DERANGEMENTS IN FINITE CLASSICAL GROUPS FOR ACTIONS RELATED TO EXTENSION FIELD AND IMPRIMITIVE SUBGROUPS AND THE SOLUTION OF THE BOSTON-SHALEV CONJECTURE.

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al. and Shalev. The conjecture asserts that there is an absolute positive constant δ such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than δ. We show that with possibly finitely many exceptions, one can take δ = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements tends to 1 as n increases and we prove similar results for families of permutation representations. [ABSTRACT FROM AUTHOR]