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On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type
About this Title
David A. Craven
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 288, Number 1434
ISBNs: 978-1-4704-6702-9 (print); 978-1-4704-7576-5 (online)
DOI: https://doi.org/10.1090/memo/1434
Published electronically: August 2, 2023
Keywords: Maximal subgroups,
exceptional groups,
finite simple groups
Table of Contents
Chapters
- 1. Introduction
- 2. Notation and preliminaries
- 3. Subgroup structure of exceptional algebraic groups
- 4. Techniques for proving the results
- 5. Modules for groups of Lie type
- 6. Rank 4 groups for $E_8$
- 7. Rank 3 groups for $E_8$
- 8. Rank 2 groups for $E_8$
- 9. Subgroups of $E_7$
- 10. Subgroups of $E_6$
- 11. Subgroups of $F_4$
- 12. Difficult cases
- 13. The trilinear form for $E_6$
Abstract
We study embeddings of groups of Lie type $H$ in characteristic $p$ into exceptional algebraic groups $\mathbf {G}$ of the same characteristic. We exclude the case where $H$ is of type $\mathrm {PSL}_2$. A subgroup of $\mathbf {G}$ is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of $\mathbf {G}$.
With a few possible exceptions, we prove that there are no Lie primitive subgroups $H$ in $\mathbf {G}$, with the conditions on $H$ and $\mathbf {G}$ given above. The exceptions are for $H$ one of $\mathrm {PSL}_3(3)$, $\mathrm {PSU}_3(3)$, $\mathrm {PSL}_3(4)$, $\mathrm {PSU}_3(4)$, $\mathrm {PSU}_3(8)$, $\mathrm {PSU}_4(2)$, $\mathrm {PSp}_4(2)β$ and ${}^2\!B_2(8)$, and $\mathbf {G}$ of type $E_8$. No examples are known of such Lie primitive embeddings.
We prove a slightly stronger result, including stability under automorphisms of $\mathbf {G}$. This has the consequence that, with the same exceptions, any almost simple group with socle $H$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$, $E_7$ and $E_8$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group.
The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.
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