Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-10T19:01:27.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Classical groups over division rings of characteristic two

Published online by Cambridge University Press:  17 April 2009

William M. Pender  and
G.E. Wall
William M. Pender
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 7 , Issue 2 , October 1972 , pp. 191 - 226
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Brauer, Richard, “On a theorem of H. Cartan”, Bull. Amer. Math. Soc. 55 (1949), 619–620.CrossRefGoogle Scholar
[2]Dieudonné, Jean, “On the structure of unitary groups”, Trans. Amer. Math. Soc. 72 (1952), 367–385.CrossRefGoogle Scholar
[3]Dieudonné, Jean, “On the structure of unitary groups (II)”, Amer. J. Math. 75 (1953), 665–678.CrossRefGoogle Scholar
[4]Dieudonné, Jean, “Sur les générateurs des groupes classiques”, Summa Brasil. Math. 3 (1955), 149–179.Google Scholar
[5]Jean Dieudonné, La géométrie des groupes classiques, 2e éd. (Springer-Verlag, Berlin, Göttingen, Heidelberg, 1963).Google Scholar
[6]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, Band 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[7]Iwasawa, Kenkiti, “Über die Einfachheit der speziellen projektiven Gruppen”, Proc. Imp. Acad. Tokyo 17 (1941), 57–59.Google Scholar
[8]Seip-Hornix, E.A.M., “Clifford algebras of quadratic quaternion forms. I”, K. Nederl. Akad. Wetensch. Proc. Ser. A 68 (1965), 326–341; “Clifford algebras of quadratic quaternion forms. II”, K. Nederl. Akad. Wetensch. Proc. Ser. A 68 (1965), 345–363.Google Scholar
[9]Tits, J., “Classification of algebraic semisimple groups”, Algebraic groups and discontinuous subgroups (Proc. Sympos. Pure Math. Boulder, Colo., 1965, 33–62. Amer. Math. Soc., Providence, Rhode Island, 1966).Google Scholar
[10]Wall, G.E., “The structure of a unitary factor group”, Publ. Math. Inst. Hautes Études Scient. 1 (1959).Google Scholar
[11]Wall, G.E., “On the conjugacy classes in the unitary, symplectic and orthogonal groups”, J. Austral. Math. Soc. 3 (1963), 1–62.CrossRefGoogle Scholar