Normal coverings of linear groups

Abstract

For a non-cyclic finite group $G$, let $γ(G)$ 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 $γ(S_n)$ and $γ(A_{n})$ are bounded above and below by linear functions of $n$. In this paper we show that if $G$ is in the range $\SL_{n}(q)\le G\le \GL_{n}(q)$ for $n>2$, then $n/π^2 < γ(G) \le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $γ(G)$ in some cases.