The base size of a primitive diagonal group

Abstract

A base B for a finite permutation group G acting on a set Ω is a subset of Ω with the property that only the identity of G can fix every point of B. We prove that a primitive diagonal group G has a base of size 2 unless the top group of G is the alternating or symmetric group acting naturally, in which case the minimal base size of G is determined up to two possible values. We also prove that the minimal base size of G satisfies a well-known conjecture of Pyber. Moreover, we prove that if the top group of G does not contain the alternating group, then the proportion of pairs of points that are bases for G tends to 1 as |G| tends to infinity. A similar result for the case when the degree of the top group is fixed is given.