On a conjecture of Degos

TL;DR

This paper proves a conjecture by Degos using Kantor's result, showing that certain polynomial-generated groups over finite fields are equal to the general linear group, under specific conditions.

Contribution

It establishes a new group-theoretic result linking primitive polynomials and polynomial conditions over finite fields, confirming Degos's conjecture.

Findings

<C_f, C_g> equals GL_n(q) under given conditions

Uses Kantor's result to prove the conjecture

Provides a new connection between polynomials and group generation

Abstract

In this note we use a result of Kantor to prove a conjecture of Degos. Specifically we prove the following: let $\mathbb{F}$ be a finite field of order $q$ and let $f, g\in\mathbb{F}[X]$ be distinct polynomials of degree $n$ such that $f$ is primitive, and the constant term of $g$ is non-zero. Then $<C_f, C_g>=\mathrm{GL}_n(q)$.