Sporadic simple groups as flag-transitive automorphism groups of symmetric designs

TL;DR

This paper classifies symmetric designs with flag-transitive automorphism groups whose socle is a sporadic simple group, identifying exactly seven such designs, including one with a point-imprimitive group and six with point-primitive groups.

Contribution

It provides the first complete classification of symmetric designs with flag-transitive automorphism groups based on sporadic simple groups.

Findings

Exactly seven symmetric designs with such automorphism groups are identified.

One design admits a point-imprimitive automorphism group.

Six designs admit point-primitive automorphism groups.

Abstract

In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag-transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point-imprimitive automorphism group and the remaining are point-primitive.