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

Fulman, Jason; Guralnick, Robert

doi:10.1090/tran/7377

Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston–Shalev conjecture
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by Jason Fulman and Robert Guralnick PDF

Trans. Amer. Math. Soc. 370 (2018), 4601-4622 Request permission

Abstract:

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 $\delta$ 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 $\delta$. We show that with possibly finitely many exceptions, one can take $\delta = .016$. 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.

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