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

Authors: Jason Fulman and Robert Guralnick
Journal: Trans. Amer. Math. Soc. 370 (2018), 4601-4622
MSC (2010): Primary 20G40, 20B15
Published electronically: February 1, 2018
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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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Additional Information

Jason Fulman
Affiliation: Department of Mathematics University of Southern California Los Angeles, California 90089-2532

Robert Guralnick
Affiliation: Department of Mathematics University of Southern California Los Angeles, California 90089-2532

Keywords: Derangement, finite classical group, random matrix, permutation group
Received by editor(s): August 28, 2015
Received by editor(s) in revised form: September 16, 2016
Published electronically: February 1, 2018
Additional Notes: The first author was partially supported by NSA grant H98230-13-1-0219 and Simons Foundation Grant 400528
The second author was partially supported by NSF grant DMS-1302886
We thank the referee for a careful reading and interesting comments
Article copyright: © Copyright 2018 American Mathematical Society

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