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In this paper we show that a flag-transitive automorphism group $G$ of anon-trivial $2$-$(v,k,\lambda)$ design with $\lambda\geq (r, \lambda)^2$ is notof product action type. In conclusion, $G$ is a primitive group of affine oralmost simple type.

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