An equivalence between a conjecture of Neumann-Praeger on Kronecker classes and a conjecture on cliques of derangement graphs

TL;DR

The paper establishes a surprising equivalence between the Neumann–Praeger conjecture on Kronecker classes and a conjecture about derangement-graph cliques in transitive groups. It introduces a combinatorial bound for the degree in terms of the maximum clique size via a normal imprimitivity length parameter $\ell$, and proves the equivalence by combining group-theoretic and number-theoretic tools, including results on simple groups of Lie type, the Monster, and prime-divisibility phenomena. The work uncovers deep links between algebraic number theory (Kronecker equivalence) and permutation-group combinatorics, while providing a controllable weak bound depending on $\ell$ that informs the overall equivalence. The methods highlight the roles of derangement structure, normal imprimitivity series, and CFSG-based classification in bounding degrees from clique constraints.

Abstract

We prove an equivalence between a conjecture of Neumann and Praeger on Kronecker classes in algebraic number fields, and a conjecture on cliques of derangement graphs in combinatorics.

Figures (2)

• Figure 1: Example: Normal systems of imprimitivity $\Sigma_{i_{j-1}}$ (thick lines) and $\Sigma_{i_j-1}$ (thin lines).
• Figure 2: Example: Here, $\kappa=4$, $r=6$, $\pi_{\Delta}$ is the partition of $\{0,\dots,23\}$ represented in the picture above. $\pi_{\Delta,1}=\{\{0,1,2\},\{3,4,5\}\},\pi_{\Delta,3}=\{\{12,15\},\{13,16\},\{14,17\}\},\pi_{\Delta,2}=\{\{6,7,8\},\{9,10,11\}\},\pi_{\Delta,4}=\{\{i\}\mid 18 \le i \le 23\}.$

Theorems & Definitions (16)

• Conjecture 1.1
• Conjecture 1.2: Neumann-Praeger, see Praeger2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Saxl
• Theorem 2.2: AS
• Theorem 2.3: FPS, Theorem 1.1
• Lemma 2.4: Siegel
• Theorem 2.5: LPS
• Lemma 2.6