Groups of Finite Morley rank with Strongly Embedded Sub-groups

JOUR NAL OF ALGEBR A AR TICLE NO . 180, 778]807 Ž1996. 0093 Groups of Finite Morley R ank with Strongly Embedded Subgroups Tuna Altinel Department of Mathematics, Rutgers Uni¨ ersity, New Brunswick, New Jersey 08903 Communicated by Gernot Stroth R eceived February 5, 1995 1. INTR ODUCTION The central problem in the area of groups of finite Morley rank is the Cherlin ]Z il’ber conjecture, which states that an infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. This conjecture can be divided into three parts. Conjecture 1. There exist no nonsolvable connected groups of finite Morley rank all of whose proper definable connected subgroups are nilpotent; such groups are called bad groups. Conjecture 2. There exists no structure ² K, q, ? , A: of finite Morley rank where K is an algebraically closed field and A is an infinite proper definable subgroup of the multiplicative group of K; such structures are called bad fields. Conjecture 3. A simple group of finite Morley rank which does not have definable bad sections and in which no bad fields are interpretable is an algebraic group over an algebraically closed field. In connection with Conjecture 3, one makes the following definition: D EFINITION 1.1. A group of finite Morley rank which does not have definable bad sections and in which no bad fields are interpretable is called a tame group. In order to classify simple tame groups of finite Morley rank a program based on ideas borrowed from finite group theory has been developed by Alexander Borovik. This paper is part of this program. 778 0021-8693r96 $18.00 Copyright Q 1996 by Academ ic P ress, Inc. All rights of reproduction in any form reserved. GR OUPS OF FINITE MOR LEY R ANK 779 There are striking similarities between finite groups and groups of finite Morley rank. Indeed, the very first step in attacking Conjecture 3 is borrowed from finite group theory; one analyzes a counterexample of minimal rank to the statement of Conjecture 3. The proper, infinite, definable, simple sections of such a counterexample are algebraic groups over algebraically closed fields. The following definitions are convenient: D EFINITION 1.2. A group of finite Morley rank whose infinite, definable, simple sections are algebraic groups over algebraically closed fields is called a K-group. D EFINITION 1.3. A group of finite Morley rank whose proper, definable subgroups are K-groups is called a K *-group. Therefore, in order to have an affirmative answer to Conjecture 3, it is enough to prove that the following conjecture is true: Conjecture 4. A simple, tame, K *-group of finite Morley rank is an algebraic group over an algebraically closed field. Local group theoretic methods and notions from finite group theory have natural generalizations to the context of groups of finite Morley rank and they are expected to be very useful in the analysis of simple tame K *-groups. One such concept is that of a strongly embedded subgroup: D EFINITION 1.4. A proper definable subgroup M of a group G of finite Morley rank is said to be a strongly embedded subgroup if it satisfies the following conditions: Ži. M contains involutions. Žii. For every g g G _ M, M l M g does not contain involutions. Finite simple groups with strongly embedded subgroups were classified by Bender in w4x. In the situation of groups of finite Morley rank, the natural conjecture is the following: Conjecture 5. An infinite simple group of finite Morley rank with a strongly embedded subgroup is isomorphic to PSL2 Ž K ., where K is an algebraically closed field of characteristic 2. In this paper we prove the following theorem: T HEOR EM 1.5. Let G be an infinite, simple, K *-group of finite Morley rank with a strongly embedded subgroup M. Assume that the Sylow 2-subgroups of G ha¨ e infinitely many commuting in¨ olutions. Then M is sol¨ able. If, in addition, G is tame, then it is isomorphic to PSL2 Ž K ., where K is an algebraically closed field of characteristic 2. 