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<p>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 <i>G</i> 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 <i>B</i>, for the set of <i>G</i>-images of <i>B</i> to form the block-set of a <i>G</i>-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 <i>N</i>-poset with four partitions. (Posets with two proper partitions have been treated previously.) This suggests the problem of finding explicit examples for other posets.</p>

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Block-transitive designs with a poset of imprimitive partitions

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  • 发表日期：

    2026年7月22日

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  • Carmen Amarra,
  • Alice Devillers,
  • Cheryl E. Praeger

摘要

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.