On the regularity number of a finite group and other base-related invariants

A 𝑘-tuple (𝐻 1 , … , 𝐻𝑘 ) of core-free subgroups of a finitegroup 𝐺 is said to be regular if 𝐺 has a regular orbiton the Cartesian product 𝐺∕𝐻1 × ⋯ × 𝐺∕𝐻𝑘. The reg-ularity number of 𝐺, denoted by 𝑅(𝐺), is the smallestpositive integer 𝑘 with the property that every such 𝑘-tuple is regular. In this paper, we develop some generalmethods for studying the regularity of subgroup tuplesin arbitrary finite groups, and we determine the pre-cise regularity number of all almost simple groups withan alternating or sporadic socle. For example, we provethat 𝑅(𝑆 𝑛 ) = 𝑛 − 1 and 𝑅(𝐴 𝑛 ) = 𝑛 − 2. We also formu-late and investigate natural generalisations of severalwell-studied problems on base sizes for finite permu-tation groups, including conjectures due to Cameron,Pyber and Vdovin. For instance, we extend earlier workof Burness, O’Brien and Wilson by proving that 𝑅(𝐺) ⩽ 7for every almost simple sporadic group, with equality ifand only if 𝐺 is the Mathieu group M 24. We also showthat every triple of soluble subgroups in an almost sim-ple sporadic group is regular, which generalises recentwork of Burness on base sizes for transitive actions ofsporadic groups with soluble point stabilisers.