Base sizes of primitive permutation groups

TL;DR

This paper investigates the minimal base size of primitive permutation groups, establishing bounds for non-large base groups and demonstrating the optimality of these bounds with infinite examples.

Contribution

It proves that primitive groups not classified as large base have a base size at most logarithmic in degree, except for the Mathieu group M24, and shows this bound is tight.

Findings

Primitive groups not large base have base size ≤ log n + 1

Mathieu group M24 is an exception with larger base size

There are infinitely many non-large base primitive groups with base size > log n + 1

Abstract

Let G be a permutation group, acting on a set \Omega of size n. A subset B of \Omega is a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r \geq 1 such that Alt(m)^r \unlhd G \leq Sym(m) \wr Sym(r), where the action of Sym(m) is on k-element subsets of {1,...,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) \leq \lceil \log n\rceil+1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.