Derangements in subspace actions of finite classical groups

Fulman, Jason; Guralnick, Robert

doi:10.1090/tran/6721

Derangements in subspace actions of finite classical groups
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by Jason Fulman and Robert Guralnick PDF

Trans. Amer. Math. Soc. 369 (2017), 2521-2572 Request permission

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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