Maximal Textrm {PSL}_2 Subgroups of Exceptional Groups of Lie Type

Format:

Book

Author/Creator:

Craven, David A.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

Lie groups.

Maximal subgroups.

Exceptional Lie algebras.

Physical Description:

1 online resource (168 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We study embeddings of PSL2(pa) into exceptional groups G(pb) for G = F4, E6, 2E6, E7, and p a prime with a, b positive integers. With a few possible exceptions, we prove that any almost simple group with socle PSL2(pa), that is maximal inside an almost simple exceptional group of Lie type F44, E6, 2E6 and E7, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A1 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 PSL2(pa) inside these finite almost simple groups are known, with three possible exceptions (pa = 7, 8, 25 for E7). In the three remaining cases we provide considerable information about a potential maximal subgroup"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Chapter 2. Notation and Preliminaries

Chapter 3. Maximal Subgroups

Chapter 4. Maximal Subgroups and Subspace Stabilizers

Chapter 5. Blueprint Theorems for Semisimple Elements

5.1. Preliminary Results

5.2. Determination of the Bounds for the Minimal Module

5.3. Consequences for Maximal Subgroups

Chapter 6. Unipotent and Semisimple Elements

6.1. Actions of Unipotent Elements

6.2. Blueprints and Element Orders

6.3. Blueprints inside Subgroups of Type ₁

6.4. Traces of Modules for \PGL₂

6.5. The Graph Automorphism of ₄

6.6. Rank-1 Subalgebras of the Lie Algebra

Chapter 7. Modules for \SL₂

7.1. Modules for \SL₂(2^{ })

7.2. Modules for \SL₂(3^{ })

7.3. Modules for \SL₂( )

7.4. Modules for \SL₂( ^{ }) for ≥5 and &gt

1

Chapter 8. Some \PSL₂s inside ₆ in Characteristic 3

Chapter 9. Proof of the Theorems: Strategy

Chapter 10. The Proof for ₄

10.1. Characteristic 2

10.2. Characteristic 3

10.3. Characteristic At Least 5

Chapter 11. The Proof for ₆

11.1. Characteristic 2

11.2. Characteristic 3

11.3. Characteristic At Least 5

Chapter 12. The Proof for ₇ in Characteristic 2

Chapter 13. The Proof for ₇ in Odd Characteristic: \PSL₂ Embedding

13.1. Characteristic 3

13.2. Characteristic At Least 5

Chapter 14. The Proof for ₇ in Odd Characteristic: \SL₂ Embedding

14.1. Characteristic 3

14.2. Characteristic At Least 5

Appendix A. Actions of Maximal Positive-Dimensional Subgroups on Minimal and Adjoint Modules

Appendix B. Traces of Small-Order Semisimple Elements

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Craven, David A. Maximal

ISBN:

9781470470210

OCLC:

1306205295