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Imprimitive irreducible modules for finite quasisimple groups


About this Title

Gerhard Hiss, William J. Husen and Kay Magaard

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1104
ISBNs: 978-1-4704-0960-9 (print); 978-1-4704-2031-4 (online)
DOI: http://dx.doi.org/10.1090/memo/1104
Published electronically: August 1, 2014
Keywords:Finite quasisimple group, maximal subgroup, finite classical group, $\mathcal {C}_2$-subgroup

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Table of Contents


Chapters

  • Acknowledgements
  • Chapter 1. Introduction
  • Chapter 2. Generalities
  • Chapter 3. Sporadic Groups and the Tits Group
  • Chapter 4. Alternating Groups
  • Chapter 5. Exceptional Schur Multipliers and Exceptional Isomorphisms
  • Chapter 6. Groups of Lie type: Induction from non-parabolic subgroups
  • Chapter 7. Groups of Lie type: Induction from parabolic subgroups
  • Chapter 8. Groups of Lie type: char$(K) = 0$
  • Chapter 9. Classical groups: $\text {char}(K) = 0$
  • Chapter 10. Exceptional groups

Abstract


Motivated by the maximal subgroup problem of the finite classical groups we begin the classification of imprimitive irreducible modules of finite quasisimple groups over algebraically closed fields . A module of a group over is imprimitive, if it is induced from a module of a proper subgroup of . We obtain our strongest results when char, although much of our analysis carries over into positive characteristic. If is a finite quasisimple group of Lie type, we prove that an imprimitive irreducible -module is Harish-Chandra induced. This being true for char different from the defining characteristic of , we specialize to the case char and apply Harish-Chandra philosophy to classify irreducible Harish-Chandra induced modules in terms of Harish-Chandra series, as well as in terms of Lusztig series. We determine the asymptotic proportion of the irreducible imprimitive -modules, when runs through a series groups of fixed (twisted) Lie type. One of the surprising outcomes of our investigations is the fact that these proportions tend to , if the Lie rank of the groups tends to infinity. For exceptional groups of Lie type of small rank, and for sporadic groups , we determine all irreducible imprimitive -modules for arbitrary characteristic of .

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