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18 results

  • 5 - Groups in Class J: defining characteristic
  • John N. Bray, Queen Mary University of London, Derek F. Holt, University of Warwick, Colva M. Roney-Dougal, University of St Andrews, Scotland
  • Book: The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
  • Published online: 05 July 2013
  • Print publication: 25 July 2013, pp 267-321
  • View extract

    Summary

    The theory of representations of finite simple groups of Lie type in defining characteristic is somewhat advanced. The representations arise from those of the associated algebraic groups, and so some familiarity with the theory of algebraic groups is necessary in order to understand it. For an introduction to this theory see, for example, the survey article by Humphreys [51]. The enthusiastic reader may wish to consult Jantzen [58] for a more detailed exposition. Humphrey's classic book [50] provide a general exposition of the theory of algebraic groups and their representations, whilst Malle and Testerman's book [91] gives an excellent introduction to the general theory, subgroup structure, and representation theory of the finite and algebraic groups of Lie type, including a fuller discussion of all of the introductory material in this chapter.

    In many respects, the study of the J2-candidates is easier than that of the J1-candidates, simply because there are far fewer of them: we just need to know about the representations in dimensions up to 12, and to be able to determine some of their properties, such as forms preserved and their behaviour under the actions of group and field automorphisms. Fortunately it is possible to extract this information starting from a superficial familiarity with the main results of the theory, principally the Steinberg Tensor Product Theorems. These theorems, together with the tables in [84], suffice to determine the representations.

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
  • The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
  • John N. Bray, Derek F. Holt, Colva M. Roney-Dougal
  • Published online: 05 July 2013
  • Print publication: 25 July 2013
  • View description
    This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.
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