780 TUNA ALTINEL In the classification of finite simple groups, strongly embedded subgroups were an important tool in analyzing various configurations. A similar use of the concept of strongly embedded subgroup is anticipated in the classification of simple tame groups of finite Morley rank and these expectations have been to some extent justified by recent work of Borovik, Cherlin, and the present author on the elimination of simple, tame, K *-groups of finite Morley rank of mixed type. It is known that the connected component of a Sylow 2-subgroup of a group of finite Morley rank is the central product of a definable, nilpotent group of bounded exponent Ža unipotent-2 group., and a divisible abelian 2-group Ža 2-torus.. Accordingly, we make the following definitions: D EFINITION 1.6. Ži. A group of finite Morley rank is said to be of e¨ en type if its Sylow 2-subgroups are of bounded exponent. Žii. A group of finite Morley rank is said to be of odd type if the connected component of any Sylow 2-subgroup is a 2-torus. Žiii. A group of finite Morley rank is said to be of mixed type if it is not of one of the above types. Simple algebraic groups over algebraically closed fields are either of even type or of odd type depending on whether the characteristic of their base field is 2 or not. In particular, they are not of mixed type. The work toward the elimination of simple, tame, K *-groups of finite Morley rank of mixed type is based on the construction of a strongly embedded subgroup in a hypothetical simple group of mixed type. This construction yields a contradiction because groups of finite Morley rank with strongly embedded subgroups cannot be of mixed type. The organizations of the paper is as follows. In the second section we review the axiomatization of groups of finite Morley rank as introduced by Borovik and then cover the necessary background from group theory and in particular the theory of groups of finite Morley rank. The third section is devoted to a discussion of some basic properties of groups of finite Morley rank with strongly embedded subgroups. The proof of Theorem 1.5 starts in the fourth section, where we prove the solvability of the strongly embedded subgroup. In the fifth section we prove that the strongly embedded subgroup does not have an infinite, definable, normal, 2 H subgroup. In the sixth section we finish the proof using ideas and results from w11x. In particular Fact 2.56 below will enable us to recognize the group PSL2 Ž K . at the end. GR OUPS OF FINITE MOR LEY R ANK 781 2. PR ELIMINAR IES In this section we will review the necessary background from model theory and group theory. Our reference for general model theoretic notions, such as first-order theories, models, definability, interpretability, and v-stability is w14x. For the existing theory of groups of finite Morley rank we refer the reader to w8, 21x. We will occasionally use notions and results from the theory of linear algebraic groups; our reference for this is w15x. A group G is said to be of finite Morley rank if it is interpretable in a structure of finite Morley rank Ži.e., a structure whose first-order theory has finite Morley rank .. It is also possible to give an axiomatic treatment of groups of finite Morley rank, as follows. Let M be a structure with underlying set M. A function rk which assigns a natural number to every interpretable subset of M n is called a rank function if it satisfies the following axioms: Axiom A Ž Monotonicity of rank .. rkŽ A. G n q 1 if and only if there are infinitely many pairwise disjoint, nonempty, interpretable