Year: 2026
Type: preprint
Abstract: Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.
Source: arXiv (Cornell University)
Authors: Anzanello, Jessica, Eberhard, Sean
Institutions: University of Milano-Bicocca, University of Warwick
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of Warwick
OpenAlex ID: W7168304072
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