Construction of finite planes

Abstract

The only finite projective and affine planes that we have actually defined so far (cf. Section 1.4) are the desarguesian planes P(q) and A(q). We have mentioned repeatedly that there exist non-desarguesian finite planes as well, and in this chapter we shall present all known such planes. The known construction techniques for finite planes all use a finite vector space in a more or less obvious, but always essential way. This is the reason that these constructions always lead to planes of prime-power order. It is one of the major unsolved problems of the theory whether or not there also exist planes of non-prime-power order.