subsets of A each of rank at least n. Axiom B Ž Definability of rank .. If f : A ª B is an interpretable function, then for each n g N the set  b g B : rkŽ fy1 Ž b .. s n4 is interpretable. Axiom C Ž Additi¨ ity of rank .. If f : A ª B is a surjective interpretable function, and if there exists n g N such that for all b g B, rkŽ fy1 Ž b .. s n then rkŽ A. s rkŽ B . q n. Axiom D Ž Elimination of infinite quantifiers.. If f : A ª B is an interpretable function, then there is m g N such that for any b g B, fy1 Ž b . is infinite whenever it contains at least m elements. A structure which admits such a rank function is called a ranked structure. A ranked structure M s ² G, ? , y1 , 1, . . . :, where ² G, ? , y1 , 1: is a group, is a ranked group. Poizat proved ŽCorollaire 2.14 and Theoreme ´ ` 2.15 of w21x. that ranked groups and groups of finite Morely rank coincide. In this paper, we will use the above axioms as the basis for our analysis of groups of finite Morley rank. In the context of algebraic geometry Morley rank coincides with algebraic dimension. There is also a rough analog of the number of irreducible components, called the degree of an interpretable set: D EFINITION 2.1. A nonempty interpretable set is said to be of degree 1 if for any interpretable subset B : A, either rkŽ B . - rkŽ A. or rkŽ A _ B . - rkŽ A.. A is said to be of degree d if it can be written as the disjoint union of d interpretable sets of degree 1 and of rank rkŽ A.. 782 TUNA ALTINEL An interpretable set is said to be irreducible if its degree is 1. We quote a few basic facts which follow from Axioms A]D: F ACT 2.2 w8, Lemma 4.1x. Let f : A ª B be an interpretable map, and A1 , B1 be interpretable subsets of A and B, respecti¨ ely. Then the restriction of f to A1 is an interpretable map and fy1 Ž B1 . is an interpretable set. F ACT 2.3 w8, Lemma 4.10x. If B and C are interpretable sets then rk Ž B j C . s maxŽ rk Ž B . , rk Ž C . . . F ACT 2.4 w8, Exercise 14, p. 65x. Then Let f : A ª B be an interpretable map. rk Ž A . F rk Ž B . q sup b g B rk Ž fy1 Ž b . . . Proof. Let f be as in the statement. Let n s sup b g B rkŽ fy1 Ž b ... We may assume that f is surjective. We define the sets Bi , where 1 F i F n q 1, as follows: Bi s  b g B : rkŽ fy1 Ž b .. s n y i q 14 . By Axiom B these are interpretable sets. Let A i s fy1 Ž Bi .. Each A i is interpretable by Fact 2.2. By Fact 2.2, the restriction of f to A i is interpretable. Therefore by Axiom C, on each of these sets we have rkŽ A i . s rkŽ Bi . q n y i q 1. As A s "1 F i F nq1 A i , rkŽ A. s rkŽ A j . for some j such that 1 F j F n q 1 by Fact 2.3. Then rkŽ A. s rkŽ A j . s rkŽ Bj . q n y j q 1 F rkŽ B . q n. F ACT 2.5 w8, Lemma 4.17x. Let f : A ª B be an interpretable bijection between two interpretable sets A and B. Then rkŽ A. s rkŽ B . and degŽ A. s degŽ B .. F ACT 2.6 w8, Exercise 12, p. 65x. Let G be a group of finite Morley rank. Let H be a definable subgroup of G. Then the left coset space GrH is interpretable and rkŽ G . s rkŽ GrH . q rkŽ H .. Proof. As H is a definable subgroup, being in the same left Žor right . coset of H is a definable equivalence relation. Therefore, the first assertion follows. As a result of this, the canonical map from G onto GrH is an interpretable map. The fibers of the canonical map, i.e., the cosets of H in G, are definable by Fact 2.2. As left translation is a definable bijection, by Fact 2.5, all the cosets of H in G have the same rank. We can therefore apply Axiom C to conclude that the second assertion holds. Fact 2.6 has a useful corollary: F ACT 2.7 w8, Exercise 12, p. 65x. Let G be a group of finite Morley rank and f : G ª H an interpretable group homomorphism. Then rkŽ G . s rkŽ f Ž G .. q rkŽker Ž f ... GR OUPS OF FINITE MOR LEY R ANK 783 The following fact is also