Normal coverings of linear groups

Abstract

For a noncyclic finite group $G$, let $\gamma \left(G\right)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. In this paper, we show that if $G$ is in the range ${SL}_n\left(q\right)\le G\le {GL}_n\left(q\right)$ for $n>2$, then $n∕{\pi }^2<\gamma \left(G\right)\le \left(n+1\right)∕2$. This result complements recent work of Bubboloni, Praeger and Spiga on symmetric and alternating groups. We give various alternative bounds and derive explicit formulas for $\gamma \left(G\right)$ in some cases.