Paul-Hermann Zieschang

An algebraic approach to association schemes

Basic results.- Decomposition theory.- Algebraic prerequisites.- Representation theory.- Theory of generators.

Sufficient conditions for a scheme to originate from a group

We present two different sufficient conditions for a scheme to originate from a transitive permutation group.

Trends and lines of development in scheme theory

The concept of an association scheme is one of those mathematical concepts which were utilized as technical tools in various different mathematical areas for a long time before becoming the subject of a theory in their own right. The significance of symmetric schemes, for instance, in the design of (statistical) experiments was recognized as early as the first half of the last century. Coding theory has been associated with commutative schemes for more than three decades, and polynomial schemes have provided the language in which major topics in algebraic graph theory are communicated for about twenty years. The notion of a scheme itself, however-a notion which, if considered in its full generality, generalizes not only the notion of a group but also the notion of a Moore geometry and that of a building in the sense of Jacques Tits-has been considered as the subject of an abstract theory in itself only relatively recently. It is the purpose of this article to reflect on the lines of development, the Entwicklungslinien, along which abstract scheme theory has been developed so far and along which scheme theory might be developed in the future. The emphasis will be not so much on completeness as on an attempt to show exemplarily how naturally and organically the structure theory of association schemes arises from certain aspects in group theory.

On Dihedral Configurations and their Coxeter Geometries

Within the theory of homogeneous coherent configurations, the dihedral configurations play the role which is played by the finite dihedral groups in the theory of finite groups. Imitating Tits? construction of a geometry from a set of subgroups of a given group, we assign a geometry of rank 2 to each dihedral configuration, its `Coxeter geometry?. (Each finite generalized polygon is a Coxeter geometry in this sense.)Apart from general results on the relationship between dihedral configurations and their Coxeter geometries, we settle completely the (ordinary) representation theory of the dihedral configurations of rank 7. We obtain three major classes. The Coxeter geometries of the first class are exactly the non-symmetric 2-designs withI=1. The other two classes lead to questions which require a further combinatorial treatment.

On the Normal Structure of NonCommutative Association Schemes of Rank 6

The concept of an association scheme is a far-reaching generalization of the notion of a group. Many group theoretic facts have found a natural generalization in scheme theory. One of these generalizations is the observation that, similar to groups, association schemes of finite order are commutative if they have at most five elements and not necessarily commutative if they have six elements. While there is (up to isomorphism) only one noncommutative group of order 6, there are infinitely many pairwise non-isomorphic noncommutative association schemes of finite order with six elements. (Each finite projective plane provides such a scheme, and non-isomorphic projective planes yield non-isomorphic schemes.) In this note, we investigate noncommutative schemes of finite order with six elements which have a symmetric normal closed subset with three elements. We take advantage of the classification of the finite simple groups.

Cayley graphs of finite groups

Abstract Let Γ ( G , T ) denote the Cayley graph of a finite group G with respect to a normal subset T of G −{1}. We compute explicitly the spectrum of Γ ( G , T ) in terms of complex character values. This allows us to determine the number of paths of length n between two arbitrary vertices of Γ ( G , T ) for each n ∈ N . Finally, we apply these results to obtain the following theorem. Suppose that G is a finite group which contains a cyclic self-normalizing subgroup W of order pq , where p and q are two different odd prime numbers. Define W 0 to be the set of all elements of order pq of W and let T :=∪ gϵG W 0 g . Then for any n ∈ N , the number of paths of length n between two adjacent vertices of the Cayley graph Γ ( G , T ) does not depend on the choice of the two adjacent vertices. Moreover, the rank of Γ ( G , T ) is 4 or 5.

On Maximal Closed Subsets in Association Schemes

We offer a sufficient condition for a closed subset in an association scheme to be maximal. The result generalizes naturally the well-known (group-theoretical) fact that the one-point-stabilizer of a flag transitive automorphism group of a 2-design with?=1 must be a maximal subgroup. In finite group theory, there exist a lot of (important) sufficient conditions for a subgroup to be maximal. In the theory of association schemes, the condition which we offer here seems to be the first one which guarantees that a closed subset of the set of relations of an association scheme is maximal.

Block Designs Admitting Flag Transitive Groups of Automorphisms

Publisher Summary This chapter presents block designs admitting flag transitive groups of automorphisms. The chapter discusses finite block designs with λ = 1 admitting a flag transitive group G of automorphisms such that (* ) each two point stabilizer G xy acts trivially on the unique block determined by x and y.