Two theorems on groups of characteristic 2-type

made the following conjecture: suppose that G is a finite simple group which is simultaneously of characteristic 2-type and characteristic 3-type. Then G is isomorphic to one of PSp(4:, 3), G 2 (3) or /^). In this paper, we prove two results which, taken together, yield a proof of this conjecture under the additional assumption that G has 2-local 3-rank at least 2. 1* Introduction* In this paper we study finite simple groups, all of whose 2-local and 3-local subgroups are 2-constrained and 3-constrained respectively. The results we obtain are extensions of Thompson's theorem ES, and their relation to simple groups of characteristic 2-type is entirely analogous to the relation of theorem ES to simple iV-groups.