Base sizes of primitive permutation groups

Abstract

AbstractLet G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 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 $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.