Symmetric $1$-designs from $PSL_{2}(q),$ for $q$ a power of an odd prime

Abstract

Let $G = PSL_{2}(q)$‎, ‎where $q$ is a power of an odd prime‎. ‎Let $M$ be a maximal subgroup of $G$‎. ‎Define $leftlbrace frac{|M|}{|M cap M^g|}‎: ‎g in G rightrbrace$ to be the set of orbit lengths of the primitive action of $G$ on the conjugates of a maximal subgroup $M$ of $G.$ By using a method described by Key and Moori in the literature‎, ‎we construct all primitive symmetric $1$-designs that admit $G$ as a permutation group of automorphisms‎.