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Chain-imprimitive, flag-transitive 2-designs

Carmen Amarra, Alice Devillers, Cheryl E. Praeger

Abstract

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 $s \geq 2$, 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 {\sc Magma}, seeking designs with $e_1e_2e_3$ points (where each $e_i\leq 50$) 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.

Chain-imprimitive, flag-transitive 2-designs

Abstract

We consider -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 -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 , there are infinitely many -designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length . Moreover an exhaustive computer search, using {\sc Magma}, seeking designs with points (where each ) and a partition chain of length , produced such flag-transitive designs, among which only three designs arise from our construction -- so there is still much to learn.
Paper Structure (12 sections, 12 theorems, 41 equations, 1 table)

This paper contains 12 sections, 12 theorems, 41 equations, 1 table.

Table of Contents

  1. Introduction
  2. Comments on our constructions and results
  3. Chain imprimitivity
  4. Chain structure on a set
  5. Iterated wreath product of groups
  6. Array of a subset
  7. Block-transitive, chain-imprimitive $2$-designs
  8. Point subsets with uniform array functions
  9. A general construction for flag-transitive, $s$-chain-imprimitive $2$-designs
  10. Proof of Theorem \ref{['thm:ft']}
  11. Explicit constructions for flag-transitive, $s$-chain-imprimitive $2$-designs
  12. Further examples

Key Result

Theorem 1.1

For any integer $s \geq 2$, there exist infinitely many $2$-designs ${\mathscr{D}}=({\mathscr{P}},{\mathscr{B}})$ such that some subgroup $G\leq {\mathrm{Aut}}({\mathscr{D}})$ acts flag-transitively on ${\mathscr{D}}$ and $s$-chain-imprimitively on ${\mathscr{P}}$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 13 more