An involution in a finite group is called central if it lies in the center of a 2-Sylow subgroup of G. A 2-Sylow intersection is called central if it is either trivial or contains a central involution. Suppose G is a finite simple group all of whose central 2-Sylow intersections are trivial or rank one 2-groups. It is proved that G is a known simple group. Previous results. Of basic importance is Theorem 2.1. Let z be a central involution in G. Suppose z lies in no proper 2-Sylow intersection

more »

... G. Then (z ), the normal closure ofz in G, has a center of odd order and, modulo this center, is the direct product of simple Bender groups and a 2-nilpotent group having 2-Sylow subgroup of exponent 2. z projects nontrivially on each of the components of this factor.