Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.

The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups L 3 (q) L₃ (q) with q ≡ − 1 (mod 4), U 3 (q) q - 1 4, U₃ (q) with q ≡ 1 (mod 4) q 1 4, or M 11 M₁₁. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of Z 2 n Z{₂䂞} and Z 2, n ≧ 2 Z₂, n 2.