Abundant p-singular elements in finite classical groups

TL;DR

This paper investigates the proportion of p-abundant elements in finite classical groups, providing explicit bounds and showing that their limiting proportion approaches a positive constant as the group dimension grows.

Contribution

It introduces the concept of p-abundant elements, derives bounds for their proportion, and demonstrates their asymptotic behavior in large-dimensional groups.

Findings

Explicit bounds for p-abundant elements' proportion

Asymptotic approach to a positive limit

Proportion at least a constant multiple of previous lower bound

Abstract

In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of |G| distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of (1 - 1/p)/e, where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain "large" subspaces of the natural G-module. We find explicit upper and lower bounds for the proportion of p-abundant elements in G, and prove that it approaches a (positive) limiting value as the dimension of G tends to infinity. It turns out that the limiting proportion of p-abundant elements is at least a constant multiple of the Isaacs-Kantor-Spaltenstein lower bound for the proportion of all p-singular elements.