Multiple transitivity except  for a system of imprimitivity

Colin D. Reid

doi:10.1515/jgth-2023-0062

Published online by De Gruyter January 19, 2024

Multiple transitivity except for a system of imprimitivity

Colin D. Reid

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2023-0062

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Abstract

Let Ω be a set equipped with an equivalence relation ∼ ; we refer to the equivalence classes as blocks of Ω. A permutation group G ≤ Sym ⁢ ( Ω ) is 𝑘-by-block-transitive if ∼ is 𝐺-invariant, with at least 𝑘 blocks, and 𝐺 is transitive on the set of 𝑘-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article, we classify the finite block-faithful 2-by-block-transitive actions. We also show that, for k ≥ 3 , there are no finite block-faithful 𝑘-by-block-transitive actions with nontrivial blocks.

Funding source: Australian Research Council

Award Identifier / Grant number: FL170100032

Funding statement: Research supported in part by ARC grant FL170100032.

Acknowledgements

I thank Tom De Medts, Michael Giudici, Bernhard Mühlherr and Hendrik Van Maldeghem for helpful comments related to this article. I also thank the anonymous referee who suggested a number of useful references and improvements.

Communicated by: Timothy C. Burness

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Received: 2023-04-24

Revised: 2023-10-15

Published Online: 2024-01-19