Base sizes of primitive permutation groups

Mariapia Moscatiello*, Colva M. Roney-Dougal

*Corresponding author for this work

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Abstract

Let G be a permutation group, acting on a set Ω of size n. A subset B of Ω 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 ≥ 1 such that Alt (m)r ... G ≤ 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) ≤ ⌈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, so our bound is optimal.
Original languageEnglish
Number of pages33
JournalMonatshefte für Mathematik
VolumeFirst Online
Early online date5 Aug 2021
DOIs
Publication statusE-pub ahead of print - 5 Aug 2021

Keywords

  • Primitive groups
  • Base size
  • Classical groups
  • Simple groups

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