Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs

TL;DR

This paper classifies symmetric designs with prime λ that have flag-transitive, point-primitive affine automorphism groups, identifying specific groups and parameters, and linking some to projective spaces and affine linear groups.

Contribution

It provides a classification of such symmetric designs, detailing the automorphism groups and parameters, and extends understanding of their structure and symmetry properties.

Findings

Automorphism group is either $2^{6}{:} ext{S}_6$ with parameters (16,6,2) or a subgroup of $ ext{A} ext{G} ext{L}_1(q)$.

Classified designs include certain projective spaces with specific parameters.

Designs are either projective spaces or have parameters related to prime powers with affine automorphism groups.

Abstract

In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,\lambda)$ design with $\lambda$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,\lambda)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $\lambda$ prime, which says that such an incidence structure is a projective space $\mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$.