International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 12, Pages 757-762
doi:10.1155/S0161171201006184
On some hyperbolic planes from finite projective planes
Department of Mathematics, Faculty of Sciences and Art, Uludag University, Bursa, Görükle 16059, Turkey
Received 5 October 1999; Revised 15 November 2000
Copyright © 2001 Basri Celik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let Π=(P,L,I) be a finite projective plane of order n, and let Π′=(P′,L′,I′) be a subplane of Π with
order m which is not a Baer subplane (i.e., n≥m2+m). Consider the substructure Π0=(P0,L0,I0) with P0=P\{X∈P|XIl, l∈L′}, L0=L\L′ where I0 stands for the restriction of I to P0×L0. It is shown that every Π0 is a hyperbolic plane, in
the sense of Graves, if n≥m2+m+1+m2+m+2. Also we give some combinatorial properties of the line
classes in Π0 hyperbolic planes, and some
relations between its points and lines.