Derangements in finite classical groups and characteristic polynomials of random matrices

TL;DR

This paper provides bounds on the proportion of elements with specific characteristic polynomials in finite classical groups, showing that elements in proper subgroups become negligible as the dimension grows, and studies random generation probabilities.

Contribution

It establishes explicit bounds for element proportions in classical groups and proves that the likelihood of random elements generating the entire group diminishes with increasing dimension.

Findings

Proportion of elements with a given characteristic polynomial is explicitly bounded.

Probability that three random elements generate the entire group tends to zero as dimension increases.

Elements in proper irreducible subgroups become negligible in large-dimensional classical groups.

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.