Normal coverings and pairwise generation of finite alternating and symmetric groups

Abstract

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$ such that, for $G$ a symmetric group $\Sym(n)$ or an alternating group $\Alt(n)$, $\gamma(G)\geq cn$. This improves results of the first two authors who had earlier proved that $a\varphi(n)\leq\gamma(G)\leq 2n/3,$ for some positive constant $a$, where $\varphi$ is the Euler totient function. Bounds are also obtained for the maximum size $\kappa(G)$ of a set $X$ of conjugacy classes of $G=\Sym(n)$ or $\Alt(n)$ such that any pair of elements from distinct classes in $X$ generates $G$, namely $cn\leq \kappa(G)\leq 2n/3$.