In the class of abelian groups a subclass of two-divisible-groups is singled out, and in the latter, a subclass of uniquely-two-divisible-groups. With every such a group a special geometrical structure, more precisely the structure of “congruence of vectors” is correlated. The notion of “affine vector space” (denoted by AffVect) is introduced. This term is defined by means of suitable axiom sys...

Let X be a p-divisible group over a regular scheme S such that the Newton polygon in each geometric point of S is the same. Then there is a p-divisible group isogenous to X which has a slope filtration.

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs. © 2006 Elsevier Inc. All rights reserved.

Let (G, ·) be a 2-divisible group and let (G∗,+) be a uniquely divisible abelian group. We show that the alternative quadratic functional equation f (x y−1)+ f (x y) =±2( f (x)+ f (y)) is equivalent to the quadratic functional equation f (x y−1)+ f (x y) = 2 f (x)+2 f (y) for the class of functions f : G→ G∗.

We give a complete solution for the existence problem of group divisible designs (or PBIBDs) with blocks of size k = 3, four groups of size (n, n, n, 1), and any two indices (λ1, λ2). Moreover, we introduce a construction of infinitely many group divisible designs with t groups of size n and one group of size 1. The construction technique utilizes our main result, together with some other known...

Chen, D., CC. Lindner and D.R. Stinson, Further results on large sets of disjoint group-divisible designs, Discrete Mathematics 110 (1992) 35-42. This paper is a continuation of a recent paper by Chen and Stinson, where some recursive constructions for large sets of group-divisible design with block size 3 arc presented. In this paper, we give two new recursive constructions. In particular, we ...

The linked systems of symmetric group divisible designs of type II is introduced, and several examples are obtained from affine resolvable designs and mutually UFS Latin squares. Furthermore, an equivalence between such symmetric group divisible designs and some association schemes with 5-classes is provided.