Derangements
Peter J. Cameron, Queen Mary, University of London
Saturday May 20, 9.30-10.15, Groups in Galway 2006
A derangement is a permutation without fixed points. Well over a century
ago, Jordan showed that any finite transitive permutation group of degree
greater than 1 contains a derangement.
In the talk I will discuss three issues concerning derangements. First,
there is a randomized algorithm which efficiently finds a derangement
in an arbitrary transitive troup, but no deterministic algorithm is
known. Second, asking for a derangement of prime-power (or prime) order
leads quickly to some very difficult questions, mostly still unresolved.
Finally, in connection with the character theory of loops, I pose a
question about derangements and random Latin squares, and present some
numerical evidence.