Almost simple groups as flag-transitive automorphism groups of 2-designs with {\lambda} = 2

TL;DR

This paper proves the non-existence of certain 2-designs with specific symmetry properties involving almost simple groups, and classifies all such designs with flag-transitive automorphism groups.

Contribution

It establishes the non-existence of 2-designs with =2 admitting flag-transitive almost simple automorphism groups with exceptional Lie type socles, and classifies all such designs.

Findings

No such 2-designs exist with exceptional Lie type socles.

Classified all 2-designs with =2 and flag-transitive almost simple automorphism groups.

Identified infinite family of 2-designs with specific parameters and automorphism groups.

Abstract

In this article, we study $2$-designs with $\lambda=2$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type, and we prove that such a $2$-design does not exist. In conclusion, we present a classification of $2$-designs with $\lambda=2$ admitting flag-transitive and point-primitive automorphism groups of almost simple type, which states that such a $2$-design belongs to an infinite family of $2$-designs with parameter set $((3^n-1)/2,3,2)$ and $X=PSL_n(3)$ for some $n\geq 3$, or it is isomorphic to the $2$-design with parameter set $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(10,4,2)$, $(11,5,2)$, $(28,7,2)$, $(28,3,2)$, $(36,6,2)$, $(126,6,2)$ or $(176,8,2)$.