Flag-Transitive, Point-Imprimitive 2-Designs and Direct Products of Symmetric Groups

Consider the direct product of symmetric groups Sc Sₙ and its natural action on P=C N, where |C|=c and |N|=n. We characterize the structure of 2-designs with point set P admitting flag-transitive, point-imprimitive automorphism groups H Sc Sₙ. As an example of its applications, we show that H cannot be any subgroup of D₂₂ Sₙ or Sc D₂₍. Besides, some families of 2-designs admitting flag-transitive automorphism groups Sc Sₙ are constructed by using complete bipartite graphs and cycles. Two families of these also admit flag-transitive, point-primitive automorphism groups Sc S₂, a family of which attain the Cameron-Praeger upper bound v= (k-2) ².