Normal coverings and pairwise generation of finite alternating and symmetric groups

TL;DR

This paper establishes linear lower bounds on the normal covering number and pairwise generation sets of symmetric and alternating groups, advancing understanding of their subgroup structures and generation properties.

Contribution

The authors prove that the normal covering number and the maximum size of pairwise generating conjugacy class sets grow linearly with the degree of the groups, improving previous bounds.

Findings

Normal covering number    grows linearly with n.

Maximum size of conjugacy class sets    is bounded linearly between cn and 2n/3.

Established new bounds improving prior results on group generation and coverings.

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 $cn\leq \kappa(G)\leq 2n/3$.