Cyclic affine planes and Paley difference sets

Arasu, K.T. and A. Pott, Cyclic affine planes and Paley difference sets, Discrete Mathematics 106/107 (1992) 19-23. The existence of a cyclic affine plane implies the existence of a Paley type difference set. We use the existence of this difference set to give the following condition on the existence of cyclic affine planes of order n: If n 8 mod 16 then n 1 must be a prime. We discuss the structure of the Paley type difference set constructed from the plane. One of the famous unsolved problems in finite geometry is to prove or disprove the conjecture that the order of a finite projective plane must be a power of some prime p. It seems that we are still far away from an answer to this question. In the following we will refer to it as the prime power conjecture (PPC). We can expect to be able to say more if we assume that the plane has some symmetry properties, i.e., admits a certain type of automorphism group. A lot of work has been done on projective planes admitting a quasiregular collineation group, i.e., a group acting in such a way that the stabilizer of each point and line is a normal subgroup. A classification of planes admitting quasiregular collineation groups G with lG( > (n” + II + 1)/2 was given by Dembowski and Piper [5]. Let us assume that the quasiregular group is abelian. Then in one case of the Dembowski-Piper classification the prime power conjecture is trivially satisfied, namely in the case of translation and dual translation planes. In two cases of the Dembowski-Piper Correspondence to: A. Pott, Mathematisches Institut der Universitlt GieOen, ArndtstraBe 2, 6300 Gie&n, Germany. * Research partially supported by NSA grant MDA 904-90-H-4008 and by an Alexander-vonHumboldt fellowship. The author thanks the Mathematisches Institut der Universitst GieBen for its hospitality during the time of this research. 0012-365X/92/$05.00