Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type

Abstract

Let \({\mathcal {D}}\) be a non-trivial G-block-transitive 3-(v, k, 1) design, where \(T\le G \le \textrm{Aut}(T)\) for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group \({}^2B_2(q)\) or \(G_2(q)\) . Furthermore, if \(T={}^2B_2(q)\) then the design \({\mathcal {D}}\) has parameters \(v=q^2+1\) and \(k=q+1\) , and so \({\mathcal {D}}\) is an inverse plane of order q, and if \(T=G_2(q)\) then the point stabilizer in T is either \(\textrm{SL}_3(q).2\) or \(\textrm{SU}_3(q).2\) , and the parameter k satisfies very restricted conditions.