in the same spirit. F ACT 2.8 w8, Exercise 13, p. 65x. Let G be a group of finite Morley rank. Let x g G. Then rkŽ G . s rkŽ x G . q rkŽ CG Ž x ... A group G of finite Morley rank satisfies the descending chain condition on definable subgroups. This makes it possible to define the minimal definable subgroup of finite index in G, called the connected component of G, which is denoted by G 0 . G 0 is stable under the action of definable group automorphisms. A group G of finite Morley rank is said to be connected if G s G 0 . The following result shows that connected groups are actually irreducible sets: F ACT 2.9 w10x. G is connected if and only if degŽ G . s 1. In the proof of Theorem 1.5, we will also have to deal with subsets of a group of finite Morley rank which are not definable using the definable closure. D EFINITION 2.10. Let X be a subset of G. The intersection of all the definable subgroups of G containing X is called the definable closure of X and is denoted by dŽ X .. By the descending chain condition on definable subgroups, the definable closure of a subset exists and is definable. Using the definable closure one can also talk about the connected component of a subgroup which is not necessarily definable: D EFINITION 2.11. If X is any subgroup of G then the connected component of X, denoted by X 0 , is defined to be X l dŽ X . 0 . A subgroup H of a group G of finite Morley rank is said to be connected if H s H 0 . The proof of Theorem 1.5 makes use of a variety of facts about groups of finite Morley rank, notably Zil’ber’s Indecomposability Theorem. D EFINITION 2.12. Let X be a definable subset of G. Then X is said to be indecomposable if for every definable subgroup H of G, either the cosets of H in G partition X into infinitely many pieces or X is contained in a single coset of H. F ACT 2.13 ŽZ il’ber’s Indecomposability Theorem w22x.. Let A i be a family of indecomposable subsets of a group of finite Morley rank G, each of which contains the identity element of G. Then the subgroup generated by Di A i is definable and connected. Furthermore, there are finitely many indices 1 1 i1 , . . . , i m such that ²Di A i : s A" ??? A" for some finite m. i1 im 784 TUNA ALTINEL This theorem has several useful corollaries: F ACT 2.14 w22x. Let G be a group of finite Morley rank. The subgroup generated by a set of definable connected subgroups of G is definable and connected and it is the setwise product of finitely many of them. F ACT 2.15 w22x. Let G be a group of finite Morley rank. Let H F G be a definable connected subgroup. Let X : G be any subset. Then the subgroup w H, X x is definable and connected. F ACT 2.16 w22x. Let G be a group of finite Morley rank. Then G n and G Ž n. are definable. If G is connected, then G n and G Ž n. are connected. There is a structure theorem for abelian groups of finite Morley rank: F ACT 2.17 w16x. Let A be an abelian group of finite Morley rank. Then A s DB Ž central product ., with D and B 0-definable, D di¨ isible, and B of bounded exponent. Certain facts about nilpotent and solvable groups of finite Morley rank will be needed in the sequel: F ACT 2.18. Then: Let G be an infinite nilpotent group of finite Morley rank. Ži. w8, Lemma 6.2x ZŽ G . is infinite. Žii. w8, Lemma 6.3x If H - G is a definable group of infinite index then Ž NG H .rH is infinite. Žiii. w8, Exercise 5, p. 98x Any infinite normal subgroup of G contains infinitely many central elements of G. Proof. We give a proof of part Žiii.. Let G be as in the statement and H an infinite normal subgroup of G. Suppose H l ZŽ G . is a finite group. We may assume that G is of nilpotency class 2. For a fixed element g g G, consider the group homomorphism H ª H l ZŽ G . which assigns w g, h x to h g H. Its kernel CH Ž g . is of finite index in H. By the descending chain condition on definable subgroups, H l ZŽ G . s H l CG Ž G . s H l n n ŽFis1 CG Ž g i .. s Fis1 CH Ž g i . for some g 1 , . . . , g n g G. But this implies that H l ZŽ G . is of finite index in H and therefore H l ZŽ G . is infinite, contrary to the assumption. F ACT 2.19 w13x. Let G be a nilpotent group. For a gi¨ en prime p, G has a unique Sylow p-subgroup. If all elements of G are of finite order then G is the direct sum of its Sylow p-subgroups. 