Kronecker Classes and Cliques in Derangement Graphs

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. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that $|A : G| = 3$ and U is a subgroup of G satisfying $G = \bigcup _{a \in A} U^a$ , then $|G : U| \leq 10$ . This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.