A Note on the Construction of Projective Planes from Groups

Chapter 4 - Projective Planes

Analysis of genetic divergence between closely related lines of chickens

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

We prove that a non-degenerate homomorphic image of a projective plane is determined to within isomorphism by the inverse image of any one point. An application gives conditions for the preservation of central collineations by a homomorphism.

We prove that a spread S over a locally compact nondlscrete field F defines a topological translation plane if and only if the spread is compact. For F=R, this is implicit in Breuning's thesis [Bre], cf. [B 2]. For the proof, we describe the point set of the projective translation plane as a quotient space of some projective space, with identifications taking place in one hyperplane. This is new even for F=R.

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in Ω+(8, q)-spaces to ovoids in corresponding Ω+(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in Ω+(6, q)-spaces, there are corresponding translation planes of order q 2 and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1. Finally, we show how certain projective planes of order q 2 of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

CHAPTER 1 - BASIC CONCEPTS CONCERNING FINITE GEOMETRIES

Let Π be a finite projective plane of order n , and let G be a large abelian (or, more generally, quasiregular) collineation group of Π; to be specific, we assume ∣ G ∣ ( n 2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings. Introduction A projective plane is a geometry consisting of points and lines such that any two distinct lines meet in exactly one point, any two distinct points are on exactly one common line, and there are four points no three of which are collinear. Well-known examples are provided by the desarguesian (or classical ) projective planes PG(2, q ), q a prime power, which can be defined over the finite field GF( q ) as follows: points and lines are the 1- and 2-dimensional subspaces of the vector space GF( q ) 3 , and a point is incident with a line if and only if it is a subset of the line. A standard reference for arbitrary projective planes is Hughes and Piper [104]; for the desarguesian case, see Hirschfeld [89].

We consider a class of symmetric divisible designs D which are almost projective planes in the following sense: Given any point p, there is a unique point p’ such that p and p’ are on two lines, whereas any other point is joined to p by exactly one line; and dually. We note that either the block size k or k — 2 is a perfect square, and exhibit examples for k =3 and k = 4. Then we add the condition that D should admit an abelian Singer group, so that we may study the associated divisible difference sets. Under this additional assumption, we show that k is a square (unless k = 3) and that the only possible prime divisors of k are 2 and 3.

Introduction. It has been shown in a recent paper (see [5]) that any countable group can be embedded in a group generated by two elements. We show here that any countable loop (quasigroup, groupoid) can be embedded in a loop (quasigroup, groupoid) generated by one element, any countable semigroup can be embedded in a semigroup generated by two elements, and any countable projective plane can be embedded in a projective plane generated by four points. No such embedding theorem exists for some systems, such as abelian groups, and some light is thrown on the general problem by the following theorem. Let ?f be a class of algebras, (E1) the property that there is a number m such that any countable 2-algebra can be embedded in an 21-algebra generated by m elements, (E2) the property that there is a number n such that the free W-algebra on n generators contains a free subalgebra on a countable number of generators. Then if 2I has property (E1), it has also property (E2) and n ? m.

An (n, k) configuration is a set of n points and n lines, with k of the points on each line and k of the lines through each point. Motivated by the group law on cubic curves, here we investigate the problem of representing such abstract configurations by abelian groups in such a way that whenever k points P1, P2,..., Pk are collinear in the given configuration then the sum P1 + P2 +,..., + Pk = 0 in the group. In this note, we show how Bill McCune’s Prover9 can be successfully employed to determine the structure of the potential group in which the PG(2, 3), projective plane of order 3, gets embedded.KeywordsProjective planesgroup embeddingsautomated deductionProver9

The article is a contribution to the classification of all 4-dimensional flexible compact projective planes. We assume that the collineation group is a 6-dimensional solvable Lie group which fixes some flag. If, moreover, the nilradical of the collineation group is 5-dimensional, then we get 4 families of new planes which are neither translation planes nor shift planes.

Shult and Thas have shown in [13] that m-systems of certain finite classical polar spaces give rise to strongly regular graphs and two weight codes. The main result of this paper is to show that maximal arcs in symplectic translation planes may be obtained from certain m-systems of finite symplectic polar spaces. Many new examples of maximal arcs are then constructed. Examples of m-systems are also constructed in Q(-)(2n + 1,q) and W2n+1(q). A method different from that of Shult and Thas is used to construct strongly regular graphs using differences of m-systems.

Construction of translation planes from t-spread sets