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Maximal $\mathrm {PSL}_2$ Subgroups of Exceptional Groups of Lie Type
About this Title
David A. Craven
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 276, Number 1355
ISBNs: 978-1-4704-5119-6 (print); 978-1-4704-7021-0 (online)
DOI: https://doi.org/10.1090/memo/1355
Published electronically: February 25, 2022
Keywords: Maximal subgroups,
exceptional groups,
finite simple groups
Table of Contents
Chapters
- 1. Introduction
- 2. Notation and Preliminaries
- 3. Maximal Subgroups
- 4. Maximal Subgroups and Subspace Stabilizers
- 5. Blueprint Theorems for Semisimple Elements
- 6. Unipotent and Semisimple Elements
- 7. Modules for $\mathrm {SL}_2$
- 8. Some $\mathrm {PSL}_2$s inside $E_6$ in Characteristic 3
- 9. Proof of the Theorems: Strategy
- 10. The Proof for $F_4$
- 11. The Proof for $E_6$
- 12. The Proof for $E_7$ in Characteristic 2
- 13. The Proof for $E_7$ in Odd Characteristic: $\mathrm {PSL}_2$ Embedding
- 14. The Proof for $E_7$ in Odd Characteristic: $\mathrm {SL}_2$ Embedding
- A. Actions of Maximal Positive-Dimensional Subgroups on Minimal and Adjoint Modules
- B. Traces of Small-Order Semisimple Elements
Abstract
We study embeddings of $\mathrm {PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm {PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group.
Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves that all maximal subgroups with socle $\mathrm {PSL}_2(p^a)$ inside these finite almost simple groups are known, with three possible exceptions ($p^a=7,8,25$ for $E_7$).
In the three remaining cases we provide considerable information about a potential maximal subgroup.
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