Finite Projective Planes with Affine Subplanes

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following: If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m. In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).