Last week and this week I have been at the University of Southampton visiting Tim Burness to work on our big project concerning derangements. Despite it being summer here, the weather has been more like a Perth winter. I am staying in a self-catered hall of residence so the cooked breakfasts have come to an end and it is back to cornflakes.
A derangement is a permutation that moves all the points of the set which it acts on. I have only recently been converted to the term derangement, have previously preferred the term fixed point free element. Peter Cameron has had a bit to say about derangements in several posts on his blog.
By the Orbit-Counting Lemma, the average number of fixed points of an element of a transitive permutation group is 1. Since the identity permutation fixes all the elements of the set, a transitive permutation group on a set of size at least two must therefore contain a derangement. This leads to the natural questions that have attracted a lot of attention:
- how many derangements are there?
- what properties do the derangements have?
The project with Tim focuses on the second.
Fein, Kantor and Schacher proved that every transitive permutation group contains a derangement of prime power order. The result was motivated by an application to number theory as the existence of a derangement of prime power order is equivalent to the relative Brauer group of any nontrivial field extension of global fields being infinite. Whereas the existence of a derangement is an elementary observation, the proof of the existence of one of prime power order relies heavily on the Classification of Finite Simple Groups. The proof though does not give information about which primes or which powers.
In fact, not every transitive permutation group contains a derangement of prime order. My favourite example is the Mathieu group in its 3-transitive action on 12 points. Note that if a group G acts transitively on a set with the stabiliser of some point being the subgroup H, then an element g fixes a point if and only if it is conjugate to an element of H. Since contains a unique conjugacy class of elements of order 2 and a unique class of elements of order 3, and 6 divides the order of the stabiliser of a point (), it follows that all elements of order 2 and 3 fix a point.
A transitive permutation group is called elusive if it has no derangements of prime order. In my PhD, I showed that the only elusive primitive permutation groups are acting on points in its natural product action for any and K a transitive subgroup of . (I in fact proved something stronger than this: that these are the only elusive permutation groups with a transitive minimal normal subgroup.) Elusive groups are interesting from a graph theory perspective as there is the polycirculant conjecture that states that every vertex-transitive digraph has an automorphism that is a derangement of prime order. (This is perhaps a topic for another day.)
Again, this result was not quantitative and it would be useful to have results determining the primes for which there are derangements. This has been the focus of my work with Tim. Since all cycles of an element of prime order r have length r, there can only be derangements of prime order for the primes dividing the size of the set. For a prime r, we call a transitive permutation group G acting on a set , r-elusive if r divides the size of and all elements of order r in G have fixed points. A group is then elusive if it is r-elusive for all primes r dividing the degree. Our first paper in the project was joint work with Rob Wilson, and has just appeared online in the Journal of Algebra.
In the paper we reduce the study of r-elusive primitive groups to studying the almost simple ones. We then determine the r-elusive primitive actions of the almost simple groups whose socle is either the alternating group with or a sporadic simple group. In the spirit of the original result of Fein, Kantor and Schacher, for the r-elusive actions we also determined if there is a derangement of order a power of r. The paper also contains lots of information about which conjugacy classes of elements of the sporadics meet which maximal subgroup. In particular, for each primitive action of the Baby Monster we determined the specific conjugacy classes of elements of prime order that are derangements. We also gave similar information for most of the primitive actions of the Monster.