Finite planar spaces with isomorphic planes

Abstract

A π-space is a planar space all of whose planes are isomorphic to a given linear space π. The only known nontrivial finite π-spaces in which all lines have the same size are the finite projective and affine spaces, the Hall spaces on 3n points and the Steiner systems S(3,|π|,v). Only one family of finite π-spaces with lines of different sizes is known at present: these spaces consist of 2k points lying on two dispoint lines of k points and their planes are degenerate projective planes. Rather restrictive relations on the parameters of such spaces are given, which are particularly strong if there are more than two line sizes. We prove that the number of points of any finite nontrivial π-space having at least two line sizes is uniquely determined by some of the parameters of π. A table of possible parameters of planes having only two line sizes and containing at most 40 points is also provided. The smallest value of |π| is 7 and corresponds to two planes; one of them is ruled out, the problem is open for the other one.