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The Automorphism Group of an Octonion Algebra

  • Tonny A. Springer
  • Ferdinand D. Veldkamp
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this chapter we study the group G = Aut(C) of automorphisms of an octonion algebra C over a field k. By “automorphism” we will in this chapter always understand a linear automorphism. Since automorphisms leave the norm invariant, Aut(C) is a subgroup of the orthogonal group O(N) of the norm of C.

Keywords

Automorphism Group Algebraic Group Invariant Subspace Maximal Torus Orthogonal Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Historical Notes

  1. [Ca 14]
    E. Cartan: Les groupes réels simples, finis et continus. Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355. Oeuvres I, 1, 399–491.Google Scholar
  2. [Fr 51]
    H. Freudenthal: Oktaven, Ausnahmegruppen und Oktavengeometrie. Mimeogr. notes, Math. Inst. Utrecht, 1951, 1960. Reprinted in Geom. Dedicata 19 (1985), 7–63.MATHMathSciNetGoogle Scholar
  3. [Ja 39]
    N. Jacobson: Cayley numbers and normal simple Lie algebras of type G. Duke Math. J. 5 (1939), 775–783.Google Scholar
  4. [Ba]
    E. Bannow: Die Automorphismengruppen der Cayley-Zahlen. Abh. Math. Sem. Univ. Hamburg 13 (1940), 240–256.CrossRefGoogle Scholar
  5. [Di Ola]
    L.E. Dickson: Theory of linear groups in an arbitrary field. Trans. Amer. Math. Soc. 2 (1901), 363–394. Math. papers II, 43–74.Google Scholar
  6. [Di 05]
    Di 05] L.E. Dickson: A new system of simple groups. Math. Ann. 60 (1905), 400417.Google Scholar
  7. [Di 08] L.E. Dickson: A class of groups in an arbitrary realm connected with the configuration of 27 lines on a cubic surface (second paper).
    Quart. J. Pure Appl. Math. 39 (1908), 205–209. Math. papers VI, 145–149.Google Scholar
  8. [Dieu]
    J. Dieudonné: La géométrie des groupes classiques. Ergebnisse der Math. und ihrer Grenzgebiete, Neue Folge, Band 5. Springer, Berlin etc., 1955, Second ed. 1963.Google Scholar
  9. [Che 55]
    C. Chevalley: Sur certains groupes simples. Tôhoku Math. J. (2) 7 (1955), 14–66.CrossRefMATHMathSciNetGoogle Scholar
  10. [Ja 58]
    N. Jacobson: Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo (2) 7 (1958), 55–80.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Tonny A. Springer
    • 1
  • Ferdinand D. Veldkamp
  1. 1.Mathematisch InstituutUtrechtThe Netherlands

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