Advertisement

Journal of Geometry

, Volume 16, Issue 1, pp 30–40 | Cite as

Maximal subgroups of the orthogonal group

  • Stephen Pierce
Article

Abstract

Let V be an n-dimensional regular quadratic space over a field K of characteristic not 2. Assume n ≥ 4. Let W be a regular hyperplane and v a nonzero vector orthogonal to W. Suppose every regular hyperplane in W is universal. If σ is an isometry of V not leaving W invariant, then σ, together with the isometries of W, generate the orthogonal group of V, with one exception.

Keywords

Maximal Subgroup Orthogonal Group Nonzero Vector Quadratic Space Regular Quadratic Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Artin,Geometric Algebra, Interscience, 1957.Google Scholar
  2. [2]
    E.W. Ellers,Decomposition of orthogonal, symplectic, and unitary isometries into simple isometries, Abh. Math. Sem. Univ. Hamburg46 (1977), 97–127.Google Scholar
  3. [3]
    E.W. Ellers,The length of a unitary transformation, J. Reine Angew. Math.281 (1976), 1–5.Google Scholar
  4. [4]
    W. Kantor,Subgroups of classical groups generated by long root elements, Trans. A.M.S.248 (1979), 347–379.Google Scholar
  5. [5]
    O.T. O'Meara,Introduction to Quadratic Forms, Academic Press, 1963.Google Scholar
  6. [6]
    S. Pierce and W. Watkins,Isometries of E 2, J. Reine Angew. Math.305 (1979), 60–64.Google Scholar
  7. [7]
    W.J. Wong,Generators and relations for classical groups, J. Alg.32 (1974), 529–553.CrossRefGoogle Scholar
  8. [8]
    W.J. Wong,A theorem on generation of finite orthogonal groups, J. Austral. Math. Soc.17 (1973), 495–506.Google Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • Stephen Pierce
    • 1
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

Personalised recommendations