Projective Planes I : PG(2,q)

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).

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