Abstract
In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.
Highlights
A symmetric (v, k, λ) design is an incidence structure D = (P, B) consisting of a set P of v points and a set B of v blocks such that every point is incident with exactly k blocks, and every pair of blocks is incident with exactly λ points
An automorphism of a symmetric design D is a permutation of the points permuting the blocks and preserving the incidence relation
An automorphism group G of D is called flag-transitive if it is transitive on the set of flags of D
Summary
Zhou in [37] proved that if D is a nontrivial symmetric (v, k, λ) designs with λ prime and G is a flag-transitive and point-primitive automorphism group of D, G must be of affine or almost simple type. We have studied nontrivial symmetric (v, k, λ) design with k prime admitting flagtransitive almost simple automorphism groups [2], and proved that such a design is either a projective space, or it has a parameters set (11, 5, 2). Praeger and Zhou [34] study symmetric (v, k, λ) designs admitting flag-transitive and point-imprimitive designs, and running through the potential parameters, we can only exclude one possibility, and so Corollary 2 below is an immediate consequence of their result [34, Theorem 1.1].
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