Normal coverings of linear groups

For a non-cyclic finite group $G$, let $\gamma (G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently established that $\gamma (S_n)$ and $\gamma (A_n)$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range ${\text{SL}}_n(q)\le G\le {\text{GL}}_n(q)$ for $n>2$, then $n/\pi^2 < \gamma(G) \le (n+1)/2$\mathrm{.}Wegivevariousalternativebounds,andderiveexplicitformulasfor$\gamma(G)$ in some cases.