On finite projective planes in Lenz-Barlotti class at least I.3

Abstract

Abstract. We establish the connections between finite projective planes admitting a collineation group of Lenz-Barlotti type 1.3 or 1.4, partially transitive planes of type (3) in the sense of Hughes, and planes admitting a quasiregular collineation group of type (g) in the DembowskiPiper classification; our main tool is an equivalent description by a certain type of difference set relative to disjoint subgroups which we will call a neo-difference set. We then discuss geometric properties and restrictions for the existence of planes of Lenz-Barlotti class 1.4. As a side result, we also obtain a new synthetic description of projective triangles in desarguesian planes.

Lenz-Barlotti I.4 perspectivity groups are abelian

“…There is also a dual partition of lines, and together with the ordinary incidence structure (P, L, F := F * ∩P ×L) they uniquely determine all incidences for the plane Π. Thus Π is uniquely determined by the incidence relation between the ordinary points and ordinary lines [1].…”

“…In this section we show that the incidence relation between ordinary points and ordinary lines appears in the internal coset geometry as the left regular representation of a subset ∆ of G. When Π is finite, the set ∆ is called a neo-difference set [1].…”

A Non-Existence Result For Finite Projective Planes In Lenz-Barlotti Class I.4

“…As Π admits a group of type I. 4, it may be represented by an abelian neo-difference set, see [5]. Using group ring notation -a standard approach in the study of any type of difference set, see [1] or our survey [4] for background and notation -D is a subset of a group G = X×X satisfying the equation…”

“…Equivalently, G is a (necessarily abelian) quasiregular collineation group of type (g) in the Dembowski-Piper classification [3]. For a proof of this fact, more background and historical references, we refer the reader to our recent systematic treatment of planes in class I.4 [5].The only known finite planes admitting a group of type I.4 are the Desarguesian planes. Any other example would necessarily be in Lenz-Barlotti class I.4, and it is widely conjectured that there are no finite planes in this class (but there do exist infinite examples).…”

“…Equivalently, G is a (necessarily abelian) quasiregular collineation group of type (g) in the Dembowski-Piper classification [3]. For a proof of this fact, more background and historical references, we refer the reader to our recent systematic treatment of planes in class I.4 [5].…”