On fixed points of elements in primitive permutation groups

The fixity of a finite permutation group is the maximal number of fixed points of a non-identity element. We study the fixity of primitive groups of degree n  , showing that apart from a short list of exceptions, the fixity of such groups is at least n1/6n1/6. We also prove that there is usually an involution fixing at least n1/6n1/6 points.