Construction of Projective Plane Up to Order 5

Question

How does one construct projective planes of order 5? What are the references of projective geometry that describe the construction of projective planes of order at least up to 5?

Answer 1

For a finite projective plane of order $n$, take a finite field $\mathbb F_n$ of order $n$. Then the set of points can be described as

$$\mathbb F_n\mathbb P^2=\frac{\mathbb F_n^3\setminus\left\{{\scriptsize\begin{pmatrix}0\\0\\0\end{pmatrix}}\right\}}{\mathbb F_n\setminus\{0\}}$$

So you take the set of all three-element vectors with elements from this finite field. You exclude the null vector, and you identify scalar multiples. The resulting equivalence classes represent the points of the plane.

The lines can be represented by equivalence classes of the same form. So although they look the same algebraically, they are usually considered two disjoint instances of the same structure. A point is incident with a line if and only if the scalar product between the two is zero.

The above works as the template to construct a projective plane (finite or infinite) over any field. But there are projective planes which don't have an underlying field. In some of them, Desargues' theorem doesn't hold; these are the so-called non-Desarguesian planes. In others, Desargues' theorem does hold, but Pappos' theorem does not. These are planes over skew fields which are not fields. Since there are no finite skew fields which are not also fields, every finite projective plane is either a non-Desarguesian plane or a plane over a finite field.