Kronecker classes and cliques in derangement graphs
Marina Cazzola, Louis Gogniat, Pablo Spiga
TL;DR
The paper addresses the structure of derangement graphs of finite transitive permutation groups and their implications for covering subgroups related to Kronecker classes. By developing a hypergraph-analytic framework and a stepwise inductive approach grounded in imprimitivity and the O'Nan–Scott classification, it proves that for degree $| Omega|>30$, the derangement graph necessarily contains a $K_4$. As a corollary, it obtains a bound $|G:U| le 10$ when $G rianglelefteq A$, $|A:G|=3$, and $G=igcup_{a\nin A} U^a$, providing evidence toward Neumann–Praeger’s conjecture on Kronecker classes. The argument combines structural group-theoretic reasoning with computer-assisted verification of a handful of small exceptional cases, yielding a sharp dichotomy between large-degree behavior and a finite list of low-degree configurations. Overall, the work connects clique structure in derangement graphs with covering properties of subgroups, reinforcing conjectures in the theory of Kronecker classes.
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.