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 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 76.PSU3(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.
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BibTeX
@misc{freedman2026infinite,
title = {{An infinite family of counterexamples to the Polycirculant Conjecture}},
author = {Saul D. Freedman and Melissa Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.23423},
eprint = {2607.23423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.23423},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An infinite family of counterexamples to the Polycirculant Conjecture
AU - Saul D. Freedman
AU - Melissa Lee
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2607.23423
UR - https://arxiv.org/abs/2607.23423
AB - 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.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.
ER -