On flag-transitive imprimitive 2-designs

TL;DR

This paper explicitly bounds the parameters of flag-transitive, point-imprimitive 2-designs and classifies small examples, including a previously unknown design with specific automorphism group.

Contribution

It derives explicit polynomial bounds on parameters of flag-transitive, point-imprimitive 2-designs and classifies all small cases with fewer than 100 points.

Findings

Explicit polynomial bounds for $k$ and $v$ in terms of $\lambda$.

Complete classification of 2-designs with fewer than 100 points.

Identification of a unique, previously unknown design with automorphism group ${ m Sym}(6)$.

Abstract

In 1987, Huw Davies proved that, for a flag-transitive point-imprimitive $2$-$(v,k,\lambda)$ design, both the block-size $k$ and the number $v$ of points are bounded by functions of $\lambda$, but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of $\lambda$ bounding $k$ and $v$. For $\lambda\leq 4$ we obtain a list of `numerically feasible' parameter sets $v, k, \lambda$ together with the number of parts and part-size of an invariant point-partition and the size of a nontrivial block-part intersection. Moreover from these parameter sets we determine all examples with fewer than $100$ points. There are exactly eleven such examples, and for one of these designs, a flag-regular, point-imprimitive $2-(36,8,4)$ design with automorphism group ${\rm Sym}(6)$, there seems to be no construction previously available in the literature.