On projective planes of Lenz-Barlotti class I 6

Trans. Amer. Math. Soc, 1962

Combinatorica, 2007

Let Π be a projective plane of order n in Lenz-Barlotti class I.4, and assume that n is a multiple of 3. Then either n = 3 or n is a multiple of 9.

Advances in Geometry, 2000

We establish the connections between finite projective planes admitting a collineation group of Lenz-Barlotti type I.3 or I.4, partially transitive planes of type (3) in the sense of Hughes, and planes admitting a quasiregular collineation group of type (g) in the Dembowski-Piper classification; our main tool is an equivalent description by a certain type of di¤erence set relative to disjoint subgroups which we will call a neo-di¤erence set. We then discuss geometric properties and restrictions for the existence of planes of Lenz-Barlotti class I.4. As a side result, we also obtain a new synthetic description of projective triangles in desarguesian planes.

Proceedings of The American Mathematical Society, 2010

We extend a 1972 result of Kantor and Pankin and give a new elementary proof of the assertion in the title for projective planes of arbitrary order. The main tool appears in the very first book on group theory by Jordan in 1870. The Lenz-Barlotti classification of projective planes is based on the possible configurations of point-line pairs for which Desargues theorem holds. Desargues theorem holds for such a pair (p, l) if and only if the plane admits a full group of perspectivities with center p and axis l. By definition, such a perspectivity group fixes all points on l, all lines on p and is maximally transitive consistent with these conditions. The Lenz-Barlotti figure of a projective plane Π is the set of point-line pairs for which Π is (p, l)-transitive and determines the Lenz-Barlotti class of Π. The plane Π is of class I.4 (respectively I.3) if its Lenz-Barlotti figure consists of the three non-incident point-line pairs of a triangle (respectively two of these pairs). These classes are two of the five for which existence questions remain open [1]. 1 The purpose of this paper is to give a new elementary proof of: Theorem 1 Let Π be a projective plane of Lenz-Barlotti type I.4. Then its three transitive perspectivity groups are isomorphic and Abelian.

2013

Abstract: In this paper we investigate the non-existence of a projective plane of order 12 by using the relationship between Latin squares and projective planes. We arrive at a conjecture: if the sum of all divisors of a positive integer n, (n)> 2n, then there is no finite projective plane of order n. Key words: Projective plane Latin square Abundant numbers

Journal of Combinatorial Theory, Series A, 1988

Linear spaces are investigated using the general theory of "Rings of Geometries I." By detining geometries and ring structures in several different ways, formulae for linear spaces embedded in finite projective and affine planes are obtained. Several "fundamental theorems" of counting in finite projective planes are proved which show why configurations with at least three points per line and at least three lines through every point are important. These theorems are illustrated by finding the formulae for the number of k-arcs in a projective plane of order q for all k < 8 and also by finding a formula for the number of blocking sets. A quick proof that a projective plane of order 6 does not exist follows from the formula for the number of ?-arcs in such a plane. Ic?

Rendiconti del Seminario Matematico della Università di Padova, 1993

The aim of this paper is to consider the structure and other properties of some of the triangle groups 4(L, m, n) for positive integers l,m,n &#x3E; 2. The triangle group J(l, m, n) is defined by the presentation It is the group of tesselation of a space with a triangle [7]. The group m, n) is finite iff the corresponding space is compact. This implies that IJ(l, m, n) I 00 iff + + &#x3E; 1. [7]. We get the following three cases for J(l, m, n). 1) The Euclidean case if + + = 1. This equation has the solution (3, 3, 3), (2, 3, 6) and (2, 4, 4). 2) The elliptic case if + 1/m + 1/n &#x3E; 1. This inequality has the following solutions (2, 2, n), (2, 3, 3), (2, 3, 4), (2, 3, 5) for n &#x3E; 2. 3) The hyperbolic case if + + 1. This inequality has an infinite number of solutions. REMARK 1. J(1, m, n) = m, n) = J(m, l, n). The group m, n) depends only on the absolute values of l, m, n and not on their order or sign. THEOREM 1. The group A(l, m, n) is finite iff 1/1 + 1/m + 1/n &#x3E; 1. PR...

Proceedings of the American Mathematical Society, 2007

Journal of Algebra, 1998

Journal of the London Mathematical Society, 1996