Reduction for flag-transitive symmetric designs with $k>\lambda(\lambda-2)$

TL;DR

This paper classifies flag-transitive automorphism groups of symmetric designs with specific parameter constraints, showing they are either affine or almost simple types when the group is point-primitive.

Contribution

It extends previous results by analyzing the case where the automorphism group is point-primitive, using the O'Nan-Scott Theorem to classify the group types.

Findings

G must be of affine or almost simple type when point-primitive

Confirmed the structure of automorphism groups under given parameter conditions

Extended classification results for symmetric designs with flag-transitive automorphisms

Abstract

Let $G$ be a flag-transitive automorphism group of a $(v,k,\lambda)$ symmetric design $\mathcal{D}$ with $k>\lambda(\lambda-2)$. O'Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters $(v,k,\lambda)=(\lambda^2(\lambda+2),\lambda(\lambda+1),\lambda)$. In the present paper, we consider the case that $G$ is point-primitive. By applying the O'Nan-Scott Theorem, we prove that $G$ must be of affine type or almost simple type.