This is an elementary description of the finite desarguesian projective plane and its automorphism group . 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 where is necessarily some power of a prime . An automorphism of a field is a permutation of the field elements such that
and the collection of all automorphisms forms a group. The automorphism group of is the cyclic group of order generated by the automorphism .
Next we construct the three-dimensional vector space with vectors being triples of elements of which we shall view as row-vectors. If is an invertible matrix with entries in , then the map is a permutation of (fixing ). The collection of all such invertible matrices forms a group called the general linear group and denoted . We can build a matrix in by picking an arbitrary non-zero vector for the first row, then choosing any vector that is not a multiple of for the second row and then any vector not in the span of for the third row. Therefore the order of the general linear group is given by