Block-transitive designs with a poset of imprimitive partitions

TL;DR

This paper investigates block designs with automorphism groups that are transitive on points and blocks, invariant under a finite poset of partitions, using wreath product theory to establish conditions for such designs and providing explicit examples.

Contribution

It generalizes previous results by characterizing block-transitive designs with automorphisms preserving a poset of partitions using wreath product theory.

Findings

Necessary and sufficient conditions for block sets to form 2-designs under the automorphism group.

Explicit infinite families of 2-designs for specific posets.

Extension of known results to more complex poset structures.

Abstract

We study block designs which admit an automorphism group that is transitive on blocks and points, and leaves invariant every partition in a given finite poset of partitions of the point set. The full stabiliser $G$ of all the partitions in the poset is a generalised wreath product. We use the theory of generalised wreath products to give necessary and sufficient conditions, in terms of the `array' of a point-subset $B$, for the set of $G$-images of $B$ to form the block-set of a $G$-block-transitive $2$-design. This generalises previous results for the special cases where the poset is a chain or an anti-chain. We also give explicit infinite families of examples of $2$-designs for each poset involving three proper partitions, and for the famous $N$-poset with four partitions. (Posets with two proper partitions have been treated previously.) This suggests the problem of finding explicit examples for other posets.