Automorphisms of pentagonal geometries with block size 3

Abstract

A pentagonal geometry PENT(k,r) is a k-uniform, r-regular partial linear space with the property that the set of points not collinear with a given point x is itself a line in the geometry. The existence spectrum of these geometries is known; we begin the study of automorphisms of these structures in the case k=3. We determine the spectrum of pentagonal geometries PENT(3,r) with either a cyclic or reverse automorphism to be r = 1 or 10 (mod 12). In addition, we show that there are 584 pairwise non-isomorphic PENT(3,7)s and classify them by the orders of their automorphism groups.