On a uniformly distributed phenomenon in matrix groups

TL;DR

This paper demonstrates that a known uniform distribution phenomenon for elements and their inverses in modular groups also applies to matrix groups over finite fields, extending classical results to a new algebraic setting.

Contribution

It introduces a novel analogy of uniform distribution phenomena from modular groups to matrix groups over finite fields, expanding understanding of distribution properties in algebraic structures.

Findings

Uniform distribution phenomenon extends from modular groups to matrix groups over finite fields.

Analogy of uniform distribution on modular hyperbolas is established in $ extrm{GL}_{n}( ext{finite field})$.

Results suggest broader applicability of distribution phenomena in algebraic groups.

Abstract

We show that a classical uniformly distributed phenomenon for an element and its inverse in ($\mathbb{Z}/n\mathbb{Z})^{*}$ also exists in $\textrm{GL}_{n}(\mathbb{F}_{p})$. A $\textrm{GL}_{n}(\mathbb{F}_{p})$ analogy of the uniform distribution on modular hyperbolas has also been considered.