Kronecker classes and cliques in derangement graphs

TL;DR

This paper investigates derangement graphs of permutation groups, proving the existence of large cliques in transitive groups of degree over 30, and explores implications for subgroup indices related to a conjecture on Kronecker classes.

Contribution

It establishes the presence of 4-cliques in derangement graphs of large transitive groups and derives bounds on subgroup indices supporting a conjecture on Kronecker classes.

Findings

Derangement graphs of transitive groups with degree > 30 contain 4-cliques.

If G is a normal subgroup with index 3 in A, then certain subgroup index bounds hold.

Supports a conjecture by Neumann and Praeger on 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.