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Overgroups of root groups in classical groups


About this Title

Michael Aschbacher

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 241, Number 1140
ISBNs: 978-1-4704-1845-8 (print); 978-1-4704-2873-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1140
Published electronically: December 10, 2015
Keywords:Finite groups, linear groups

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. 3-transpositions
  • Chapter 2. The $(V,f)$-setup
  • Chapter 3. Direct sum decompositions
  • Chapter 4. Subfield structures
  • Chapter 5. Modules for alternating groups
  • Chapter 6. Modules with $p=2$
  • Chapter 7. The orthogonal space $\mathbf F_2^n$
  • Chapter 8. Overgroups of long root subgroups
  • Chapter 9. Maximal overgroups of long root subgroups
  • Chapter 10. Subgroups containing long root elements
  • Chapter 11. Overgroups of short root subgroups
  • Chapter 12. Short root subgroups in symplectic groups of characteristic 2
  • Chapter 13. Overgroups of subgroups in $\mathbf R_c$ in III
  • Chapter 14. Overgroups of subgroups in $\mathbf R_c$ in III when $q>3$
  • Chapter 15. A special case for $q=3$ in III
  • Chapter 16. Overgroups of subgroups in $\mathbf R_c$ in III when $q=3$
  • Chapter 17. A result of Stellmacher
  • Chapter 18. More case III with $q=3$
  • Chapter 19. The proof of Theorem 1
  • Chapter 20. A characterization of alternating groups
  • Chapter 21. Orthogonal groups with $q=2$
  • Chapter 22. The proof of Theorem 2
  • Chapter 23. Symplectic and unitary groups
  • Chapter 24. Symplectic and unitary groups with $q$ odd
  • Chapter 25. The proof of Theorem 3
  • Chapter 26. Unitary groups with $q$ even
  • Chapter 27. The proofs of Theorems A and B
  • References

Abstract


We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.

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