Normal coverings of linear groups

Abstract

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 $\SL_{n}(q)\le G\le \GL_{n}(q)$ for $n>2$, then $n/{\pi }^2<\gamma (G)\le (n+1)/2$. We give various alternative bounds, and derive explicit formulas for $\gamma (G)$ in some cases.

18 pages

References in corpus (1)

• 2-Coverings of classical groups

Cited by in corpus (1)

• Normal coverings and pairwise generation of finite alternating and symmetric groups