Base sizes of primitive permutation groups

Mariapia Moscatiello; Colva M. Roney-Dougal

doi:10.1007/s00605-021-01599-5

Base sizes of primitive permutation groups

Mariapia Moscatiello^*, Colva M. Roney-Dougal

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 language	English
Number of pages	33
Journal	Monatshefte für Mathematik
Volume	First Online
Early online date	5 Aug 2021
DOIs	https://doi.org/10.1007/s00605-021-01599-5
Publication status	E-pub ahead of print - 5 Aug 2021

Keywords

Primitive groups
Base size
Classical groups
Simple groups

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Moscatiello_2021_MM_Basesizes_CC
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Final published version, 432 KBLicence: CC BY

CoDiMa: CoDiMa (CCP in the area of Computational Discrete Mathematics)
Linton, S. A. (PI) & Konovalov, O. (CoI)
EPSRC
1/03/15 → 29/02/20
Project: Standard

Cite this

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title = "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.",

keywords = "Primitive groups, Base size, Classical groups, Simple groups",

author = "Mariapia Moscatiello and Roney-Dougal, {Colva M.}",

note = "This work was supported by: EPSRC Grant Numbers EP/R014604/1 and EP/M022641/1.",

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AU - Moscatiello, Mariapia

AU - Roney-Dougal, Colva M.

N1 - This work was supported by: EPSRC Grant Numbers EP/R014604/1 and EP/M022641/1.

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Y1 - 2021/8/5

N2 - 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.

AB - 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.

KW - Primitive groups

KW - Base size

KW - Classical groups

KW - Simple groups

U2 - 10.1007/s00605-021-01599-5

DO - 10.1007/s00605-021-01599-5

M3 - Article

SN - 0026-9255

VL - First Online

JO - Monatshefte für Mathematik

JF - Monatshefte für Mathematik

ER -

Base sizes of primitive permutation groups

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CoDiMa: CoDiMa (CCP in the area of Computational Discrete Mathematics)

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