More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size , could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of . 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.

中文翻译：

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具有原始点划分链的块传递 2 设计

30 多年前，Delandtsheer 和 Doyen 证明，块传递 2 设计的自同构群（块大小为 ）可以使非平凡点划分保持不变，但前提是点的数量受 限制。从那时起，已经发现了存在两个非平凡点分区的示例，它们要么形成分区链，要么在点集上形成网格结构。我们通过构建无限族设计证明，块传递 2 设计的不变点划分链的长度没有限制。我们引入了点集“数组”的概念，它描述了该集合如何与各个分区的部分相互作用，并且我们根据点集的“数组”相对于分区获得了必要和充分的条件链，因为它是这样设计的一个块。