On subplane partitions of a finite projective plane

Abstract

Bruck (1960) notes that every finite Desarguesian plane of order m 2 can be partitioned into disjoint subplanes of order m. His method follows directly from the discovery by Singer (1938) that every such plane is cyclic and may be derived from a difference set. The present paper begins investigating the existence of other partitions which do not follow naturally from the difference set representation of the plane. The procedure is based on the idea, defined herein, of conjugacy with respect to a given subplane. When m = 2, only Bruck's partition is possible, but new results have been obtained in the cases m = 3, 4, 5, 7. This paper treats the case m = 3 in some detail.