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On Non-Generic Finite Subgroups of Exceptional Algebraic Groups
About this Title
Alastair J. Litterick, University of Auckland, Auckland, New Zealand
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
MSC: Primary 20G15, 20E07
Table of Contents
Chapters
- 1. Introduction and Results
- 2. Background
- 3. Calculating and Utilising Feasible Characters
- 4. Normaliser Stability
- 5. Complete Reducibility
- 6. Tables of Feasible Characters
- A. Auxiliary Data
Abstract
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, 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 $\operatorname {Alt}_{n}$, $\operatorname {Sym}_{n}$ for $n \ge 10$, do not occur as Lie primitive subgroups of an exceptional algebraic group.
A subgroup of $G$ is called $G$-completely reducible if, whenever it lies in a parabolic subgroup of $G$, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-$G$-completely reducible, non-generic simple subgroups.
As an intermediate result, for each simply connected $G$ of exceptional type, and each non-generic finite simple group $H$ which embeds into $G/Z(G)$, we derive a set of feasible characters, which restrict the possible composition factors of $V \downarrow S$, whenever $S$ is a subgroup of $G$ with image $H$ in $G/Z(G)$, and $V$ is either the Lie algebra of $G$ or a non-trivial Weyl module for $G$ of least dimension.
This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for $n \ge 10$, $\operatorname {Alt}_n$ and $\operatorname {Sym}_n$, 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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