Block-transitive \(3\)-\((v, k, 1)\) designs on exceptional groups of Lie type (Q6570592)

A \(t\)-\((v, k, \lambda)\) design \(\mathcal{D}\) is a set of \(v\) points \(\mathcal{P},\) together with a collection of \(k\)-element subsets of \(\mathcal{P}\) called blocks, such that any \(t\)-subset of \(\mathcal{P}\) lie together in precisely \(\lambda\) blocks. The number of blocks is denoted by \(b\) and the number of blocks containing a point by \(r.\)\N\NLet \(\mathcal{D}\) be a non-trivial \(G\)-block-transitive \(3\)-\((v, k, 1)\) design, where \(S \leq G \leq \mathrm{Aut}(S)\) for some finite non-abelian simple group \(S\). The paper under review proves that if \(S\) is a simple exceptional group of Lie type, then \(S\) is either the Suzuki group \(^2 B_2(q)\) or \(G_2(q).\) Using a reduction result the authors prove that for \(S={^2B}_2(q)\) the design \(\mathcal{D}\) is an inversive plane of order \(q\) and its parameters are given. In the case \(S = G_2(q),\) the point stabilizer in \(S\) is shown to be either \(\mathrm{SL}_3(q) \cdot 2\) or \(\mathrm{SU}_3(q) \cdot 2\) and the parameter \(k\) of \(\mathcal{D}\) satisfies certain conditions.