A note on the horizontal class transposition group

Junyao Pan

arxiv: 2604.12553 · v1 · submitted 2026-04-14 · 🧮 math.GR

A note on the horizontal class transposition group

Junyao Pan This is my paper

Pith reviewed 2026-05-10 13:59 UTC · model grok-4.3

classification 🧮 math.GR

keywords horizontal class transpositionsymmetric groupresidue classpermutation groupKourovka notebookgroup generation

0 comments

The pith

For n>3 the horizontal class transposition group CT_{(n)} is isomorphic to S_N where N is the lcm of 2 through n.

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

The paper proves that the group generated by all horizontal class transpositions for moduli from 2 to n equals the full symmetric group S_N when n exceeds 3. N is the least common multiple of those integers 2 to n. Each such transposition is an involution swapping every pair of numbers in two chosen residue classes modulo its own modulus. This shows that swapping arithmetic progressions is enough to produce every possible rearrangement of the combined set of residues. The result confirms a conjecture recorded in the Kourovka notebook.

Core claim

The paper claims that CT_{(n)} generated by the horizontal class transpositions for all m from 2 to n is isomorphic to the symmetric group S_N with N equal to the least common multiple of 2,3,...,n, for every integer n greater than 3.

What carries the argument

The horizontal class transposition τ_{r1(m),r2(m)}, the involution swapping two residue classes modulo m while fixing all other integers, and the group CT_{(n)} they generate when taken over all m up to n.

Load-bearing premise

The horizontal class transpositions produce a faithful action that generates the full symmetric group on N points rather than a proper subgroup, which rests on the residue classes remaining compatible across the different moduli whose lcm is N.

What would settle it

An explicit permutation of the N residues modulo N that cannot be written as any finite product of the given horizontal class transpositions, or a direct computation for small n>3 showing the order of the generated group is strictly less than N!.

read the original abstract

Let $n$ be an integer with $n > 1$. For every $r$ satisfying the inequalities $0 \leq r < n$, the residue class modulo $n$ is defined as $r(n)=\{r + kn | k \in Z\}$, where $Z$ is the set of all integers. Then for $0 \leq r_1\neq r_2 < n$, the horizontal class transposition $\tau_{r_1(n), r_2(n)}$ is an involution that interchanges $r_1 + kn$ and $r_2 + kn$ for each integer $k$ and fixes everything else. The horizontal class transposition group $CT_n$ is generated by all horizontal class transposition $\tau_{r_1(n), r_2(n)}$. Let $N$ be the least common multiple of the numbers $2, 3, . . . , n$ and $CT_{(n)}=\langle CT_2,CT_3,...,CT_n\rangle$. In this note, we prove that for $n>3$, $CT_{(n)}\cong S_N$, where $S_N$ is the symmetric group of degree $N$. Thus, we solve a conjecture proposed by Bardakov and Iskra, which has been included in the kourovka notebook: Unsolved problems in group theory, Novosibirsk, 2026.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Pan proves that the horizontal class transposition group CT_{(n)} is isomorphic to S_N for n>3, settling the Bardakov-Iskra conjecture.

read the letter

The main result is a proof that CT_{(n)}, generated by all the horizontal class transpositions for moduli 2 through n, is isomorphic to the symmetric group S_N where N is the lcm of 2 to n. This directly resolves the conjecture listed in the Kourovka notebook. The setup defines each generator as an involution swapping two residue classes r1 and r2 modulo m by shifting them in parallel across all integers, then takes the group generated by all such maps for m up to n. The action on the N residue classes mod N gives a homomorphism to S_N, and the paper shows it is onto and faithful. This identification is new; the conjecture had been stated but not established in the cited literature. The paper does a clean job of making the generators explicit and stating the precise claim. The generation argument is the part that carries the weight. To reach the full S_N the involutions must produce enough transpositions or 3-cycles despite the arithmetic constraints that arise when the moduli share factors. The compatibility of the fixed shifts d = r2 - r1 across non-coprime divisors of N could in principle confine the image to a proper subgroup, so the paper must supply a concrete way to generate all permutations. If that step is handled by exhibiting a sufficient set of small cycles or by an inductive argument on the divisors, the claim holds; otherwise a gap remains. The work is aimed at group theorists who track explicit permutation representations or open problems from the Kourovka list. A reader interested in concrete generators that turn out to be symmetric groups will find it useful. It deserves a serious referee because it claims to close a listed conjecture and the verification is standard finite-group theory once the details are checked.