785 GR OUPS OF FINITE MOR LEY R ANK F ACT 2.20 w8, Exercises 11, 12, pp. 13, 14x. p-groups. Then the following hold: Let G be a nilpotent-by-finite Ži. ZŽ G . / 1. Žii. If H is a nontri¨ ial normal subgroup of G then ZŽ G . l H / 1. Žiii. For any X - G, X - NG Ž X .; that is, G satisfies the normalizer condition. Proof. The reader is referred to the copious hints given for the above mentioned exercises. D EFINITION 2.21. Let G be a group of finite Morley rank. A subgroup of G is said to be G-minimal if it is definable, infinite, normalized by G, and minimal with respect to these properties. F ACT 2.22 w23x. Let G s A i H be a group of finite Morley rank, where A and H are infinite definable abelian subgroups and A is H-minimal. Assume CH Ž A. s 1. Then the following hold: Ži. The subring K s Zw H xrann Zw H xŽ A. of End Ž A. is a definable algebraically closed field; in fact, there is an integer l such that e¨ ery element of K can be represented as the endomorphism Ýlis1 h i , where Ž h i g H .. Žii. A ( Kq, H is isomorphic to a subgroup T of K *, and H acts on A by multiplication; in other words, GsAiH( ½ž / t 0 5 a : t g T, a g K . 1 Žiii. In particular, H acts freely on A, K s T q ??? qT Ž l times. and with the additi¨ e notation A s  Ýlis1 h i a : h i g H 4 for any a g A*. This has the following consequence: F ACT 2.23 w8, Theorem 9.7x. Let A i G be a group of finite Morley rank such that A is abelian and CG Ž A. s 1. Let H eG1 eG be definable subgroups with G1 connected and H infinite abelian. Assume also that A is G1-minimal. Then 0 K s Z Z Ž G . rann Z w ZŽG .0xŽ A . is a definable algebraically closed field, A is a finite dimensional ¨ ector space o¨ er K, G acts on A as ¨ ector space automorphisms, and H acts scalarly. In particular, G F GLnŽ K . for some n, H F ZŽ G ., and CAŽ G . s 1. F ACT 2.24 w8, Theorem 9.8x. Let A i G be a sol¨ able group of finite Morley rank with A abelian and definable, G definable and connected. Let B F A be either G9- or G-minimal. Then G9 centralizes B. 786 TUNA ALTINEL F ACT 2.25 w22, 17x. Let G be a connected sol¨ able group of finite Morley rank. Then G9 is nilpotent. D EFINITION 2.26. Let p be a set of primes. A Hall p-subgroup of a group G is a maximal p-subgroup. In our context, there are several versions of the Schur ]Z assenhaus theorem. The ones which will be needed are as follows. F ACT 2.27. Ži. w5x. In a sol¨ able group of finite Morley rank, a normal Hall p-subgroup has a complement. Žii. w6x. Let G be a sol¨ able group of finite Morley rank and H a normal Hall p-subgroup of G. If H has bounded exponent then the complements of H in G are definable and conjugate to each other. F ACT 2.28 w5x. Let G be a connected sol¨ able group of finite Morley rank. Then the Hall p-subgroups of G are connected. F ACT 2.29 w2; 8, Theorem 9.35x. Let p be a set of primes. Any two Hall p-subgroups of a sol¨ able group of finite Morley rank are conjugate F ACT 2.30 w2x. Let G be a sol¨ able group of finite Morley rank, N eG, and let H be a Hall p-subgroup of G for some set p of primes. Then: Ži. H l N is a Hall p-subgroup of N, and all Hall p-subgroups of N are of this form. Žii. If N is definable then HNrN is a Hall p-subgroup of GrN, and all Hall p-subgroups of GrN are of this form. Proof. Part Ži. is a consequence of Fact 2.29. So is the second half of part