Derangements in intransitive groups

Abstract

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and this can happen even if $G$ has only two orbits, both of which have size $(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture when $G$ acts primitively on at least one of the orbits. Equivalently, we conjecture that a finite group is never the union of conjugates of two proper subgroups of the same order, and we prove this conjecture when at least one of the subgroups is maximal. (Feldman also implicitly raised this conjecture on StackExchange.) We also prove the conjecture for soluble groups, almost simple groups and groups of order at most 50000, and we reduce the conjecture to perfect groups. Along the way, we prove a linear variant on Isbell's conjecture regarding derangements of prime-power order, and we highlight connections with intersecting families of permutations and roots of polynomials modulo primes.