Flag-transitive symmetric $2$-designs]{The symmetric $2$-$(v,k,\lambda )$ designs, with $k>\lambda \left(\lambda-3 \right)/2$, admitting a flag-transitive, point imprimitive automorphism group are known

TL;DR

This paper classifies symmetric 2-designs with specific parameters that admit a flag-transitive, point-imprimitive automorphism group, identifying only four known designs with those properties.

Contribution

It provides a complete classification of symmetric 2-designs with given parameters and automorphism group properties, confirming they are among four known designs.

Findings

Identified four known symmetric 2-designs with specified parameters.

Proved no other designs with these properties exist.

Confirmed the classification is complete for the given conditions.

Abstract

The symmetric $2$-$(v,k,\lambda )$ designs, with $k>\lambda \left(\lambda-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.