Irredundant bases for the symmetric group

Colva Roney-Dougal; Peiran Wu

doi:10.1112/blms.13027

Irredundant bases for the symmetric group

Colva Roney-Dougal, Peiran Wu^*

^*Corresponding author for this work

Pure Mathematics

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Abstract

An irredundant base of a group 𝐺 acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser sub-groups in 𝐺,terminating at the trivial subgroup. Suppose that 𝐺 is S_𝑛 or A_𝑛 acting primitively on Γ, and that the point stabiliser is primitive in its natural action on𝑛points. We prove that the maximum size of an irredundant base of 𝐺 is 𝑂 (√𝑛),and in most cases 𝑂((log𝑛)²). We also show that these bounds are best possible.

Original language	English
Number of pages	15
Journal	Bulletin of the London Mathematical Society
Volume	Early View
Early online date	20 Mar 2024
DOIs	https://doi.org/10.1112/blms.13027
Publication status	E-pub ahead of print - 20 Mar 2024

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Roney-Dougal_2024_BLMS_Irredundant-bases_CC
Copyright © 2024 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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abstract = "An irredundant base of a group 𝐺 acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser sub-groups in 𝐺,terminating at the trivial subgroup. Suppose that 𝐺 is S𝑛 or A𝑛 acting primitively on Γ, and that the point stabiliser is primitive in its natural action on𝑛points. We prove that the maximum size of an irredundant base of 𝐺 is 𝑂 (√𝑛),and in most cases 𝑂((log𝑛)2). We also show that these bounds are best possible.",

author = "Colva Roney-Dougal and Peiran Wu",

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T1 - Irredundant bases for the symmetric group

AU - Roney-Dougal, Colva

AU - Wu, Peiran

N1 - Funding: This work was supported by EPSRC grant No EP/R014604/1, and also partially supported by a grant from the Simons Foundation.

PY - 2024/3/20

Y1 - 2024/3/20

N2 - An irredundant base of a group 𝐺 acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser sub-groups in 𝐺,terminating at the trivial subgroup. Suppose that 𝐺 is S𝑛 or A𝑛 acting primitively on Γ, and that the point stabiliser is primitive in its natural action on𝑛points. We prove that the maximum size of an irredundant base of 𝐺 is 𝑂 (√𝑛),and in most cases 𝑂((log𝑛)2). We also show that these bounds are best possible.

AB - An irredundant base of a group 𝐺 acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser sub-groups in 𝐺,terminating at the trivial subgroup. Suppose that 𝐺 is S𝑛 or A𝑛 acting primitively on Γ, and that the point stabiliser is primitive in its natural action on𝑛points. We prove that the maximum size of an irredundant base of 𝐺 is 𝑂 (√𝑛),and in most cases 𝑂((log𝑛)2). We also show that these bounds are best possible.

U2 - 10.1112/blms.13027

DO - 10.1112/blms.13027

M3 - Article

SN - 0024-6093

VL - Early View

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

ER -

Irredundant bases for the symmetric group

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