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

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.