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

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 $(v,k,\hat{I}»)$ design with $\hat{I}»$ prime is point-primitive of affine type, then $G=2^6:{\mathrm{S}}_6$ and $(v,k,\hat{I}»)=(16,6,2)$, or $G$ is a subgroup of ${\mathrm{A}\hat{\mathrm{I}}“\mathrm{L}}_1(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $\hat{I}»$ 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 ${\mathrm{A}\hat{\mathrm{I}}“\mathrm{L}}_1(q)$.