An infinite family of counterexamples to the Polycirculant Conjecture

Abstract

We disprove the Polycirculant Conjecture, which states that every transitive 2-closed permutation group is non-elusive, i.e. contains a derangement of prime order. In fact, we prove a stronger result, answering a long-standing question of Maru\v{s}i\v{c} and Jordan: there exists a vertex-transitive graph admitting no semiregular automorphism. To do so, we employ recently developed methods of Chen et al. for constructing elusive groups via non-split extensions, allowing us to construct an elusive group $7^6.\mathrm{PSU}_3(3)$ of degree 16,464. We show that this group is the full automorphism group of seven of its orbital graphs and hence is 2-closed. Our example extends to infinitely many counterexamples of the Polycirculant Conjecture, and infinitely many vertex-transitive graphs admitting no semiregular automorphism.