The order of a typical matrix with entries in a finite field (Q1903122)

An estimation for the order or minimum period \(T_n (A)\) of a (typical) matrix \(A \in {\mathcal G}_n\) is given, where \({\mathcal G}_n\) is the group of invertible matrices over a finite field \(\mathbb{F}_q\). It is shown that \(T_n (A) = q^{n - (\log n)^{2 + o(1)}}\) for almost every \(A\). This result is a consequence of a similar theorem which is proved for the order of a monic polynomial and it is obtained by using an approximation theorem of \textit{J. C. Hansen} and the author [Math. Proc. Camb. Philos. Soc. 114, No. 3, 507-515 (1993; Zbl 0793.15013)].