The base size of the symmetric group acting on subsets

Coen del Valle; Colva Roney-Dougal

doi:10.5802/alco.370

The base size of the symmetric group acting on subsets

Coen del Valle^*, Colva Roney-Dougal

^*Corresponding author for this work

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

Abstract

A base for a permutation group G acting on a set Ω is a subset B of Ω such that the pointwise stabiliser G(_B) is trivial. Let n and r be positive integers with n > 2r. The symmetric and alternating groups S_n and A_n admit natural primitive actions on the set of r-element subsets of {1,2,..., n}. Building on work of Halasi [8], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of S_n and A_n.

Original language	English
Pages (from-to)	959-967
Number of pages	8
Journal	Algebraic Combinatorics
Volume	7
Issue number	4
DOIs	https://doi.org/10.5802/alco.370
Publication status	Published - 3 Sept 2024

Keywords

Symmetric group
Base size
Hypergraph

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10.5802/alco.370Licence: CC BY

del_Val_2024_The_Base_size_ALCO_2024__7_4_959_CCBY
Copyright © The author(s), 2024. This article is licensed under the Creative Commons Attribution (CC-BY) 4.0 license. http://creativecommons.org/licenses/by/4.0/.
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title = "The base size of the symmetric group acting on subsets",

abstract = "A base for a permutation group G acting on a set Ω is a subset B of Ω such that the pointwise stabiliser G(B) is trivial. Let n and r be positive integers with n > 2r. The symmetric and alternating groups Sn and An admit natural primitive actions on the set of r-element subsets of \{1,2,..., n\}. Building on work of Halasi [8], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of Sn and An.",

keywords = "Symmetric group, Base size, Hypergraph",

author = "\{del Valle\}, Coen and Colva Roney-Dougal",

note = "Funding: Research of Coen del Valle is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number PGSD-577816-2023], as well as a University of St Andrews School of Mathematics and Statistics Scholarship.",

year = "2024",

month = sep,

day = "3",

doi = "10.5802/alco.370",

language = "English",

volume = "7",

pages = "959--967",

journal = "Algebraic Combinatorics",

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T1 - The base size of the symmetric group acting on subsets

AU - del Valle, Coen

AU - Roney-Dougal, Colva

N1 - Funding: Research of Coen del Valle is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number PGSD-577816-2023], as well as a University of St Andrews School of Mathematics and Statistics Scholarship.

PY - 2024/9/3

Y1 - 2024/9/3

N2 - A base for a permutation group G acting on a set Ω is a subset B of Ω such that the pointwise stabiliser G(B) is trivial. Let n and r be positive integers with n > 2r. The symmetric and alternating groups Sn and An admit natural primitive actions on the set of r-element subsets of {1,2,..., n}. Building on work of Halasi [8], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of Sn and An.

AB - A base for a permutation group G acting on a set Ω is a subset B of Ω such that the pointwise stabiliser G(B) is trivial. Let n and r be positive integers with n > 2r. The symmetric and alternating groups Sn and An admit natural primitive actions on the set of r-element subsets of {1,2,..., n}. Building on work of Halasi [8], we provide explicit expressions for the base sizes of all of these actions, and hence determine the base size of all primitive actions of Sn and An.

KW - Symmetric group

KW - Base size

KW - Hypergraph

UR - https://www.scopus.com/pages/publications/85203284896

U2 - 10.5802/alco.370

DO - 10.5802/alco.370

M3 - Article

SN - 2589-5486

VL - 7

SP - 959

EP - 967

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

IS - 4

ER -

The base size of the symmetric group acting on subsets

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The base size of the symmetric group acting on subsets

Abstract

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All about that base

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