Flag-transitive, point-imprimitive 2-designs and direct products of symmetric groups

Jianfu Chen¹ ; Shenglin Zhou¹ ; Jiaxin Shen²

¹School of Mathematics, South China University of Technology
²School of Mathematics and Computational Science, Wuyi University

The electronic journal of combinatorics, Tome 31 (2024) no. 2

Résumé

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 $v=(k-2)^2$.

Détail

DOI : 10.37236/11002

Classification : 05E16, 05B05, 05B25, 20B25
Mots-clés : structure of 2-designs, flag-transitive automorphism groups