Finite projective plane

Description

A finite projective plane is a finite set of points with a family of subsets, called lines, with three properties. Every two distinct points lie on exactly one line, and every two distinct lines meet in exactly one point. There are four points with no three on a line. Each such plane has an order: an integer $n \ge 2$ such that each line has $n + 1$ points and each point lies on $n + 1$ lines. A plane of order $n$ has $n^2 + n + 1$ points and $n^2 + n + 1$ lines.

The standard construction is $PG(2, q)$ for a prime power $q$. Let $\F_q$ be the finite field with $q$ elements. The points are the $1$-dimensional subspaces of $\F_q^3$ and the lines are its $2$-dimensional subspaces, with incidence given by inclusion. Check: there are $(q^3 - 1)/(q - 1) = q^2 + q + 1$ points. Two distinct $1$-dimensional subspaces span exactly one $2$-dimensional subspace. Two distinct $2$-dimensional subspaces $U, W$ satisfy $\dim(U \cap W) = 2 + 2 - \dim(U + W) = 1$, so they meet in exactly one point. The four points spanned by $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(1,1,1)$ have no three on a line. So $PG(2, q)$ is a projective plane of order $q$.