The maximal subgroups of the exceptional groups F4(q), E6(q) and 2E6(q) and related almost simple groups

• 成果类型: Article
• 署名作者: Craven, David A. A.
• 署名单位: University of Birmingham
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01208-2
• 发表日期: 2023
• 页码: 637-719
• 关键词: lie type; 27-dimensional module; finite subgroups; EMBEDDINGS; rank; j3

摘要

This article produces a complete list of all maximal subgroups of the finite simple groups of type F-4, E-6 and twisted E-6 over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maxi-mal subgroup of F-2(4)(8), correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilin-ear form on the 27-dimensional minimal module for E-6. We provide a collection of supplementary Magma files that prove the author's computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text.