A probabilistic approach toward the finite general linear and unitary groups

TL;DR

This paper introduces probabilistic algorithms to prove theorems about finite general linear and unitary groups, offering an alternative to traditional algebraic techniques like character theory.

Contribution

It presents a novel probabilistic approach to establish key theorems in the theory of finite groups, replacing classical algebraic proofs.

Findings

Probabilistic algorithms successfully prove theorems about unipotent elements.

The approach provides new proofs for fixed space properties of random matrices.

It offers a new perspective on counting nilpotent matrices of a given rank.

Abstract

Probabilistic algorithms are applied to prove theorems about the finite general linear and unitary groups which are typically proved by techniques such as character theory and Moebius inversion. Among the theorems studied are Steinberg's count of unipotent elements, Rudvalis and Shindoda's work on the fixed space of a random matrix, and Lusztig's work on counting nilpotent matrices of a given rank.