Flag-transitive automorphism groups of $2$-designs with $\lambda \ge (r,\lambda {)}^2$ are not product type

In this paper we show that a flag-transitive automorphism group $G$ of a non-trivial $2$-$(v,k,\lambda )$ design with $\lambda \ge (r,\lambda {)}^2$ is not of product action type. In conclusion, $G$ is a primitive group of affine or almost simple type.