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Derangements in finite classical groups and characteristic polynomials of random matrices

Jason Fulman, Robert Guralnick

Abstract

We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which lie in a proper irreducible subgroup tends to 0 as the dimension of the natural module goes to infinity. This result is analogous to the result of Luczak and Pyber [15] that the proportion of elements of the symmetric group S_n which are contained in a proper transitive subgroup other than the alternating group goes to 0 as n goes to infinity. We also show that the probability that 3 random elements of SL(n,q) invariably generate goes to 0 as n goes to infinity.

Derangements in finite classical groups and characteristic polynomials of random matrices

Abstract

We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which lie in a proper irreducible subgroup tends to 0 as the dimension of the natural module goes to infinity. This result is analogous to the result of Luczak and Pyber [15] that the proportion of elements of the symmetric group S_n which are contained in a proper transitive subgroup other than the alternating group goes to 0 as n goes to infinity. We also show that the probability that 3 random elements of SL(n,q) invariably generate goes to 0 as n goes to infinity.

Paper Structure

This paper contains 14 sections, 30 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Characteristic polynomials
  3. $GL(n,q)$
  4. $GU(n,q)$
  5. $Sp(2n,q)$
  6. $O^{\pm}(n,q)$
  7. Derangements for extension field subgroups
  8. $GL(n,q)$
  9. $GU(n,q)$
  10. $Sp(2n,q)$
  11. $O^{\pm}(n,q)$
  12. Other cases
  13. Bounded Rank and Lower Bounds
  14. Maximal Tori and Invariable generation

Key Result

Theorem 1.1

For any polynomial $\phi$, the chance that a random element of $GL(n,q)$ has characteristic polynomial $\phi$ is at most

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • ...and 34 more