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REVIEW 3 major objections 4 minor 41 references

This paper resolves eight open problems from the Kourovka Notebook, offering explicit constructions and counterexamples, and reports that every proof is verified by a computer.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 17:51 UTC pith:GC523VWZ

load-bearing objection Eight real solutions to Kourovka problems, Lean-backed, but Theorem 2.1's printed proof has a subscript error that must be fixed before the paper works as a readable informal account. the 3 major comments →

arxiv 2607.17477 v2 pith:GC523VWZ submitted 2026-07-20 math.GR math.CO

On Some Problems from the Kourovka Notebook

Wouter van Doorn , Elias Judin , Pietro Monticone , Daniel Morrison This is my paper
classification math.GR math.CO MSC 68V2005C2505E1606F1517B3820B3020D0620D15
keywords Kourovka Notebookgroup theorylocally soluble groupspermuted productsRota-Baxter operatorsclass transpositionspower graphsformal verification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reports solutions to eight open problems from the Kourovka Notebook, a standard collection of open questions in group theory. The answers are explicit constructions or counterexamples: a group with exactly two maximal locally soluble normal subgroups; for each 1 ≤ k ≤ n! a group whose n! ordered products of n chosen elements take exactly k values; a simple and a non-simple group with the same order and same totient sum; a surjective but non-injective Rota–Baxter operator on a non-abelian group; the identification of a class-transposition group with a finite symmetric group; the statement that cograph power graphs are chordal; right-relatively convex subgroups that do not form a sublattice; and a p-group extension where a natural rank inequality fails. The paper also reports that every proof was developed and checked by an automated formal reasoning system, so the results come with machine-verified formalizations.

Core claim

The paper claims to settle all eight open problems, each with a machine-checked proof. The results are explicit: a group with exactly two maximal locally soluble normal subgroups; for every 1 ≤ k ≤ n! a group whose n! ordered products of n chosen elements take exactly k values; groups of order 6048 and equal totient sum 23984 with different simplicity; a surjective non-injective Rota–Baxter operator on a non-abelian group; CT(k) ≅ S_{lcm(2,...,k)}; cograph power graphs are chordal; right-relatively convex subgroups need not form a sublattice; and a p-group extension in which rank(Z(H)∩H′) exceeds rank(B).

What carries the argument

The constructions differ by section. The permuted-products theorem is carried by a central extension Z^n × Z/kZ whose multiplication has the bilinear factor B(u,v)=Σ_{i<j} u_j v_i (j−1)! mod k; this turns the inversion vector of a permutation into a single residue. The local-solubility result uses a semidirect product of unipotent matrices over Z² with lattice translations, with a commutator estimate showing [U_d,U_e]≤U_{d+e}. The simplicity pair uses the group PSU(3,3) acting primitively on 28 isotropic points. The class-transposition theorem uses a standard theorem on primitive permutation groups containing a 3-cycle. The cograph/chordal proof uses a cyclic-subgroup containment dichotomy;

Load-bearing premise

The results stand on finite checks that the paper does not reproduce — twelve conjugation identities in the proof of Theorem 6.1, the element-order distributions in Section 4, and machine-checked proof files whose exact version is not given — so any undetected error in these checks would undo the corresponding theorem.

