Classification of flag-transitive primitive symmetric $(v,k,\lambda)$ designs with $PSL(2,q)$ as socle

TL;DR

This paper classifies certain symmetric designs with specific automorphism group actions, showing that only a few parameter sets are possible when the group has socle PSL(2,q) and acts flag-transitively and point-primitively.

Contribution

It provides a complete classification of symmetric designs with automorphism group socle PSL(2,q) under flag-transitivity and point-primitivity conditions.

Findings

Identifies five possible parameter sets for such designs.

Establishes constraints on the structure of automorphism groups.

Advances understanding of symmetry in combinatorial designs.

Abstract

Let $\mathcal D$ be a nontrivial symmetric $(v,k,\lambda)$ design, and $G$ be a subgroup of the full automorphism group of $\mathcal D$. In this paper we prove that if $G$ acts flag-transitively, point-primitively on $\mathcal D$ and $Soc(G)= PSL(2,q)$, then D has parameters $(7, 3, 1)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 3)$ or $(15, 8, 4)$.