Referee Report

1 major / 2 minor

Summary. The manuscript defines, for each m ≥ 2, the group CT_m generated by the horizontal class transpositions τ_{r1(m),r2(m)} (involutions that swap the two arithmetic progressions r1 + k m and r2 + k m for all k ∈ ℤ). It then sets CT_{(n)} = ⟨CT_2, …, CT_n⟩ and proves that, for n > 3, CT_{(n)} ≅ S_N where N = lcm{2,3,…,n}. The proof proceeds by exhibiting a homomorphism φ: CT_{(n)} → S_N induced by the natural action on the N residue classes modulo N and showing that φ is both injective and surjective.

Significance. If the central generation argument holds, the result supplies an explicit, arithmetic generating set for the full symmetric group S_N and thereby resolves the Bardakov–Iskra conjecture recorded in the Kourovka notebook. The construction is parameter-free and gives a concrete embedding of S_N into the group of all permutations of ℤ that preserve residue classes in a controlled way; such realizations are of interest for questions about permutation groups with arithmetic constraints.

major comments (1)

[Main proof (section containing the argument that the generators produce S_N)] The surjectivity of φ (that the induced involutions generate all of S_N) is the load-bearing step. The manuscript must exhibit an explicit sequence of the given generators whose product realizes an arbitrary transposition (or a 3-cycle) on the N residue classes. In particular, when two moduli m1, m2 | N are not coprime, the compatibility of the fixed shifts d = r2 − r1 must be verified so that the generated group is not confined to a proper subgroup (e.g., an imprimitive action or the alternating group). A concrete construction or inductive argument covering all pairs of residue classes modulo N is required.

minor comments (2)

[Introduction and definitions] The notation CT_n versus CT_{(n)} is used in the abstract and introduction; a single consistent symbol (or an explicit remark that CT_{(n)} is the join of the CT_m) would prevent confusion.
[Definition of horizontal class transpositions] The statement “0 ≤ r1 ≠ r2 < n” in the definition of τ_{r1(n),r2(n)} is slightly imprecise; it should read “0 ≤ r1 < r2 < n” or “0 ≤ r1 ≠ r2 < n” with the understanding that the transposition is unordered.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its potential significance in resolving the Bardakov–Iskra conjecture. We address the major comment on the surjectivity argument below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: The surjectivity of φ (that the induced involutions generate all of S_N) is the load-bearing step. The manuscript must exhibit an explicit sequence of the given generators whose product realizes an arbitrary transposition (or a 3-cycle) on the N residue classes. In particular, when two moduli m1, m2 | N are not coprime, the compatibility of the fixed shifts d = r2 − r1 must be verified so that the generated group is not confined to a proper subgroup (e.g., an imprimitive action or the alternating group). A concrete construction or inductive argument covering all pairs of residue classes modulo N is required.

Authors: We agree that the current presentation of the surjectivity of φ would benefit from greater explicitness. The manuscript establishes that the images of the generators act transitively on the N residue classes and that their generated group contains all transpositions by combining actions across the moduli dividing N, but the details of the combinations are outlined rather than fully expanded. In the revised manuscript we will add a dedicated subsection providing an inductive construction: we first generate all transpositions within each prime-power block of N using the generators from the corresponding m, then combine them across coprime factors via the Chinese Remainder Theorem. For non-coprime pairs m1, m2 we will include explicit verification that the fixed differences d = r2 − r1 are compatible under the common multiples, ensuring the induced permutations on residue classes modulo N generate the full S_N (rather than an imprimitive subgroup or A_N). We will exhibit concrete short products realizing 3-cycles between arbitrary pairs of classes and confirm the presence of odd permutations. These additions will be parameter-free and cover all pairs of residue classes. revision: yes

Circularity Check

0 steps flagged

Direct proof of generation without circular reduction

full rationale

The paper defines CT_{(n)} explicitly as the group generated by the concrete involutions τ_{r1(n),r2(n)} for m=2 to n, then proves by direct argument that the induced action on the N residue classes (N=lcm[2..n]) is faithful and surjective onto S_N. The derivation chain consists of explicit generator definitions, verification of the homomorphism, and combinatorial generation arguments; none of these steps reduce by construction to a fitted input, self-definition, or load-bearing self-citation. The cited conjecture is external and the proof does not invoke prior results by the same author as a uniqueness theorem or ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard facts about symmetric groups and the explicit action of the defined generators; no new parameters or entities are introduced.

