Carmen Amarra , Alice Devillers , Cheryl E. Praeger
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引用次数: 0
摘要
30 多年前,德兰切尔和多延证明,大小为 k 块的块变换 2 设计的自动形群可以使一个非难点分区保持不变,但前提是点的数量以 k 为界。自那时起,人们发现了有两个非难点分区的例子,它们要么形成了分区链,要么在点集中形成了网格结构。我们通过构建无穷设计族证明,对于块过渡 2 设计,不变点分区链的长度没有限制。我们引入了点集 "阵列 "的概念,它描述了点集如何与不同分区的部分相互作用,我们还获得了点集 "阵列 "相对于分区链的必要条件和充分条件,从而使其成为这种设计的区块。
Block-transitive 2-designs with a chain of imprimitive point-partitions
More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.