On affine translation divisible designs
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.
scientific article; zbMATH DE number 4210150