{"title":"Base sizes of primitive groups of diagonal type","authors":"Hong Yi Huang","doi":"10.1017/fms.2023.121","DOIUrl":null,"url":null,"abstract":"<p>Let <span>G</span> be a permutation group on a finite set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104065504114-0233:S2050509423001214:S2050509423001214_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\Omega $</span></span></img></span></span>. The base size of <span>G</span> is the minimal size of a subset of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104065504114-0233:S2050509423001214:S2050509423001214_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\Omega $</span></span></img></span></span> with trivial pointwise stabiliser in <span>G</span>. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"8 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum of Mathematics Sigma","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fms.2023.121","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a permutation group on a finite set $\Omega $. The base size of G is the minimal size of a subset of $\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.
让 G 是有限集合 $\Omega $ 上的一个置换群。 G 的基大小是在 G 中具有琐碎点稳定器的 $\Omega $ 子集的最小大小。在本文中，我们扩展了 Fawcett 早期的工作，确定了每个对角型有限基元置换群的精确基大小。特别是，这是奥南-斯科特定理中出现的第一个在所有情况下都计算出精确基大小的原始群族。我们的方法还允许我们确定所有对角类型的原始群，它们都有一个唯一的正则子轨道。
期刊介绍：
Forum of Mathematics, Sigma is the open access alternative to the leading specialist mathematics journals. Editorial decisions are made by dedicated clusters of editors concentrated in the following areas: foundations of mathematics, discrete mathematics, algebra, number theory, algebraic and complex geometry, differential geometry and geometric analysis, topology, analysis, probability, differential equations, computational mathematics, applied analysis, mathematical physics, and theoretical computer science. This classification exists to aid the peer review process. Contributions which do not neatly fit within these categories are still welcome.
Forum of Mathematics, Pi and Forum of Mathematics, Sigma are an exciting new development in journal publishing. Together they offer fully open access publication combined with peer-review standards set by an international editorial board of the highest calibre, and all backed by Cambridge University Press and our commitment to quality. Strong research papers from all parts of pure mathematics and related areas will be welcomed. All published papers will be free online to readers in perpetuity.
$\Omega $. The base size of G is the minimal size of a subset of 
$\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
