Transactions on Combinatorics | 2021
Symmetric $1$-designs from $PSL_{2}(q),$ for $q$ a power of an odd prime
Abstract
Let $G = PSL_{2}(q)$\u200e, \u200ewhere $q$ is a power of an odd prime\u200e. \u200eLet $M$ be a maximal subgroup of $G$\u200e. \u200eDefine $leftlbrace frac{|M|}{|M cap M^g|}\u200e: \u200eg in G rightrbrace$ 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\u200e, \u200ewe construct all primitive symmetric $1$-designs that admit $G$ as a permutation group of automorphisms\u200e.