04652nam 22005653 450 991095755830332120231110214833.09781470470210(electronic bk.)9781470451196(MiAaPQ)EBC6939728(Au-PeEL)EBL6939728(CKB)21420569300041(OCoLC)1306205295(EXLCZ)992142056930004120220327d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierMaximal Textrm {PSL}_2 Subgroups of Exceptional Groups of Lie Type1st ed.Providence :American Mathematical Society,2022.©2022.1 online resource (168 pages)Memoirs of the American Mathematical Society ;v.276Print version: Craven, David A. Maximal Providence : American Mathematical Society,c2022 9781470451196 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."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.Memoirs of the American Mathematical Society Lie groupsMaximal subgroupsExceptional Lie algebrasGroup theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie typemscGroup theory and generalizations -- Linear algebraic groups and related topics -- Exceptional groupsmscLie groups.Maximal subgroups.Exceptional Lie algebras.Group theory and generalizations -- Abstract finite groups -- Simple groups: alternating groups and groups of Lie type.Group theory and generalizations -- Linear algebraic groups and related topics -- Exceptional groups.512/.482512.48220D0620G41mscCraven David A511976MiAaPQMiAaPQMiAaPQ9910957558303321Maximal Textrm {PSL}_2 Subgroups of Exceptional Groups of Lie Type4346576UNINA