Year: 2026
Type: preprint
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 Marušič 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.
Source: arXiv (Cornell University)
Authors: Freedman, Saul D., Lee, Melissa
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171570267