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Derangements in subspace actions of finite classical groups


Authors: Jason Fulman and Robert Guralnick
Journal: Trans. Amer. Math. Soc. 369 (2017), 2521-2572
MSC (2010): Primary 20G40, 20B15
DOI: https://doi.org/10.1090/tran/6721
Published electronically: June 20, 2016
MathSciNet review: 3592520
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Abstract: This is the third in a series of four papers in which we prove a conjecture made by Boston et al. and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.


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Additional Information

Jason Fulman
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: fulman@usc.edu

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@math.usc.edu

DOI: https://doi.org/10.1090/tran/6721
Keywords: Derangement, finite classical group, random matrix, random permutation
Received by editor(s): April 8, 2013
Received by editor(s) in revised form: July 28, 2014, and April 14, 2015
Published electronically: June 20, 2016
Additional Notes: The first author was partially supported by NSA grants H98230-13-1-0219 and by Simons Foundation Fellowship 229181
The second author was partially supported by NSF grants DMS-1001962 and DMS-1302886 and by Simons Foundation Fellowship 224965
Article copyright: © Copyright 2016 American Mathematical Society