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

Abstract

In this article, we investigate symmetric designs admitting flag-transitive and point-primitive affine automorphism groups. We prove that if a flag-transitive automorphism group G of a symmetric $(v,k,\lambda )$ design with λ prime is point-primitive of affine type, then $G=2^6:S_6$ and $(v,k,\lambda )=(16,6,2)$, or G is a subgroup of $A\Gamma L_1\left(q\right)$ for some odd prime power q. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with λ 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 $A\Gamma L_1\left(q\right)$.