The Classification of Extremely Primitive Groups

Timothy C. Burness, Adam R. Thomas
{"title":"The Classification of Extremely Primitive Groups","authors":"Timothy C. Burness, Adam R. Thomas","doi":"10.1093/IMRN/RNAA369","DOIUrl":null,"url":null,"abstract":"Let $G$ be a finite primitive permutation group on a set $\\Omega$ with nontrivial point stabilizer $G_{\\alpha}$. We say that $G$ is extremely primitive if $G_{\\alpha}$ acts primitively on each of its orbits in $\\Omega \\setminus \\{\\alpha\\}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Group Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1093/IMRN/RNAA369","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 12

Abstract

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
极原始群的分类
设$G$是集$\Omega$上具有非平凡点稳定子$G_{\alpha}$的有限原始置换群。我们说$G$非常原始如果$G_{\alpha}$对$\Omega \setminus \{\alpha\}$的每个轨道都有原始的作用。这些群体在几种不同的背景下自然出现,他们的研究可以追溯到20世纪20年代曼宁的工作。在本文中,我们确定了一类几乎简单的极原始群,并给出了一类李型的例外群。通过将这一结果与Burness, Praeger和Seress的早期工作相结合,完成了几乎简单的极原始群的分类。此外,根据Mann, Praeger和Seress的结果,我们的主要定理给出了所有有限极原始群的完全分类,直到有限多个仿射例外(并且推测没有例外)。在此过程中,我们还建立了一些新的结果,这些结果是关于异常群的原始动作的基大小,这可能是独立的兴趣。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
Galois descent of equivalences between blocks of 𝑝-nilpotent groups Onto extensions of free groups. Finite totally k-closed groups Shrinking braids and left distributive monoid Calculating Subgroups with GAP
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1