofs before the note was written; run C is the second independent package and the third run in all).

| Item | Run C |
|---|---|
| Statement against the sources | CONFIRMED (arXiv v1 of 2014, v46, v48; corpus record; Problem 1.1 faithful) |
| Lemmas 2.1–2.3, Proposition 3.1(a)–(d), Remark 3.2, Example 3.3 | CONFIRMED (re-derived from the text as written: indices, conjugations, unitarisation of the intertwiner) |
| First proof; Lemmas 5.1, 5.2 with strictness; Theorem 1.2(a)–(c) | CONFIRMED (both directions of (b)) |
| Remark 6.1, Remark 6.2, Example 6.3 (added when the note was written) | CONFIRMED (re-derived; Bors's Section 2.2 compared, including his definition of L_{-1}; the 28 symmetric invertible matrices counted directly) |
| Quoted statements: arXiv:2605.23195 (Lemma 2.19 verbatim, Lemma 2.20, Theorem 3.1 and its use of the absolute value, Theorem 3.5, Corollary 3.6), arXiv:2605.22127 (Lemma 2.6, Corollary 2.4, Theorem 2.7), Bors (Prop. 2.2.2, Thm 2.2.3, Cor. 2.2.5), Vinroot (Section 2.2, Lemma 2.1, Theorem 10.1 and its sources [37] = TV18, [42] = Vin05), zbMATH reviews of KM90 and BG04 | CONFIRMED against the texts; the wording on Lemma 2.19 is accurate and neutral |
| Bibliography | CONFIRMED (Crossref, all 9 DOIs; arXiv API: both 2026 preprints still version 1 on 2026-10-18) |
| Program of the note | CONFIRMED: the certification argument (class-algebra identities, χ(1) = d, norm identity, distinct rows) is valid; re-run from the archived source.zip with byte-identical output |
| Own computation | 59 groups, 7,832 automorphisms, 6,326 explicit intertwiners, negative controls; 0 failures; 42/42 figures of the recorded output reproduced |
| Run B's totals | recomputed from its result files (4,319,984 / 3,923,398 / 120,818 / 4,261 and 4,263); the capped groups are C_2⁵, C_4 × C_2⁴, C_2⁶ (Hillar–Rhea count) |
| Literature | searches repeated on 2026-10-18 (arXiv: 8 queries; zbMATH Open: 4; Crossref; Semantic Scholar and OpenAlex: not available): nothing new; novelty claim not stronger than the evidence |

Run C found no mathematical error. Corrections it required, all made: (1) the Verification paragraph of the note
describes run C, and Remark 6.1, Remark 6.2, Example 6.3(3) and the program are no longer listed as unexamined;
(2) Section 7 says that the program checks T(G × H) = T(G)T(H) for eight direct products (not for all products of
the list); (3) the Scope paragraph records the repeated search of 18 October 2026; (4) this report and the
reproducibility README describe run C and its package. Observation not acted on: the proof of Lemma 2.3 of
arXiv:2605.22127 for odd l omits a one-line parity step; the statement is true and is a special case of Theorem 1.2.

## Relation to the literature, novelty and scope
- **Searches (17 October 2026).** arXiv API (twisted Frobenius–Schur indicators; automorphisms inverting many
  elements; MacHale; sums of character degrees with automorphisms; twisted involutions; Kourovka Notebook);
  zbMATH Open (all records of D. MacHale; topic queries; reviews of Kawanaka–Matsuyama, Bump–Ginzburg,
  Berkovich–Mann and others); Crossref; Semantic Scholar; the Notebook's website; six web searches. Bibliographic
  data of the references were checked again with Crossref and the arXiv API when the note was written.
  - Apart from arXiv:2605.23195 (statement with an incomplete proof), no printed answer to the question was found
    in the sources searched. A search that finds nothing is not a proof of novelty.
  - Semantic Scholar listed no work citing either 2026 preprint, and 49, 46 and 18 works citing Kawanaka–Matsuyama,
    Bump–Ginzburg and Vinroot 2005 (titles scanned; apart from the two preprints, none concerns the question).
    Later requests were rate-limited, so only the count for arXiv:2605.23195 was checked a second time; the other
    counts are unverified. OpenAlex refused all requests and was not used.
  - The Notebook's website showed no post on Problem 16.60 on 17 October 2026.
- **What was read.** arXiv:2605.23195 and arXiv:2605.22127 completely (version 1); Bors, Section 2.2, in arXiv
  version 3 (the journal version was not compared); Vinroot 2020, Section 1, Section 2.2 and Theorem 10.1, in the
  arXiv version (the journal version was not compared); only the zbMATH reviews of Kawanaka–Matsuyama,
  Bump–Ginzburg and Berkovich–Mann; of Vinroot 2005 and Taylor–Vinroot 2018 only what Vinroot 2020 says about them.
  The full text of Kawanaka–Matsuyama was not read (the publisher's page refused automated access). The proofs of
  the note do not depend on any of these texts.
- **Scope.** Complete answer (yes), with the characterisation of equality. No priority is claimed; the novelty is
  modest. Under the policy of the corpus (solved only when peer-reviewed), the record stays `open` with a note on
  this unrefereed result.

## 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/