Žii.. Therefore, it remains to prove the first half of part Žii.. As N is a solvable group of finite Morley rank, by Fact 2.16, N has a finite normal series of definable subgroups such that the corresponding sections are abelian. Using Fact 2.17 this series can be refined so that each section is either elementary abelian of exponent p for some p g p or a p-divisible abelian group. By induction on the length of this series we may assume that N is either an elementary abelian p-group or p-divisible. In the first case the conclusion follows immediately. For the second case the reader is referred to the proof of Fact 2.29 in w8x. D EFINITION 2.31. For any prime number p, a p H -element is an element whose order is either infinite or relatively prime to p, and a p H -group is a group all of whose elements are p H . D EFINITION 2.32. Let G be a group of finite Morley rank. O Ž G . Ž odd part of G . is the largest, normal, definable, connected, 2 H -subgroup of G. GR OUPS OF FINITE MOR LEY R ANK 787 O 2 Ž G . is the largest normal 2-subgroup of G. s Ž G . is the subgroup generated by all the normal solvable subgroups of G, and F Ž G . is the subgroup generated by all the normal nilpotent subgroups of G. s Ž G . is called the sol¨ able radical of G and F Ž G . the Fitting subgroup of G. F ACT 2.33 w3, 20x. Let G be a group of finite Morley rank. Then the Fitting subgroup and the sol¨ able radical of G are definable, and they are nilpotent and sol¨ able, respecti¨ ely. F ACT 2.34 w19x. Let G be a connected sol¨ able group of finite Morley rank. Then GrF Ž G . 0 Ž hence, GrF Ž G .. is a di¨ isible abelian group. We need some facts about elements of finite order. F ACT 2.35 w8, Exercise 11, p. 72x. Let G be a p H -group of finite Morley rank, where p is a prime number. Then G is p-di¨ isible. Proof. Let g g G. By considering the definable subgroup ZŽ CG Ž g .., we may assume that G is abelian. By Fact 2.17, G s DB, where D is divisible and B is of bounded exponent. Then g s db, where d g D and b g B. By the assumption b m s 1 for some m relatively prime to p. Hence, we may assume g s db p. But D is divisible; therefore, we may also assume that we have g s d p b p. Thus db is a pth root of g. F ACT 2.36 w5x. Let G be a group of finite Morley rank and H be a definable normal subgroup of G. If x is an element of G such that x is a p-element of G s GrH then the coset xH contains a p-element. F ACT 2.37 w5x. Let K and L be definable p H -subgroups of a group G of finite Morley rank. Assume K normalizes L. Then KL is also a p H -subgroup. D EFINITION 2.38. A p-subgroup of a group of finite Morley rank is said to be a p-torus if it is divisible and abelian. F ACT 2.39 w9x. Let G be a group of finite Morley rank. Let D be a di¨ isible abelian subgroup of G. Then for e¨ ery prime p, D has finitely many elements of order p. Now, we go over some results on Sylow 2-theory. For any subset X of any group G, I Ž X . will denote the set of involutions in X. We start with a general crucial property of involutions. F ACT 2.40. product. If i and j are in¨ olutions in a group G then they in¨ ert their The following is an analog of an elementary result about finite groups: F ACT 2.41 w8, Proposition 10.2x. Let G be a group of finite Morley rank and i, j g I Ž G .. Then either i and j are dŽ ij .-conjugate or they commute with a third in¨ olution of dŽ ij .. 