Symmetric $۱$-designs from $PSL_{۲}(q),$ for $q$ a power of an odd prime

Let $G = \PSL_{۲}(q)$‎, ‎where $q$ is a power of an odd prime‎. ‎Let $M$ be a maximal subgroup of $G$‎. ‎Define $\left\lbrace \frac{|M|}{|M \cap M^g|}‎: ‎g \in G \right\rbrace$ 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 $۱$-designs that admit $G$ as a permutation group of automorphisms‎.

Keywords:

‎۱-design‎ , ‎symmetric designs‎ , ‎primitive design‎ , ‎projective special linear group