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

Let D be a non-trivial G-block-transitive 3-(v, k, 1) design, where T ≤ G ≤ 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 2 B 2 (q) or G 2 (q) . Furthermore, if T = 2 B 2 (q) then the design D has parameters v = q 2 + 1 and k = q + 1 , and so D is an inverse plane of order q, and if T = G 2 (q) then the point stabilizer in T is either SL 3 (q). 2 or SU 3 (q). 2 , and the parameter k satisfies very restricted conditions. [ABSTRACT FROM AUTHOR]