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 $\left\lbrace \frac{|M|}{|M \cap M^g|}‎: ‎g \in G \right\rbrace$ 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 $1$-designs that admit $G$ as a permutation group of automorphisms‎.

Keywords

• ‎1-design‎
• ‎symmetric designs‎
• ‎primitive design‎
• ‎projective special linear group