The base size of a primitive diagonal group

TL;DR

This paper investigates the minimal base size of primitive diagonal groups, establishing bounds and asymptotic properties, and confirms a conjecture of Pyber for these groups.

Contribution

It provides a tight bound for the base size of primitive diagonal groups and verifies Pyber's conjecture in this context.

Findings

Primitive diagonal groups have a base size of 2 unless the top group is symmetric or alternating.

The proportion of point pairs forming bases tends to 1 as group size increases, if the top group lacks the alternating group.

A similar asymptotic result holds when the top group's degree is fixed.

Abstract

A base B for a finite permutation group G acting on a set X is a subset of X 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 a tight bound for the minimal base size of G is given. This bound also 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.