Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T01:03:05.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Bireflectionality in Absolute Geometry

Published online by Cambridge University Press:  20 November 2018

Dragoslav Ljubić
Dragoslav Ljubić*
Affiliation:
University of Washington, Seattle, Washington
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.

If G is any group then g ∊ G is called an involution if g ≠ 1 and g o g = 1. A group G is called bireflectional if every element in G is a product of two involutions. It is known that 2- dimensional, 3- dimensional, and some types of n-dimensional (n > 3) absolute geometries (in the sense of H. Kinder) are bireflectional. In this article the author proves the general result that every n-dimensional absolute geometry is bireflectional.

Keywords

51F1020G15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 54 - 63
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Ahrens, J., Begrundung der absoluten Géométrie des Raumes aus dem Spiegelungsbegriff, Math. Z. 71 (1959), 154–185. Google Scholar
2. Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbe griff, 2nd edition, Springer- Verlag, New York-Heidelberg-Berlin, 1973.Google Scholar
3. Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974.Google Scholar
4. Djokovic, D. Z., Product of two involutions, Arch. Math. XVIII (1967), 582–584.Google Scholar
5. Ellers, E. W., Bireflectionality in classical groups, Can. J. Math XXIX (1977), 1157–1162.Google Scholar
6. Ellers, E. W. and W. Nolte, Bireflectionality of orthogonal and symplectic groups, Arch. Math. 39 (1982), 113–118. Google Scholar
7. Ellers, E. W., Bireflectionality, Annals of Discrete Mathematics 18 (1983), 333–334. Google Scholar
8. Ewald, G., Spiegelungsgeometrische Kennzeichnung euklidischer und nichteuklidischer Raurne beliebiger Dimension, Abh. Math. Sem. Hamburg 41 (1974), 224–251. Google Scholar
9. Ewald, G., Normal Forms of Isometries, The Geometric Vein, The Coxeter Festschrift (1981), 471— 476.Google Scholar
10. Kinder, H., Begrundung der n-dimensionalen absoluten Géométrie aus dem Spiegelungsbe griff, Dissertation, Kiel 1965.Google Scholar
11. Ljubic, D., Normal Forms of Isometries, in preparation.Google Scholar
12. Ljubic, D., Direct Limit of Isometry Groups, in preparation.Google Scholar
13. Wonenburger, M. J., Transformations which are products of two involutions, J. Math. Mech. 16 (1966), 327–338. Google Scholar