Finite Projective Planes

Abstract

We propose graph theoretic equivalents for existence of a finite projective plane. We then develop a new approach and see that the problem of existence of a finite projective plane of order n is linked up with a subset of sharply 2 transitive permutations. If n is prime power then it is well known that there exists a finite field and existence of this field implies existence of MOLS which further implies existence of fpp. We show that by assuming the existence of MOLS the existence of a group made up of sharply 2 transitive permutations can be implied through transforming the given MOLS to suitable form. From a known results it then follows that when such group exists the order n has to be a prime power. We then see the relation between MOLS and determinantal monomials and between MOLS and a cyclic group that permutes the rows of MOLS. Finally, we conclude the paper with some important remarks.