A subplane of a projective plane is a subset of the points of the plane which themselves form a projective plane with the same incidence relations.
In the finite desarguesian planes PG(2,pn), the subplanes have orders which are the orders of the subfields of the finite field GF(pn), that is, pi where i is a divisor of n. In non-desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of order M in a plane of order N with M2 + M = N is an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order.
Read more about this topic: Projective Plane
Other articles related to "subplanes, subplane":
... A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2 ... The name Fano for this subplane is really a misnomer ... A Fano subplane however violates Fano's Axiom ...