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

Abstract A ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.