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).

The group {\Gamma L(3,q)} is the semi-direct product of {GL(3,q)} and the cyclic group {\text{Aut}(GF(q)) = \langle \rho \rangle} of field automorphisms. Each element of this group consists of a pair {(A, \sigma)} where {A \in GL(3,q)} and {\sigma \in \text{Aut}(GF(q))} and each element of this group determines a permutation of vectors in {V} given by:

\displaystyle  (A,\sigma): v \rightarrow (vA)^\sigma

where {\sigma} acts coordinate-wise on the elements of {vA}. The order of {\Gamma L(3,p^h)} is therefore given by

\displaystyle  |\Gamma L(3,p^h)| = h |GL(3,p^h)|.

The desarguesian projective plane {PG(2,q)} is a point-line incidence structure that is constructed from a three-dimensional vector space over a finite field. The points of {PG(2,q)} are the 1-dimensional subspaces of {V}, and the lines of {PG(2,q)} are the 2-dimensional subspaces of {V}. A point and a line are declared to be incident if the point is contained in the line (when both are viewed as subspaces). The following properties follow immediately from elementary linear algebra.

  • Two distinct points are mutually incident with a unique line, and
  • Two distinct lines are mutually incident with a unique point.

Finite geometers usually drop the rather formal language of incidence, instead using more geometric language where lines are viewed as joining points and points viewed as lying on lines. So in this language, we just say that

  • There is a unique line on any two points, and
  • Any two lines meet in a unique point.

An automorphism of a projective plane, including {PG(2,q)} is a permutation of the points and lines that maps points to points, lines to lines and preserves the incidence relation. Because it is incidence preserving, an automorphism is determined by its action on the points alone and often it will be given in that way, or even defined in that way i.e., as a permutation of the points that preserves lines. On the other hand a duality is a permutation of the points and lines that maps points to lines, lines to points and preserves the incidence relation.

Any element of {\Gamma L(3,q)} permutes the vectors in {V}, a maps subspaces to subspaces. In fact, an element {(A,\sigma)} maps a subspace with basis {\{v_1, v_2, \ldots, v_k\}} to the subspace with basis {\{(v_1A)^\sigma, (v_2A)^\sigma, \ldots, (v_kA)^\sigma\}}. Hence every element of {\Gamma L(3,q)} induces an automorphism of {PG(2,q)}. A classical theorem called The Fundamental Theorem of Projective Geometry says that there are no other permutations of {V} that map subspaces to subspaces.

We have not quite determined the automorphism group of {PG(2,q)} however, because not every element of {\Gamma L(3,q)} induces a different automorphism of {PG(2,q)}. In particular, any matrix in the set of scalar matrices

\displaystyle Z = \{\lambda I \mid \lambda \not= 0\}

fixes every subspace of {V} (although permuting the vectors contained in the subspace). Therefore the elements in any coset of {Z} all induce the same automorphism of {PG(2,q)}. The group {P\Gamma L(3,q)} is the quotient group {\Gamma L(3,q) / Z} and it is this group that is the full automorphism group of {PG(2,q)}. Therefore we conclude that

\displaystyle  P\Gamma L(3,q) = \Gamma L(3,q) / Z = \text {Aut}(PG(2,q)).


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