Bireflectionality in Absolute Geometry
Canadian mathematical bulletin, Tome 32 (1989) no. 1, pp. 54-63

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1989-008-6
Mots-clés : 51F10, 20G15
Ljubić, Dragoslav. Bireflectionality in Absolute Geometry. Canadian mathematical bulletin, Tome 32 (1989) no. 1, pp. 54-63. doi: 10.4153/CMB-1989-008-6
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[1] 1. Ahrens, J., Begrundung der absoluten Géométrie des Raumes aus dem Spiegelungsbegriff, Math. Z. 71 (1959), 154–185. Google Scholar

[2] 2. Bachmann, F., Aufbau der Geometrie aus dem Spiegelungsbe griff, 2nd edition, Springer- Verlag, New York-Heidelberg-Berlin, 1973. Google Scholar

[3] 3. Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974. Google Scholar

[4] 4. Djokovic, D. Z., Product of two involutions, Arch. Math. XVIII (1967), 582–584. Google Scholar

[5] 5. Ellers, E. W., Bireflectionality in classical groups, Can. J. Math XXIX (1977), 1157–1162. Google Scholar

[6] 6. Ellers, E. W. and W. Nolte, Bireflectionality of orthogonal and symplectic groups, Arch. Math. 39 (1982), 113–118. Google Scholar

[7] 7. Ellers, E. W., Bireflectionality, Annals of Discrete Mathematics 18 (1983), 333–334. Google Scholar

[8] 8. Ewald, G., Spiegelungsgeometrische Kennzeichnung euklidischer und nichteuklidischer Raurne beliebiger Dimension, Abh. Math. Sem. Hamburg 41 (1974), 224–251. Google Scholar

[9] 9. Ewald, G., Normal Forms of Isometries, The Geometric Vein, The Coxeter Festschrift (1981), 471— 476. Google Scholar

[10] 10. Kinder, H., Begrundung der n-dimensionalen absoluten Géométrie aus dem Spiegelungsbe griff, Dissertation, Kiel 1965. Google Scholar

[11] 11. Ljubic, D., Normal Forms of Isometries, in preparation. Google Scholar

[12] 12. Ljubic, D., Direct Limit of Isometry Groups, in preparation. Google Scholar

[13] 13. Wonenburger, M. J., Transformations which are products of two involutions, J. Math. Mech. 16 (1966), 327–338. Google Scholar

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