Baer Subplanes in Finite Projective and Affine Planes

TLDR: The Baer subplane as mentioned in this paper is the largest possible proper subplane of a projective or an affine plane, and it is defined as a configuration of projective and affine planes with an improper line.

Abstract: Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘. Let π be a finite projective (respectively, an affine) plane of order n and π 0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π 0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.