What would settle it

Independently recompute the element-order distributions of PSU(3,3), C6×S4×F, and C2×W×F and verify that each has total sum 6048 and totient sum 23984; a mismatch would refute Theorem 4.1. Alternatively, recompute the twelve conjugation identities in Section 6 to confirm that (σ(3)_{0,1}σ(4)_{0,1})^3 = (6 7); if that equation fails, the induction proving CT(k) ≅ S_{lcm(2,...,k)} collapses.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Every value 1 ≤ k ≤ n! occurs as the number of distinct products obtained by permuting n elements in some group.
  • Order and totient sum together do not characterize simplicity: PSU(3,3) and a non-simple group of order 6048 share the same totient sum 23984.
  • The group generated by horizontal class transpositions with modulus at most k is the full symmetric group on lcm(2,...,k) elements for every k ≥ 4.
  • A finite group whose power graph is a cograph has a chordal power graph.
  • In extensions of elementary abelian p-groups, the rank of Z(H)∩H′ can exceed the rank of B, so the proposed rank bound is false.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The inversion-vector central extension used for permuted products suggests a general recipe: any statistic on permutations that can be computed as a sum of fixed weights can be forced to take exactly k values by working modulo k. One could test whether the same construction works for other natural statistics, such as total displacement rather than inversion count.
  • The class-transposition theorem gives an explicit generation of every permutation of {0,...,lcm(2,...,k)-1} by modulus-bounded transpositions; a practical consequence would be an algorithm that decomposes any such permutation as a product of class transpositions, with possible use in computing with maps on residue classes that iterate affine steps.
  • The counterexample to the rank inequality has p=3, rank(B)=2 and rank(Z(H)∩H′)=3; an obvious next step is to determine, for each prime p, the largest possible gap between these ranks, or whether a rank-1 counterexample exists.
  • The workflow described in the paper — automated formal search producing a human-readable proof after verification — could be applied to other open problems whose answers are finite constructions. The main caveat, tied to the paper's own lack of reproduced computations, is that such a workflow needs its computational checks to be independently rerunnable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper presents solutions to eight open problems from the Kourovka Notebook, one dating back to 1969. The results are: a group with exactly two maximal locally soluble normal subgroups (Theorem 2.1); groups whose n! permuted products take exactly k values for every 1 ≤ k ≤ n! (Theorem 3.1); two groups of order 6048 with equal totient sums but only one simple (Theorem 4.1); a surjective non-injective Rota–Baxter operator on a non-abelian group (Theorem 5.1); identification of the group generated by horizontal class transpositions of modulus at most k with S_lcm(2,...,k) (Theorem 6.1); cograph power graphs of finite groups are chordal (Theorem 7.1); right-relatively convex subgroups need not form a sublattice (Theorem 8.1); and a counterexample to a proposed rank inequality for p-group extensions (Theorem 9.1). Each proof is accompanied by a claim of formal verification in Lean by the Aristotle agent, and an appendix describes the formalization workflow.

Significance. If the results are correct, the paper resolves eight problems from a standard open-problem collection in group theory, including one of long standing. The constructions are explicit and the statements are falsifiable; several results have already been independently obtained by other researchers (e.g., Pan for Theorem 6.1, Rundström for Theorem 7.1), which increases confidence. The strongest feature is the claim that every proof is machine-checked in Lean, and the paper is transparent about the human-AI division of labor. However, the manuscript is not fully self-contained in a few places: the proof of Theorem 2.1 contains a notational error that invalidates the printed induction, and the base case of Theorem 6.1 is delegated to an unshown computation. These issues are local and fixable, but they are load-bearing for the corresponding theorems.

major comments (3)
  1. [Section 2, paragraph after Lemma 2.5] The definitions of u_i and v_i are incompatible with the displayed identities. The text defines u_i := e_{m0,m0+2i} and v_i := e_{m0+2i,m0+2i+1}. With these definitions, g^{2i} u_i g^{-2i} = e_{m0+2i,m0+4i}, which equals v_i only for i = 0, and [u_i, v_i] = e_{m0,m0+2i+1}, not u_{i+1} = e_{m0,m0+2i+2}. Moreover, the displayed computation of [u_0,[u_0,g]] uses v_0 = e_{m0+1,m0+2}, whereas the printed definition gives v_0 = e_{m0,m0+1}. The intended definitions are clearly u_i = e_{m0,m0+2^i}, v_i = e_{m0+2^i,m0+2^{i+1}}, with conjugation by g^{2^i}. As printed, the induction proving that the derived subgroups of H are nontrivial cannot be followed. This is a load-bearing error for Theorem 2.1, though it is easily corrected; the Lean file should be checked against the corrected statement.
  2. [Section 6, base case of Theorem 6.1] The proof that CT(4) = S_12 rests on the assertion (σ^(3)_{0,1} σ^(4)_{0,1})^3 = (6 7) and on eleven displayed conjugation identities, followed by “These identities can all be verified computationally.” No calculation or derivation is provided. Because these identities supply the entire base case of the induction, the natural-language proof is not self-contained: the reader must either rerun the computation or trust the claimed Lean formalization. I recommend adding the computation in an appendix or giving a precise pointer to the verified Lean code, and pinning the repository version.
  3. [Section 4, Lemma 4.2] The element-order distribution of PSU(3,3) is stated as a table with no derivation. The totient sum 23984 depends exactly on this table, and the same is true for the distribution of H in Lemma 4.3. These are finite computations and may be routine, but since Theorem 4.1 is a counterexample based on these numerical coincidences, the text should at least indicate how the distributions were obtained (for instance, from conjugacy classes of PSU(3,3) or from the cited ATLAS file) or refer to a checked computation in the Lean repository.
minor comments (4)
  1. [Section 2, sentence before Lemma 2.5] The trichotomy “one of g_1, g_1 g_2, g_1^2 g_2 must have both translation coordinates nonzero” is unnecessary: if g_1 has nonzero first coordinate and g_2 has nonzero second coordinate, then g_1 g_2 already has both translation coordinates nonzero. This is not an error in the argument, but the wording is confusing.
  2. [Section 6, displayed formulas] Some displayed conjugation identities have unbalanced parentheses due to the line-breaking (for example, the display for (0 1)). This makes the formulas harder to check. Please reformat them.
  3. [Appendix A / GitHub link] The repository is cited by URL only, with no commit hash. Since the paper’s formal-verification claim is central, a pinned hash is essential for reproducibility and for refereeing the Lean code.
  4. [Section 8] In the proof of convexity of H_1, the step “it follows that c_y = 0” is compressed. A sentence explaining that both c-a and b-c must lie in the positive cone P is enough; currently the reader has to supply the case analysis.

