Derangements in intransitive groups

TL;DR

This paper investigates conditions under which intransitive permutation groups contain derangements, proposing conjectures and proving results for various classes of groups, with implications for group theory and permutation combinatorics.

Contribution

It introduces a conjecture about derangements in intransitive groups with two equal-sized orbits and proves it in several important cases, advancing understanding of derangements and subgroup unions.

Findings

Proves the conjecture for primitive, soluble, almost simple groups, and groups of order ≤ 50000.

Establishes the conjecture when one subgroup is maximal.

Provides a linear variant related to Isbell's conjecture and explores connections to permutation families and polynomial roots.

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.