Jessica Anzanello

Derangements in intransitive groups

A classical theorem of Jordan states that every non-trivial finite transitive permutation group contains a derangement, that is, an element with no fixed points. Fein, Kantor, and Schacher further proved that one can find a derangement of prime-power order.

In this talk, I will discuss the existence of derangements in intransitive permutation groups, focusing on the case of two orbits. Generalizing the theorem of Fein–Kantor–Schacher, we show that every finite permutation group with two orbits on more than two points contains either a derangement or an element of prime-power order with exactly one fixed point. As a corollary, we confirm and generalize a conjecture of Ellis and Harper, proving the existence of a derangement when the two orbits have sizes a and b satisfying (a, b-1)=(a-1, b)=1. This is joint work with Sean Eberhard.