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This is the fourth paper in a series. We prove a conjecture madeindependently by Boston et al and Shalev. The conjecture asserts that there isan absolute positive constant delta such that if G is a finite simple groupacting transitively on a set of size n > 1, then the proportion of derangementsin G is greater than delta. We show that with possibly finitely manyexceptions, one can take delta = .016. Indeed, we prove much stronger resultsshowing that for many actions, the proportion of derangements goes to 1 as nincreases and prove similar results for families of permutationrepresentations.

Derangements in finite classical groups for actions related to extension  field and imprimitive subgroups and the solution of the Boston-Shalev  conjecture