The Automorphism Group of an Octonion Algebra

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


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.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations