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ClassicalMaximals

Maximal subgroups of classical groups

1.0

17 May 2026

Maximilian Hauck
Email: mahauck@rhrk.uni-kl.de

Max Horn
Email: mhorn@rptu.de

Tristan Pfersdorff
Email: tristan.pfersdorff@edu.rptu.de

Christian Seeger
Email: christian.seeger@edu.rptu.de

Sergio Siccha
Email: siccha@mathematik.uni-kl.de

Contents

1.2 Overview over this manual

1.3 Feedback and Support

2 Classical Groups and Aschbacher's Theorem

2.1 Classical Forms

  2.1-1 Sesquilinear forms
  2.1-2 Quadratic forms
  2.1-3 Matrix realizations of classical forms
  2.1-4 Non-degeneracy
  2.1-5 Isometries and similarities
  2.1-6 Classification of sesquilinear forms

2.2 Classical Groups

  2.2-1 Linear Groups (Case \({\bf L}\))
  2.2-2 Symplectic Groups (Case \({\bf S}\))
  2.2-3 Unitary Groups (Case \({\bf U}\))
  2.2-4 Orthogonal groups in odd dimension (Case \({\bf O}\))
  2.2-5 Orthogonal groups in even dimension (Cases \({\bf O}^+\) and \({\bf O}^-\))
  2.2-6 Standard forms in ClassicalMaximals

2.3 Aschbacher's Theorem

3 Maximal Subgroups of Classical Groups

3.1 The Main function

3.1-1 ClassicalMaximalsGeneric

3.2 Conjugating elements in the ambient classical group

  3.2-1 GLMinusSL
  3.2-2 GUMinusSU
  3.2-3 NormSpMinusSp
  3.2-4 SOMinusOmega
  3.2-5 GOMinusSO
  3.2-6 NormGOMinusGO
  3.2-7 Warning concerning orthogonal groups of minus type

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