Almost elusive permutation groups

By a classical theorem of Jordan from the 19th century, we know that every finite transitive permutation group of degree at least two contains a fixed-point-free element, which we call a derangement. This observation leads to a number of natural questions, which have been intensively studied in recent years, with numerous applications. Emily's project will explore some new directions in this area. For example, Fein, Kantor and Schacher used the Classification of Finite Simple Groups to prove that every transitive group has a derangement of prime power order, but there are examples with no derangements of prime order; they are called "elusive groups". The elusive primitive groups were classified by Giudici in 2002 and it would be interesting to see what happens if the definition is weakened slightly, say by allowing a unique conjugacy class of derangements of prime order. Emily will investigate this class of "almost elusive groups", using the O'Nan-Scott Theorem to reduce the problem to almost simple groups, which will be the main focus of her research. I expect this will lead naturally to several other related problems. For example, it would be interesting to combine the study of derangements with problems concerning minimal and probabilistic generation (e.g, which primitive groups can be generated by two derangements?). There are also a number of interesting open problems on the proportion of derangements in primitive groups, which may form part of Emily's project.