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On Non-Generic Finite Subgroups of Exceptional Algebraic Groups


About this Title

Alastair J. Litterick

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 253, Number 1207
ISBNs: 978-1-4704-2837-2 (print); 978-1-4704-4409-9 (online)
DOI: https://doi.org/10.1090/memo/1207
Published electronically: March 29, 2018
Keywords:Algebraic groups, exceptional groups, finite simple groups, Lie primitive, subgroup structure, complete reducibility

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Table of Contents


Chapters

  • Chapter 1. Introduction and Results
  • Chapter 2. Background
  • Chapter 3. Calculating and Utilising Feasible Characters
  • Chapter 4. Normaliser Stability
  • Chapter 5. Complete Reducibility
  • Chapter 6. Tables of Feasible Characters
  • Appendix A. Auxiliary Data

Abstract


The study of finite subgroups of a simple algebraic group reduces in a sense to those which are almost simple. If an almost simple subgroup of has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of , then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic.A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups , for , do not occur as Lie primitive subgroups of an exceptional algebraic group.A subgroup of is called -completely reducible if, whenever it lies in a parabolic subgroup of , it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non--completely reducible, non-generic simple subgroups.As an intermediate result, for each simply connected of exceptional type, and each non-generic finite simple group which embeds into , we derive a set of feasible characters, which restrict the possible composition factors of , whenever is a subgroup of with image in , and is either the Lie algebra of or a non-trivial Weyl module for of least dimension.This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for , and , as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.

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