On a conjecture of Degos

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 $<C_f, C_g>=\mathrm{GL}_n(q)$.