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

Abstract

This paper is a contribution to the classification of flag-transitive 4-$(v,k,\lambda )$ designs. Let $\mathcal D=({\mathcal {P}}, {\mathcal {B}})$ be a 4-$(q+1,k,\lambda )$ design with $\lambda \ge 5$ and $q+1>k>4$, $G=PSL(2,q)$ be a flag-transitive automorphism group of $\mathcal D$, $G_x$ be the stabilizer of a point $x\in {\mathcal {P}}$, and $G_B$ be the setwise stabilizer of a block $B\in {\mathcal {B}}$. Using the fact that $G_B$ must be one of twelve kinds of subgroups of PSL(2, q), up to isomorphism we get the following two results: (i) If $10\ge \lambda \ge 5$, then with the possible exception of $(G,G_x,G_B,k,\lambda )=(PSL(2,761),{E_{761}}\rtimes {C_{380}},S_4,24,7)$ or $(PSL(2,512),{E_{512}}\rtimes {C_{511}},{D_{18}},18,8)$ which remain undecided, $\mathcal D$ is a unique 4-(24, 8, 5), 4-(9, 6, 10), 4-(8, 6, 6), 4-(9, 7, 10), 4-(9, 8, 5), 4-(10, 9, 6), 4-(12, 11, 8) or 4-(14, 13, 10) design with $(G,G_x,G_B)=(PSL(2,23),$ ${E_{23}}\rtimes {C_{11}},D_8)$, $(PSL(2,8),{E_{8}}\rtimes {C_{7}},D_6)$, $(PSL(2,7),{E_{7}}\rtimes {C_{3}},D_6)$, $(PSL(2,8),{E_{8}}\rtimes {C_{7}},D_{14})$, $(PSL(2,8),{E_{8}}\rtimes {C_{7}},{E_8}\rtimes {C_7})$, $(PSL(2,9),{E_{9}}\rtimes {C_{4}},{E_9}\rtimes {C_4})$, $(PSL(2,11),{E_{11}}\rtimes {C_{5}},{E_{11}}\rtimes {C_{5}})$ or $(PSL(2,13),{E_{13}}\rtimes {C_{6}},{E_{13}}\rtimes {C_6})$ 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