Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-20T08:47:00.618Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension Of Finite Projective Planes I. Uniform Hjelmslev Planes

Published online by Cambridge University Press:  20 November 2018

Robert T. Craig
Robert T. Craig*
Affiliation:
Fordham University, New York
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.

In his paper (3) on finite H-planes, Kleinfeld has defined invariants s and t for an H-plane π as follows: let P and k be any point and line of π such that P is incident with k, then let s be the number of non-neighbour points to P on k and let t be the number of neighbour points to P on k. He has shown that s and t are independent of the choice of P and k, t divides s, π has s2 + st + t2 points (lines), π* has order s/t, and, if t ≠ 1, s ≤ t2. In the case s = t2, π is called uniform and has the property that each pair of neighbour lines (points) has exactly t points (lines) in common.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 261 - 266
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Archbold, J. W. and Johnson, N. L., A method of constructing partially balanced incomplete block designs, Ann. Math. Stat., 27 (1956), 624–632.Google Scholar
2. Bruck, R. H. and Ryser, H. J., The nonexistence of certain finite projective planes, Can. J. Math., 1 (1949), 88–93.Google Scholar
3. Kleinfeld, Erwin, Finite Hjelmslev planes, III. J. Math., 3 (1959), 403–407.Google Scholar
4. Klingenberg, Wilhelm, Projektive und affine Ebenen mit Nachbarelementen, Math. Zeit., 60 (1954), 384–406.Google Scholar
5. Pickert, Gunter, Projektive Ebenen (Berlin, 1955).Google Scholar