Finite projective plane

Number of points in a finite projective plane

If in an abstract projective plane there exists a line incident with points, #m/thm/geo/fin

  1. every line of is incident with points;
  2. every point of is incident with lines; and
  3. contains points and the same number of lines.1

By duality, the same holds if there is a point incident with lines. Thus is called the order of a projective plane. #m/def/geo

Proof

Let the points incident with be . If then by ^P1 there exist pairwise distinct lines . Now this must exhaust all lines passing through , since each such line must intersect by ^P2 at a point. By duality, if there is a point incident with lines, then every line not through is incident with points.

If is a line distinct from then there exists by ^P4 a third line through , and by ^P3 this is incident with a point . Since , there are lines passing through , and since this yields points on . This proves ^C1, and similarly one shows ^C2.

Consider an arbitrary point , and the lines incident with it. Each of these contains points distinct from each other and , so the total number of points is . By duality, the same holds for lines.

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2020, Finite geometries, p. 6 ↩