A parametrisation for symmetric designs admitting a flag-transitive, point-primitive automorphism group with a product action

Authors

  • Eugenia O'Reilly-Regueiro Universidad Nacional Autónoma de México, México https://orcid.org/0000-0001-5867-7258
  • José Emanuel Rodríguez-Fitta Universidad Nacional Autónoma de México, México

DOI:

https://doi.org/10.26493/1855-3974.2507.a1d

Keywords:

Symmetric-designs, flag-transitivity, primitive groups, automorphism groups of designs

Abstract

We study (v,k,λ)-symmetric designs having a flag-transitive, point-primitive automorphism group, with v = m2 and (k,λ) = t > 1, and prove that if D is such a design with m even admitting a flag-transitive, point-primitive automorphism group G, then either:

  1. D is a design with parameters ((2t+s−1)2, (2t2 − (2−s)t)/s, (t2 − t)/s2)) with s ≥ 1 odd, or

  2. G does not have a non-trivial product action.

We observe that the parameters in (1), when s = 1, correspond to Menon designs.
We also prove that if D is a (v,k,λ)-symmetric design with a flag-transitive, point-primitive automorphism group of product action type with v = ml and l ≥ 2 then the complement of D does not admit a flag-transitive automorphism group.

Published

2022-11-11

Issue

Section

Articles