CIT Groups of Finite Morley Rank (I)

AbstractThis is the first of two papers whose goal is the proof of the following result: THEOREM. Let G he an infinite omega-stable group of finite rank. Assume G has involutions and that the centralizer of any ivolution is a 2-group. Then one of the following holds: (1) G has a normal, nontrivial 2-subgroup. (2) G ≃ H[formula]S where H is a definable, abelian 2′-subgroup and S is a finite Sylow 2-subgroup of G with a unique involution that acts on H hy inversion. (3) G ≃ SL2(K) for some algebraically closed field K of characteristic 2. In this paper, we show that if Case 1 fails and if the Sylow 2-subgroups are finite, then we are in the second case. We also show that when G has infinite disjoint Sylow 2-subgroups, then we are in Case 3