On affine translation divisible designs
type: Item
instance of: scholarly article
title: On affine translation divisible designs
zbMATH Open document ID: 0732.05009
DOI: 10.1007/BF01225884
author: Antonino Giorgio Spera
published in: Journal of Geometry
publication date: 1991-01-01T00:00:00Z
review text: A divisible design with parameters s, k and \(\lambda\) is an incidence structure with blocks of size \(k>1\) whose point set admits a partition into classes of size s such that two points from distinct point classes are joined by precisely \(\lambda\geq 1\) blocks and two points from the same class are joined by no blocks. A divisible design D which admits an automorphism group T which acts regularly on the set of points of D and which is such that \(B^ f=B\) or \(B^ f\cap B=\emptyset\) for every block B and every \(f\in T\), is called a translation divisible design (TDD). T is a translation group of D. (Some translation designs can be obtained from an affine translation plane by deleting a parallel class of lines.) The block orbits of T form a parallelism of D. If blocks in different orbits of T are never disjoint, D is said to be an affine TDD. In the paper under review, it is shwon that the translation group T of an affine TDD must be elementary Abelian. A construction for non-affine TDD's with an Abelian translation group is given. It is also shown that a TDD with an Abelian translation group is embeddable in a finite projective geometry in a certain way.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 51E05
zbMATH DE Number: 4210150
zbMATH Keywords: divisible design
zbMATH Keywords: translation
zbMATH Keywords: translation group
zbMATH Keywords: affine
MaRDI profile type: Publication
cites work: Partitions of a vector space
cites work: Q5545191
cites work: Transitive symmetric nets
cites work: Symmetric translation nets.
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description: scientific article; zbMATH DE number 4210150