Circularity Check

0 steps flagged

No significant circularity: the eight theorems are explicit constructions/computations with independent proofs. Self-citations are provenance, not load-bearing. The Theorem 2.1 subscript inconsistency and unpinned Lean artifact are correctness/verification gaps, not circularity.

full rationale

The paper's derivations do not reduce to their own inputs. Theorem 3.1 constructs a group in which permuted products equal R(pi) mod k, and the cardinality-k conclusion follows from the bijectivity of the inversion-code map R and elementary modular arithmetic; the outcome is engineered but not assumed. Section 4 gives explicit finite groups and computes order and totient-sum distributions; the match is a calculated equality, not a fitted parameter. Section 6 provides an induction with explicit generator identities and Jordan's theorem, Section 7 a self-contained graph argument, Section 8 explicit right orders, and Section 9 explicit commutator computations. The self-citations to the authors' Kourovka solution notes [25]–[32] are provenance: the proofs are reproduced in the paper, so none of these citations carries the mathematical argument. The Aristotle/Lean claim is an audit assertion rather than a premise of the informal derivations. The most serious in-paper issue is not circularity: in the proof of Theorem 2.1, the definitions 'ui := em0,m0+2i' and 'vi := em0+2i,m0+2i+1' are inconsistent with the claimed identity 'g^{2i} u_i g^{-2i} = v_i' (e.g., g^2 u_1 g^{-2} = e_{m0+2,m0+4} while v_1 = e_{m0+2,m0+3}), and the Lean repository is linked without a commit hash. These are correctness and reproducibility gaps, not reductions of conclusions to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters or invented entities; the central claims are direct constructions. The paper depends on standard background results (Zorn, Iwasawa, Jordan, Taylor's classical-group facts) and on finite computations that are asserted rather than fully derived in the prose.

axioms (5)
  • standard math Zorn's lemma: every group has a maximal locally soluble normal subgroup
    Used in Section 2 to assert existence of maximal locally soluble normal subgroups.
  • domain assumption Iwasawa's criterion and Taylor's classical-group results for SU(3,3)/PSU(3,3)
    Section 4 Lemma 4.2 uses them to establish simplicity, order, and transvection generation of PSU(3,3).
  • standard math Jordan's theorem on primitive permutation groups containing a q-cycle
    Section 6 uses it to show CT(k+1) contains A_{L_{k+1}}.
  • domain assumption Finite computations asserted in prose: CT(4) conjugation identities and order/totient distributions
    Section 6 says identities can be verified computationally; Section 4 tables are given without derivation; these are load-bearing for Theorems 4.1 and 6.1.
  • domain assumption Order formula for projective special unitary groups
    Section 4 uses |PSU(n,q)| formula from Taylor to compute 6048.

pith-pipeline@v1.3.0-alltime-deepseek · 23817 in / 27962 out tokens · 238745 ms · 2026-08-01T17:51:58.669921+00:00 · methodology

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Cite this review

Pith. "Pith review of On Some Problems from the Kourovka Notebook." pith.science (2026). https://pith.science/paper/GC523VWZ

@misc{pith2026260717477,
  author       = {Pith},
  title        = {Pith review of: On Some Problems from the Kourovka Notebook},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GC523VWZ}},
  note         = {Machine review of arXiv:2607.17477}
}
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read the original abstract

The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1 \le k\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic $\sum_g\varphi(\lvert g\rvert)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.

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Reference graph

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