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

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 $(96,20,4)$.