Base sizes of primitive permutation groups
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.
Citation
Moscatiello , M & Roney-Dougal , C M 2021 , ' Base sizes of primitive permutation groups ' , Monatshefte für Mathematik , vol. First Online . https://doi.org/10.1007/s00605-021-01599-5
Publication
Monatshefte für Mathematik
Status
Peer reviewed
ISSN
0026-9255Type
Journal article
Description
This work was supported by: EPSRC Grant Numbers EP/R014604/1 and EP/M022641/1.Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.