DERANGEMENTS IN PERMUTATION GROUPS WITH TWO ORBITS

Abstract A classical theorem of Jordan asserts that if a group G acts transitively on a finite set of size at least $2$ , then G contains a derangement (a fixed-point free element). Generalisations of Jordan’s theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper [‘Orbits of permutation groups with no derangements’, Preprint, 2024, arXiv:2408.16064], which says that if G has exactly two orbits and those orbits have equal length $n \geq 2$ , then G contains a derangement. We prove this conjecture in the case where n is a product of two primes, and in the case where $n=bp$ with p a prime and $2b\leq p$ . We also verify the conjecture computationally for $n \leq 30$ .