Flag-transitive $4$-designs and $PSL(2,q)$ groups

Abstract

This paper considers flag-transitive $4$-$(q+1,k,\lambda )$ designs with $\lambda \ge 5$ and $q+1>k>4$. Let the automorphism group of a design $\mathcal{D}$ be a simple group $G=PSL(2,q)$. Depend on the fact that the setwise stabilizer $G_B$ must be one of twelve kinds of subgroups, up to isomorphism we get the following two results. (i) If $10\ge \lambda \ge 5$, then except $(G,G_x,G_B,k,\lambda )=(PSL(2,761),E_{761}⋊C_{380},S_4,24,7)$ or $(PSL(2,512),E_{512}⋊C_{511},D_{18},18,8)$ undecided, $\mathcal{D}$ is a $4$-$(24,8,5)$, $4$-$(9,8,5)$, $4$-$(8,6,6)$, $4$-$(10,9,6)$, $4$-$(9,6,10)$, $4$-$(9,7,10)$, $4$-$(12,11,8)$ or $4$-$(14,13,10)$ design with $G_B=D_8$, $E_8⋊C_7$, $D_6$, $E_9⋊C_4$, $PSL(2,2)$, $D_{14}$, $E_{11}⋊C_5$ or $E_{13}⋊C_6$ respectively. (ii) If $\lambda >10$, $G_B=A_4$, $S_4$, $A_5$, $PGL(2,q_0)$($g>1$ even) or $PSL(2,q_0)$, where ${q_0}^g=q$, then there is no such design.

References in corpus (1)

• A Census Of Highly Symmetric Combinatorial Designs