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The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

Book 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.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Foreword by Martin Liebeck
  6. Preface
  7. 1. Introduction
    1. 1.1 Background
    2. 1.2 Notation
    3. 1.3 Some basic group theory
    4. 1.4 Finite fields and perfect fields
    5. 1.5 Classical forms
    6. 1.6 The classical groups and their orders
    7. 1.7 Outer automorphisms of classical groups
    8. 1.8 Representation theory
    9. 1.9 Tensor products
    10. 1.10 Small dimensions and exceptional isomorphisms
    11. 1.11 Representations of simple groups
    12. 1.12 The natural representations of the classical groups
    13. 1.13 Some results from number theory
  8. 2. The main theorem and the types of geometric subgroups
    1. 2.1 The main theorem
    2. 2.2 Introducing the geometric types
    3. 2.3 Preliminary arguments concerning maximality
  9. 3. Geometric maximal subgroups
    1. 3.1 Dimension 2
    2. 3.2 Dimension 3
    3. 3.3 Dimension 4
    4. 3.4 Dimension 5
    5. 3.5 Dimension 6
    6. 3.6 Dimension 7
    7. 3.7 Dimension 8
    8. 3.8 Dimension 9
    9. 3.9 Dimension 10
    10. 3.10 Dimension 11
    11. 3.11 Dimension 12
  10. 4. Groups in Class S: cross characteristic
    1. 4.1 Preamble
    2. 4.2 Irrationalities
    3. 4.3 Cross characteristic candidates
    4. 4.4 The type of the form and the stabilisers in Ω and C
    5. 4.5 Dimension up to 6: quasisimple and conformal groups
    6. 4.6 Determining the effects of duality and field automorphisms
    7. 4.7 Dimension up to 6: graph and field automorphisms
    8. 4.8 Dimension up to 6: containments
    9. 4.9 Dimensions greater than 6
    10. 4.10 Summary of the S1-maximal subgroups
  11. 5. Groups in Class S: defining characteristic
    1. 5.1 General theory of S2-subgroups
    2. 5.2 Symmetric and anti-symmetric powers
    3. 5.3 The groups SL2(q) = Sp2(q)
    4. 5.4 The groups SLn(q) and SUn(q) for n ≥ 3
    5. 5.5 The groups Spn(q)
    6. 5.6 The groups Onε(q), 3D4(q), and their covers
    7. 5.7 The remaining groups and their covers
    8. 5.8 Summary of S2*-candidates
    9. 5.9 Determining the effects of duality and field automorphisms
    10. 5.10 Containments
    11. 5.11 Summary of the S2*-maximals
  12. 6. Containments involving S-subgroups
    1. 6.1 Introduction
    2. 6.2 Containments between S1- and S2*-maximal subgroups
    3. 6.3 Containments between geometric and S*-maximal subgroups
  13. 7. Maximal subgroups of exceptional groups
    1. 7.1 Introduction
    2. 7.2 The maximal subgroups of Sp4(2e) and extensions
    3. 7.3 The maximal subgroups of Sz(q) and extensions
    4. 7.4 The maximal subgroups of G2(2e) and extensions
  14. 8. Tables
    1. 8.1 Description of the tables
    2. 8.2 The tables
  15. References
  16. Index