On minimal coverings and pairwise generation of some primitive groups of wreath product type

TL;DR

This paper determines the minimal covering number for certain primitive groups with a specific normal subgroup structure and explores asymptotic properties of pairwise generation, extending previous results on symmetric groups.

Contribution

It generalizes the calculation of covering numbers to a broader class of primitive groups and investigates pairwise generation asymptotics.

Findings

Calculated covering number for specific primitive groups

Extended results to groups with normal subgroup isomorphic to A_n^m

Provided asymptotic analysis of pairwise generation

Abstract

The covering number of a finite group $G$, denoted $\sigma(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $\sigma(G)$ for a family of primitive groups $G$ with a unique minimal normal subgroup $N$, isomorphic to $A_n^m$ with $n$ divisible by $6$ and $G/N$ cyclic. This is a generalization of a result of E. Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.