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The classification of almost simple extremely primitive groups
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Burness, Timothy C. and Thomas, Adam (2022) The classification of almost simple extremely primitive groups. International Mathematics Research Notices, 2022 (13). pp. 10148-10248. doi:10.1093/imrn/rnaa369 ISSN 1073-7928.
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Official URL: https://doi.org/10.1093/imrn/rnaa369
Abstract
Let G be a finite primitive permutation group on a set Ω with nontrivial point stabilizer Gα. We say that G is extremely primitive if Gα acts primitively on each of its orbits in Ω∖{α}. 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.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Finite simple groups, Representations of groups, Permutation groups | ||||||||
Journal or Publication Title: | International Mathematics Research Notices | ||||||||
Publisher: | Oxford University Press | ||||||||
ISSN: | 1073-7928 | ||||||||
Official Date: | July 2022 | ||||||||
Dates: |
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Volume: | 2022 | ||||||||
Number: | 13 | ||||||||
Page Range: | pp. 10148-10248 | ||||||||
DOI: | 10.1093/imrn/rnaa369 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Reuse Statement (publisher, data, author rights): | This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Timothy C Burness, Adam R Thomas, The Classification of Extremely Primitive Groups, International Mathematics Research Notices, 2021;, rnaa369 is available online at: https://doi.org/10.1093/imrn/rnaa369 | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 26 November 2020 | ||||||||
Date of first compliant Open Access: | 18 February 2022 | ||||||||
RIOXX Funder/Project Grant: |
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