It is well-known that if the order s of a finite projective plane is a prime or a power of a prime, then there is always a cyclic solution for the (0, 1) incidence matrix of such a plane. We raise a more difficult question. Excluding some type of border, it is possible that the incidence matrix for the plane can be partitioned into subsets of permutation matrices. Are there solutions which are, excluding the border, “cyclic” in the permutation matrices themselves? The permutation matrices themselves need not be cyclic. A minor advantage of this approach is that a specific solution can be obtained very rapidly. We show that with the standard border (defined later), there is always a “cyclic” solution if the order is prime. With a Baer subplane border, we have no new results involving “cyclic” solutions, but the fragmentary information here is given in section 3.

We say we have a “semi-cyclic” solution if, exclusive of a border, the incidence matrix remaining can be partitioned into permutation matrices such that there is a further partitioning into larger subdivisions each of which is “cyclic” in the permutation matrices. In section 3 we give a solution of this type for the plane of order 9.

With this type of approach, the isomorphism question is complex. In the last section we give a generalization of Halľs multiplier theorem that is useful along with some details from a communication from R. H. F. Denniston received after the conclusion of the conference in which he shows that the two types of planes of order 9 just described are inequivalent.