We consider 2-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on 2-designs which are block-transitive but not necessarily flag-transitive. In particular we use the concept of the “array” of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any \(s \ge 2\), there are infinitely many 2-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length s. Moreover an exhaustive computer search, using Magma, seeking designs with \(e_1e_2e_3\) points (where each \(e_i\le 50\)) and a partition chain of length \(s=3\), produced 57 such flag-transitive designs, among which only three designs arise from our construction—so there is still much to learn.

我们考虑 2-设计，它允许一组具有标志传递性的自同构，并且留下不变量的非平凡点划分链。我们以最近关于 2 设计的工作为基础，这些设计是块传递的，但不一定是标志传递的。特别是，我们使用点子集“数组”的概念来表示点分区链；该数组描述了子集中的点在每个分区的类之间的分布。我们在数组上获得必要和充分的条件，以便子集成为这种设计的一个块。通过显式构造，我们表明对于任何\(s \ge 2\)，有无限多个 2-设计承认一个标志传递群，该群保留长度为s 的点分区的不变链。此外，使用Magma进行详尽的计算机搜索，寻找具有\(e_1e_2e_3\)点（其中每个\(e_i\le 50\) ）和长度为\(s=3\)的分区链的设计，产生了 57 个这样的标志传递设计，其中只有三个设计是我们建造的，所以还有很多东西需要学习。