Some years ago I wrote a paper called On a conjecture of Degos appearing as Cah. Topol. Géom. Différ. Catég. 57, No. 3, 229-237 (2016). Some years later, Joel Brewster Lewis got in touch with me to say that the main theorem of that paper was not true when $n=2$. The correct statement should be:

Theorem: Let $f,g\in \mathbb{F}_q[x]$ be distinct monic polynomials of degree $n>2$ such that $f$ is primitive and the constant term of $g$ is non-zero. Then $\langle C_f, C_g\rangle=\textrm{GL}_n(q)$.

(Here $C_f$ means the companion matrix of f.) My error was in omitting that “$>2$”.

I should have published an erratum but I did not get to it and, in the meantime, Joel has written up a paper on a connected subject, where he explains the error completely. Joel’s paper is called $\textrm{GL}_n(\mathbb{F}_q)$-analogues of some properties of $n$-cycles in $S_n$.

I am very grateful to Joel for, first, taking the time to find and correct my error; and, second, for then writing up a proper account of it – a task that I should have done myself. At the very least, then, to show my gratitude I should advertise the main result of his paper! It is a very striking result which, as the title of his paper indicates, gives two properties of the general linear group, $\textrm{GL}_n(\mathbb{F}_q),$ which are direct analogues of properties of the symmetric group, $S_n$. Here is his result:

Theorem

1. An element $g\in \textrm{GL}_n(\mathbb{F}_q)$ is a Singer cycle if and only if the factors in every minimum-length factorization of $g$ as a product of reflections form a generating set for $\textrm{GL}_n(\mathbb{F}_q)$.
2. Suppose that $c\in \textrm{GL}_n(\mathbb{F}_q)$ is a Singer cycle and $t\in \textrm{GL}_n(\mathbb{F}_q)$ is a reflection that does not normalize the cyclic subgroup $\langle c \rangle$; then $\langle c, t\rangle=\textrm{GL}_n(\mathbb{F}_q)$.