Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-20T04:21:48.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Study of Certain Similitudes

Published online by Cambridge University Press:  20 November 2018

Maria J. Wonenburger
Maria J. Wonenburger*
Affiliation:
Queen's University, Kingston C.S.I.C., Madrid
Rights & Permissions [Opens in a new window]

Extract

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.

One important point in the determination of the automorphisms of the classical groups is the study of group-theoretic properties of the elements of order 2, that is, the involutions. The group of similitudes and the projective group of similitudes of a non-degenerate quadratic form Q are extensions of the orthogonal group and the projective orthogonal group, respectively. These extended groups may contain involutions which do not belong to the orthogonal group or the projective orthogonal group. To study the automorphisms of such groups, group-theoretic properties of some of these new involutions should be established.

The aim of this paper is to give some properties of a similitude T of ratio ρ whose square T2 is equal to the scalar multiplication by — p (it is always assumed that we are dealing with a field of characteristic ≠2). If ρ = — 1, T is an involution in the group of similitudes and for any ρ the coset of T in the projective group of similitudes is an involution.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 60 - 68
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Dieudonné, J., On the automorphisms of the classical groups, Mem. Amer. Math. Soc, No. 2 (1951).Google Scholar
2. Dieudonné, J., On the structure of unitary groups II, Amer. J. Math. 76 (1953), 665–678.Google Scholar
3. Dieudonné, J., La géométrie des groupes classiques (Berlin: Springer-Verlag, 1955).Google Scholar
4. Jacobson, N., A note on hermitian forms, Bull. Amer. Math. Soc. 46 (1940), 264–268.Google Scholar
5. Wonenburger, M. J., The Clifford algebra and the group of similitudes, Can. J. Math. 14 (1961), 000-000.Google Scholar