This is an elementary description of the finite desarguesian projective plane ${PG(2,q)}$ and its automorphism group ${P\Gamma L(3,q)}$. I needed to explain this to a non-combinatorialist, so thought I would just add it to the blog. The required background is just elementary linear algebra, elementary group theory and the concepts of a finite field and an incidence structure.

We start with the finite field ${GF(q)}$ where ${q = p^h}$ is necessarily some power of a prime ${p}$. An automorphism of a field is a permutation ${\sigma}$ of the field elements such that

$\displaystyle \sigma(x+y) = \sigma(x)+\sigma(y) \qquad \sigma(xy) = \sigma(x)\sigma(y)$

and the collection of all automorphisms forms a group. The automorphism group of ${GF(p^h)}$ is the cyclic group ${C_h}$ of order ${h}$ generated by the automorphism ${\rho: x \rightarrow x^p}$.

Next we construct the three-dimensional vector space ${V = GF(q)^3}$ with vectors being triples of elements of ${GF(q)}$ which we shall view as row-vectors. If ${A}$ is an invertible matrix with entries in ${GF(q)}$, then the map ${v \rightarrow vA}$ is a permutation of ${V}$ (fixing ${0}$). The collection of all such invertible matrices forms a group called the general linear group and denoted ${GL(3,q)}$. We can build a matrix in ${GL(3,q)}$ by picking an arbitrary non-zero vector ${v_1}$ for the first row, then choosing any vector ${v_2}$ that is not a multiple of ${v_1}$ for the second row and then any vector ${v_3}$ not in the span of ${\{v_1, v_2\}}$ for the third row. Therefore the order of the general linear group is given by

$\displaystyle |GL(3,q)| = (q^3-1)(q^3-q)(q^3-q^2).$