Number of points in a finite projective plane

If in an abstract projective plane $Π$ there exists a line $𝑒$ incident with $𝑛 + 1$ points, #m/thm/geo/fin

every line of $Π$ is incident with $𝑛 + 1$ points;
every point of $Π$ is incident with $𝑛 + 1$ lines; and
$Π$ contains $𝑛 2 + 𝑛 + 1$ points and the same number of lines.¹

By duality, the same holds if there is a point incident with $𝑛 + 1$ lines. Thus $𝑛$ is called the order of a projective plane. #m/def/geo

Proof

Let the points incident with $𝑒$ be $𝑃 1, \dots, 𝑃 𝑛 + 1$ . If $𝑄 ⧸ I 𝐸$ then by ^P1 there exist pairwise distinct lines ${𝑄 𝑃 𝑖} 𝑛 + 1 𝑖 = 1$ . Now this must exhaust all lines passing through $𝑄$ , since each such line must intersect $𝑒$ by ^P2 at a point. By duality, if there is a point $𝐸$ incident with $𝑛 + 1$ lines, then every line not through $𝐸$ is incident with $𝑛 + 1$ points.

If $𝑓$ is a line distinct from $𝑒$ then there exists by ^P4 a third line $𝑔$ through $𝑒 \cap 𝑓$ , and by ^P3 this $𝑔$ is incident with a point $𝑅 \neq 𝑒 \cap 𝑓$ . Since $𝑅 ⧸ I 𝑒$ , there are $𝑛 + 1$ lines passing through $𝑅$ , and since $𝑅 ⧸ I 𝑒$ this yields $𝑛 + 1$ points on $𝑓$ . This proves ^C1, and similarly one shows ^C2.

Consider an arbitrary point $𝑃 \in P$ , and the $𝑛 + 1$ lines incident with it. Each of these contains $𝑛$ points distinct from each other and $𝑃$ , so the total number of points is $𝑛 2 + 𝑛 + 1$ . By duality, the same holds for lines.

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

2020, Finite geometries, p. 6 ↩

Number of points in a finite projective plane

Footnotes