On the Brunnian conjecture

Let $p$ be a primer number, $n\ge 3$ and integer. Let $f(X)=X^n+a_{n-1}{X}^{n-1}+\cdots +a_{1}X+a_0\in {\mathbb{F}}_p[X]$ be a primitive polynomial of degree $n$. Let $C_f$ be the companion matrix of $f(X)$, and $G$ the companion matrix of the polynomial $X^n-1$. Define $G_1:=C_f$ and $G_{k+1}=G{G}_k{G}^{-1}$ for $0\le k\le n-1$. The so called ``Brunnian Conjecture’’ states that: the general linear group $GL(n,p)$ is generated by $G_1,G_2,\dots ,G_n$. In this paper, we prove it for $p\ge 5$ and $n$ not divisible by $p-1$.