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

Abstract

The covering number of a finite group [Formula: see text], denoted [Formula: see text], is the smallest positive integer [Formula: see text] such that [Formula: see text] is a union of [Formula: see text] proper subgroups. We calculate [Formula: see text] for a family of primitive groups [Formula: see text] with a unique minimal normal subgroup [Formula: see text], isomorphic to [Formula: see text] with [Formula: see text] divisible by [Formula: see text] and [Formula: see text] cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.