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

Abstract

Consider the direct product of symmetric groups $S_c\times S_n$ and its natural action on $\mathcal{P}=C\times N$, where $|C|=c$ and $|N|=n$. We characterize the structure of 2-designs with point set $\mathcal{P}$ admitting flag-transitive, point-imprimitive automorphism groups $H\leq S_c\times S_n$. As an example of its applications, we show that $H$ cannot be any subgroup of $D_{2c}\times S_n$ or $S_c\times D_{2n}$. Besides, some families of 2-designs admitting flag-transitive automorphism groups $S_c\times S_n$ are constructed by using complete bipartite graphs and cycles. Two families of these also admit flag-transitive, point-primitive automorphism groups $S_c\wr S_2,$ a family of which attain the Cameron-Praeger upper bound $v=(k-2)^2$.