Fixed-point-free elements in two-orbit permutation groups

Jessica Anzanello; Sean Eberhard

group theory

Fixed-point-free elements in two-orbit permutation groups

arXiv:2607.11543

summary

The paper proves that any permutation group with exactly two orbits on more than two points must contain either a derangement or an element of prime‑power order fixing exactly one point, and derives conditions guaranteeing a derangement.

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 $g cd (n_{1}, n_{2} - 1) = g cd (n_{1} - 1, n_{2}) = 1$ , 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.

9 pages

Topics & keywords

#permutation groups #derangements #two‑orbit actions #simple groups #normal 2‑coveringsderangementprime‑power orderfixed pointclassification of finite simple groupsnormal 2‑coverings