Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T17:52:09.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite Projective Geometries

Published online by Cambridge University Press:  20 November 2018

Gerald Berman
Gerald Berman*
Affiliation:
Illinois Institute of Technology
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.

James Singer [12] has shown that there exists a collineation which is transitive on the (t - 1)-spaces, that is, (t - 1)-dimensional linear subspaces, of PG(t, p n). In this paper we shall generalize this result showing that there exist t - r collineations which together are transitive on the s-spaces of PG(t, p n). An explicit construction will be given for such a set of collineations with the aid of primitive elements of Galois fields. This leads to a calculus for the linear subspaces of finite projective geometries.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 4 , 1952, pp. 302 - 313
Copyright
Copyright © Canadian Mathematical Society 1952

References

1. Bussey, W. H., Tables of Galois fields, Bull. Amer. Math. Soc, vol. 12 (1905-6), 22–38.Google Scholar
2. Bussey, W. H. Tables of Galois fields, Bull. Amer. Math. Soc, vol. 16 (1909-10), 188–206.Google Scholar
3. Carmichael, R. D., Introduction to the theory of groups of finite order (Boston, 1937).Google Scholar
4. Coxeter, H. S. M., The real projective plane (New York, 1949).Google Scholar
5. Coxeter, H. S. M., Self-dual configurations and regular graphs, Bull. Amer. Math. Soc, vol. 56 (1950), 413–456.Google Scholar
6. Fano, G., Sui postulati fondamentali delta geometria proiettiva in uno spazio lineare a un numero qualunque di dimensioni, Giornale di Mathematiche, vol. 30 (1892), 100–132.Google Scholar
7. Hall, Marshall Jr., Cyclic projective planes, Duke Math. J., vol. 14 (1947), 1079.Google Scholar
8. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, vol. 1 (Cambridge, 1947).Google Scholar
9. Rao, C. R.,Finite geometries and certain derived results in the theory of numbers, Proc Nat. Inst. Sci. India (1945), 136–149.Google Scholar
10. Rao, C. R., Difference sets and combinatorial arrangements derivable from finite geometries, Proc. Nat. Inst. Sci. India (1946), 123–135.Google Scholar
11. Robinson, G. de B., The foundations of geometry (Toronto, 1940).Google Scholar
12. Singer, J., A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc, vol. 43 (1938), 377–385.Google Scholar
13. Snapper, E., Periodic linear transformations of affine and projective geometries, Can. J. Math., vol. 2 (1950), 149–151.Google Scholar
14. Sommerville, D. M. Y., An introduction to the geometry of n dimensions (London, 1929).Google Scholar
15. van der Waerden, B. L., Moderne Algebra (Berlin, 1931).Google Scholar
16. Veblen, O. and Bussey, W. H., Finite projective geometries, Trans. Amer. Math. Soc, vol. 7 (1906), 244.Google Scholar