Block-transitive 2-designs with a chain of imprimitive point-partitions

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

doi:10.1016/j.jcta.2024.105866

Journal of Combinatorial Theory, Series A

2024

DOI: 10.1016/j.jcta.2024.105866

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Block-transitive 2-designs with a chain of imprimitive point-partitions

Carmen Amarra,

Alice Devillers,

Cheryl E. Praeger

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2024

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Chain-imprimitive, flag-transitive 2-designs

Amarra,

Devillers,

Praeger

2024

Des. Codes Cryptogr.

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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 $$s \ge 2$$ s ≥ 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 $$e_1e_2e_3$$ e 1 e 2 e 3 points (where each $$e_i\le 50$$ e i ≤ 50 ) and a partition chain of length $$s=3$$ s = 3 , produced 57 such flag-transitive designs, among which only three designs arise from our construction—so there is still much to learn.

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Chain-imprimitive, flag-transitive 2-designs

Amarra,

Devillers,

Praeger

2024

Des. Codes Cryptogr.

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View full text Add to dashboard Cite

show abstract

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Block-transitive 2-designs with a chain of imprimitive point-partitions

Cited by 1 publication

References 27 publications

Chain-imprimitive, flag-transitive 2-designs

Chain-imprimitive, flag-transitive 2-designs

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