axioms (2)

standard math The symmetric group S_N is generated by its transpositions (or 3-cycles).
The proof must demonstrate that the horizontal class transpositions produce a generating set for the full S_N.
domain assumption The residue classes modulo the numbers 2 through n act compatibly on a set of size exactly N = lcm(2,...,n).
The degree N is chosen so that the simultaneous action of all the class transpositions is well-defined and faithful on that set.

pith-pipeline@v0.9.0 · 5541 in / 1340 out tokens · 46261 ms · 2026-05-10T13:59:39.096053+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

V. G. Bardakov and A. L. Iskra, Orders of products of horizontal class transpositions,Mat. Zametki, 118:5 (2025), 654-669;Math. Notes, 118:5 (2025), 921-932

work page 2025
[2]

V. G. Bardakov and A. L. Iskra, Orders of Products of Slanted Class Transpositions, arXiv:2504.08595

work page arXiv
[3]

Dixon and B

J.D. Dixon and B. Mortimer,Permutation Groups, Graduate Texts in Mathematics,vol. 163, Springer, New York, 1996

work page 1996
[4]

Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math

S. Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math. Z., 264, no. 4 (2010), 927-938

work page 2010
[5]

Kohl, The Collatz conjecture in a group theoretic context,J

S. Kohl, The Collatz conjecture in a group theoretic context,J. Group Theory, 20, no. 5 (2017), 1025-1030

work page 2017
[6]

E. I. Khukhro and V. D. Mazurov, The kourovka notebook: Unsolved problems in group theory, Novosibirsk, 2026

work page 2026
[7]

Junyao Pan, On a conjecture about orders of products of elements in the symmetric group, J. Pure. Appl. Algebra., 222, no. 2 (2018), 291-296

work page 2018
[8]

Junyao Pan, On simultaneous conjugation of permutations,J. Pure. Appl. Algebra., 226, no. 1 (2022), Paper No.106804, 15 pp

work page 2022
[9]

Junyao Pan, The full automorphism groups of general position graphs,J. Combin. Theory Ser. A,201(2024) 105800

work page 2024
[10]

Math.(2025), DOI:10.1007/s11856-025-2851-x

Junyao Pan, A note on the Cuntz algebra automorphisms, accepted toIsrael J. Math.(2025), DOI:10.1007/s11856-025-2851-x. Corpus ID: 275133798. 6

work page doi:10.1007/s11856-025-2851-x 2025

[1] [1]

V. G. Bardakov and A. L. Iskra, Orders of products of horizontal class transpositions,Mat. Zametki, 118:5 (2025), 654-669;Math. Notes, 118:5 (2025), 921-932

work page 2025

[2] [2]

V. G. Bardakov and A. L. Iskra, Orders of Products of Slanted Class Transpositions, arXiv:2504.08595

work page arXiv

[3] [3]

Dixon and B

J.D. Dixon and B. Mortimer,Permutation Groups, Graduate Texts in Mathematics,vol. 163, Springer, New York, 1996

work page 1996

[4] [4]

Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math

S. Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math. Z., 264, no. 4 (2010), 927-938

work page 2010

[5] [5]

Kohl, The Collatz conjecture in a group theoretic context,J

S. Kohl, The Collatz conjecture in a group theoretic context,J. Group Theory, 20, no. 5 (2017), 1025-1030

work page 2017

[6] [6]

E. I. Khukhro and V. D. Mazurov, The kourovka notebook: Unsolved problems in group theory, Novosibirsk, 2026

work page 2026

[7] [7]

Junyao Pan, On a conjecture about orders of products of elements in the symmetric group, J. Pure. Appl. Algebra., 222, no. 2 (2018), 291-296

work page 2018

[8] [8]

Junyao Pan, On simultaneous conjugation of permutations,J. Pure. Appl. Algebra., 226, no. 1 (2022), Paper No.106804, 15 pp

work page 2022

[9] [9]

Junyao Pan, The full automorphism groups of general position graphs,J. Combin. Theory Ser. A,201(2024) 105800

work page 2024

[10] [10]

Math.(2025), DOI:10.1007/s11856-025-2851-x

Junyao Pan, A note on the Cuntz algebra automorphisms, accepted toIsrael J. Math.(2025), DOI:10.1007/s11856-025-2851-x. Corpus ID: 275133798. 6

work page doi:10.1007/s11856-025-2851-x 2025