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Total closure for permutation actions of finite nonabelian simple groups

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Date
02/03/2023
Author
Freedman, Saul Daniel
Giudici, Michael
Praeger, Cheryl E.
Keywords
k-closed permutation groups
Primitive groups
Base size
Simple groups
T-DAS
MCP
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Abstract
For a positive integer k, a group G is said to be totally k-closed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym(Ω) that leaves invariant each of the G-orbits in the induced action on Ω  ×···×Ω = Ωk. Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. We address the question of determining the closure number k(G) for finite simple groups G. Prior to our work it was known that k(G) = 2 for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that k(G) ≥ 3 for all other finite simple groups. We determine the value for the alternating groups, namely k(An) = n − 1. In addition, for all simple groups G, other than alternating groups and classical groups, we show that k(G) ≤ 7. Finally, if G is a finite simple classical group with natural module of dimension n, we show that k(G) ≤ n + 2 if n ≥ 14, and k(G) ≤ ⌊n/3 + 12⌋ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on n and the type of G) on the base sizes of the primitive actions of G, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.
Citation
Freedman , S D , Giudici , M & Praeger , C E 2023 , ' Total closure for permutation actions of finite nonabelian simple groups ' , Monatshefte für Mathematik . https://doi.org/10.1007/s00605-023-01822-5
Publication
Monatshefte für Mathematik
Status
Peer reviewed
DOI
https://doi.org/10.1007/s00605-023-01822-5
ISSN
0026-9255
Type
Journal article
Rights
Copyright © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Description
Funding: St Leonard’s International Doctoral Fees Scholarship (SF); Australian Research Council (DP190101024, DP190100450).
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/27098

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