Flag-transitive, point-imprimitive symmetric $2$-$(v,k,\lambda )$ designs with $k>\lambda \left(\lambda-3 \right)/2$

TL;DR

This paper classifies certain symmetric 2-designs with flag-transitive, point-imprimitive automorphism groups, showing they are isomorphic to known designs under specific parameter conditions.

Contribution

It proves that symmetric 2-designs with given parameters and conditions are isomorphic to two known designs, extending previous classification results.

Findings

Designs are isomorphic to known (45,12,3) or (96,20,4) designs.

Conditions on parameters restrict possible designs.

Classification under specific parameter bounds.

Abstract

Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a symmetric $2$-$(v,k,\lambda )$ design admitting a flag-transitive, point-imprimitive automorphism group $G$ that leaves invariant a non-trivial partition $\Sigma$ of $\mathcal{P}$. Praeger and Zhou \cite{PZ} have shown that, there is a constant $k_{0}$ such that, for each $B \in \mathcal{B}$ and $\Delta \in \Sigma$, the size of $\left\vert B \cap \Delta \right \vert$ is either $0$ or $k_{0}$. In the present paper we show that, if $k>\lambda \left(\lambda-3 \right)/2$ and $k_{0} \geq 3$, $\mathcal{D}$ is isomorphic to one of the known flag-transitive, point-imprimitive symmetric $2$-designs with parameters $(45,12,3)$ or $(96,20,4)$.