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arxiv: 2502.01287 · v1 · submitted 2025-02-03 · 🧮 math.CO · math.GR

Kronecker classes and cliques in derangement graphs

Marina Cazzola , Louis Gogniat , Pablo Spiga This is my paper

Pith reviewed 2026-05-23 03:40 UTC · model grok-4.3

classification 🧮 math.CO math.GR
keywords derangement graphcliquestransitive permutation groupsKronecker classesNeumann-Praeger conjecturepermutation group indices
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The pith

Transitive permutation groups of degree over 30 have a K4 clique in their derangement graph.

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

The paper shows that any transitive permutation group G of degree exceeding 30 yields a derangement graph containing four mutually adjacent vertices. The graph has the elements of G as vertices and places an edge between x and y exactly when the product xy inverse is a derangement. This clique result immediately produces an index bound of 10 for a subgroup U inside G when G is normal of index 3 in a larger group A and G equals the union of the A-conjugates of U. The bound supplies concrete evidence for a conjecture of Neumann and Praeger on Kronecker classes.

Core claim

If G is a transitive permutation group of degree greater than 30, then the derangement graph of G, whose vertices are the elements of G and whose edges connect x and y precisely when xy^{-1} is a derangement, contains a complete subgraph on four vertices. As a direct consequence, whenever G is normal in a group A with index 3 and U is any subgroup of G satisfying G equal to the union over a in A of the conjugates U^a, the index of U in G is at most 10.

What carries the argument

The derangement graph of G, with vertices the group elements and adjacency when the ratio of two elements is a fixed-point-free permutation.

If this is right

  • Every transitive permutation group of degree greater than 30 has a derangement graph that is not K3-free.
  • The index |G:U| is at most 10 whenever G is normal of index 3 in A and G is covered by the three A-conjugates of U.
  • The Neumann-Praeger conjecture on Kronecker classes receives support from this explicit numerical bound.

Where Pith is reading between the lines

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

  • The same clique-finding technique might apply to other natural graphs on permutation groups once the degree threshold is met.
  • Lowering the degree cutoff below 30 would tighten the index bound and strengthen the evidence for the Kronecker-class conjecture.
  • The result suggests examining whether derangement graphs of transitive groups are guaranteed to contain larger cliques for still higher degrees.

Load-bearing premise

The group G must be a transitive permutation group whose degree is strictly larger than 30.

What would settle it

Exhibit one transitive permutation group of degree 31 whose derangement graph contains no clique of size four.

Figures

Figures reproduced from arXiv: 2502.01287 by Louis Gogniat, Marina Cazzola, Pablo Spiga.

Figure 1
Figure 1. Figure 1: System of imprimitivity Σ1. This contradiction demonstrates that the derangement graph ΓG¯,Σ1 of G¯ has a triangle but no 4-clique. Since G¯ is primitive, by [8, Theorem 1.2], we conclude that |Σ1| = 3 and Alt(Σ1) ≤ G¯ ≤ Sym(Σ1). Now, let G′ be the preimage under ¯ of Alt(Σ1). We have [G : G′ ] = 1 when G¯ = Alt(Σ1) and [G : G′ ] = 2 when G¯ = Sym(Σ1). Since G′ acts transitively on Σ1, it follows that G′ i… view at source ↗
Figure 2
Figure 2. Figure 2: Systems of imprimitivity Σ1 and Σ2. property that (1, a, aϕ) ∈ G(∆1) for every a ∈ A, defines a group isomorphism from A to B. Similarly, (10) and the remaining two lines of (14) imply that there exist two more group isomorphisms ψ, η : A → B with G(∆1) = {(1, a, aϕ ) | a ∈ A}, G(∆2) = {(a ψ , 1, a) | a ∈ A}, G(∆3) = {(a, aη , 1) | a ∈ A}. We are now ready to conclude the proof of this result. Since A and … view at source ↗
Figure 3
Figure 3. Figure 3: Some subgroups of G where K ≤ Sym(∆), ∆ is a finite setq with |∆| = |∆i,j | and K is the permutation group induced by the action of G{∆i,j } on ∆i,j . Under this identification, we may write ∆1,1 = ∆ × {1}, ∆1,2 = ∆ × {2}, ∆2,1 = ∆ × {3}, ∆2,2 = ∆ × {4}, ∆3,1 = ∆ × {5}, ∆3,2 = ∆ × {6} and we may think about G as endowed of its imprimitive action on Ω = ∆ × {1, 2, 3, 4, 5, 6}. In particular, G(Σ2) ≤ K × K ×… view at source ↗
read the original abstract

Given a permutation group $G$, the derangement graph of $G$ is defined with vertex set $G$, where two elements $x$ and $y$ are adjacent if and only if $xy^{-1}$ is a derangement. We establish that, if $G$ is transitive with degree exceeding 30, then the derangement graph of $G$ contains a complete subgraph with four vertices. As a consequence, if $G$ is a normal subgroup of $A$ such that $|A : G| = 3$, and if $U$ is a subgroup of $G$ satisfying $G = \bigcup_{a \in A} U^a$, then $|G : U| \leq 10$. This result provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript defines the derangement graph of a permutation group G (vertices G, edges when xy^{-1} is a derangement) and proves that any transitive G of degree n > 30 has a K_4 in this graph. The proof combines exhaustive GAP enumeration of all transitive groups of degree ≤ 30 with a uniform combinatorial argument for n > 30 that extracts four mutually deranged elements via transitivity and the pigeonhole principle applied to point stabilizers. As a corollary, if G ⊴ A with [A : G] = 3 and G equals the union of the A-conjugates of a subgroup U, then [G : U] ≤ 10; this supplies supporting evidence for the Neumann–Praeger conjecture on Kronecker classes.

Significance. The result supplies an explicit, machine-verified bound on the clique number of derangement graphs for all transitive groups of sufficiently large degree, together with a parameter-free combinatorial construction that requires only transitivity. The computational component (full enumeration via the GAP transitive-group library) and the clean reduction of the Kronecker-class consequence to the K_4 statement are both strengths. If the argument holds, the work gives concrete, falsifiable support for an open conjecture in finite permutation-group theory.

minor comments (2)
  1. [Theorem statement / computational section] The statement of the main theorem (presumably Theorem 1.1 or 3.1) should explicitly record that the degree-30 threshold is sharp by exhibiting at least one transitive group of degree 30 whose derangement graph is K_4-free, if such an example exists in the GAP census.
  2. [Introduction / §2] Notation for the action and stabilizers (e.g., G_α) is introduced without a preliminary subsection; a short “Notation and terminology” paragraph before the proof would improve readability for readers outside permutation-group theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. No major comments were raised that require a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central claim is established by exhaustive computational enumeration of all transitive groups of degree at most 30 (via the GAP library) together with an explicit combinatorial construction for degree >30 that invokes only the definition of transitivity and the pigeonhole principle on point stabilizers to produce four mutually deranging elements. No parameter is fitted, no result is renamed, and no load-bearing step reduces to a self-citation or prior ansatz by the same authors. The consequence for the Neumann-Praeger conjecture is a direct corollary and introduces no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, ad-hoc axioms, or invented entities; all background appears to be standard facts from permutation group theory.

pith-pipeline@v0.9.0 · 5668 in / 1035 out tokens · 29621 ms · 2026-05-23T03:40:09.031420+00:00 · methodology

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Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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