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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

Continue reading “Projective Planes I : PG(2,q)”