Flag-transitive point-primitive quasi-symmetric $2$-designs and exceptional groups of Lie type

TL;DR

This paper investigates the symmetry properties of certain quasi-symmetric 2-designs, proving that their automorphism groups cannot have socles that are exceptional groups of Lie type, thus narrowing the possible group structures.

Contribution

It establishes that automorphism groups of these designs cannot have socles that are finite simple exceptional Lie type groups, refining the classification of such symmetries.

Findings

Socle of automorphism group cannot be an exceptional Lie type group

Automorphism groups are either affine or almost simple but exclude exceptional groups

Results restrict possible symmetry groups for these designs

Abstract

Let $\mathcal{D}$ be a non-trivial quasi-symmetric $2$-design with two block intersection numbers $x=0$ and $2\leq y\leq10$, and suppose that $G$ is an automorphism group of $\mathcal{D}$. If $G$ is flag-transitive and point-primitive, then it is known that $G$ is either of affine type or almost simple type. In this paper, we show that the socle of $G$ cannot be a finite simple exceptional group of Lie type.