Projective Plane - Subplanes

Subplanes

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.

(Bruck 1955) proves the following theorem. Let Π be a finite projective plane of order N with a proper subplane Π₀ of order M. Then either N = M2 or N ≥ M2 + M.

When N is a square, subplanes of order are called Baer subplanes. Every point of the plane lies on a line of a Baer subplane and every line of the plane contains a point of the Baer subplane.

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.