788 TUNA ALTINEL For groups of finite Morley rank a Sylow theorem has been proved only for the prime 2. F ACT 2.42 w9x. The Sylow 2-subgroups of a group of finite Morley rank are conjugate. There is also a structure theorem for Sylow 2-subgroups of a group of finite Morley rank. F ACT 2.43 w9x. If S is a Sylow 2-subgroup of a group of finite Morley rank then the following hold: Ži. S is nilpotent-by-finite. Žii. S 0 s B)T is the central product of a definable, connected, nilpotent subgroup B of bounded exponent and a di¨ isible, abelian 2-group T. Moreo¨ er, B and T are unique. Žiii. If S is infinite and of bounded exponent then ZŽ S . has infinitely many in¨ olutions and S is nilpotent. F ACT 2.44 w8, Exercise 1, p. 97x. An infinite nilpotent p-group of finite Morley rank and of bounded exponent has infinitely many central elements of order p. Proof. Let P be an infinite nilpotent p-group of finite Morley rank and of bounded exponent. By Fact 2.18Ži., ZŽ P . is infinite. As ZŽ P . is of bounded exponent, there is an infinite set  x i : i g N4 : ZŽ P . such that x ip s x jp for all i, j g N. Then  x 0 xy1 : j g N4 is an infinite set of elements j of ZŽ P . of order p. F ACT 2.45 w8, Exercise 2, p. 175x. Let G s Q i E be a group of finite Morley rank such that Q is normal, 2 H , definable, connected, and sol¨ able and E is a definable, connected 2-group of bounded exponent. Then w Q, E x s 1. Proof. Note that G is a connected, solvable group by Facts 2.14 and 2.43Ži.. First, assume that Q is nilpotent. Then Q F F Ž G . and GrF Ž G . is of bounded exponent. But by Fact 2.34, GrF Ž G . is divisible. Therefore, G s F Ž G . and the result follows from Fact 2.19. In the more general case where Q is solvable, G9 is a connected, nilpotent group by Facts 2.16 and 2.25. By induction on the nilpotency class of E, we may assume it is abelian. Then G9 F Q and G9 is a 2 H -group. Thus E centralizes G9 by the above paragraph. In particular, E n n centralizes w E, Q x. Hence, for any x g E and y g Q, w x 2 , y x s w x, y x 2 . n 2 H Thus, for n large enough we have w x, y x s 1. But w E, Q x is a 2 -group. Therefore, w x, y x s 1. Hence, w E, Q x s 1. GR OUPS OF FINITE MOR LEY R ANK 789 F ACT 2.46 w1x. Let G be a connected, sol¨ able group of finite Morley rank. Assume that its Sylow 2-subgroups are of bounded exponent. Then G has a unique Sylow 2-subgroup. Proof. Let S be a Sylow 2-subgroup of G. By Fact 2.28, S is connected. Hence, SF Ž G . 0rF Ž G . 0 is a connected 2-group of bounded exponent. By Fact 2.44, SF Ž G . 0rF Ž G . 0 has infinitely many involutions unless it is trivial, i.e., S F F Ž G . 0 . But by Fact 2.34, GrF Ž G . 0 is divisible abelian. Thus by Fact 2.39, GrF Ž G . 0 has finitely many involutions. This forces S F F Ž G .8. By Fact 2.19, S is the unique Sylow 2-subgroup of F Ž G . 0 and therefore of G. The Sylow theory yields the following useful result on fusion: F ACT 2.47 w8, Lemma 10.22x. Let S and T be as in Fact 2.43. If X, Y : S 0 and X g s Y, where g g G, then there exists h g NG ŽT . such that X h s Y Ž that is, NG ŽT . controls fusion in S 0 .. The following corollary is crucial for our analysis: F ACT 2.48 w1x. If a group G of finite Morley rank has a single conjugacy class of in¨ olutions then the connected component of any Sylow 2-subgroup is either of bounded exponent or di¨ isible. Proof. Let S be a Sylow 2-subgroup of G. S 0 s B)T by Fact 2.43Žii.. We will show that we cannot have B / 1 and T / 1. Suppose toward a contradiction that B / 1 and T / 1. By Fact 2.39, T contains finitely many involutions. As I Ž B . is infinite, we can find u g I Ž S 0 _ T . and ¨ g I ŽT .. Since these are conjugate in G, by Fact 2.47, there exists g g NG ŽT . such that u s ¨ g . But this implies that u g T, contradicting the choice of u. The following facts about the actions of involutions on definable subgroups will be useful in the sequel. F ACT 2.49 w18x. Let a be a definable in¨ oluti¨ e automorphism of a group of finite Morley rank G. If a has finitely many fixed points then G has a definable normal subgroup of finite index which is abelian and in¨ erted by a . The following properties of tame groups of finite Morley rank will be needed: F ACT 2.50 w8, Theorem B.1x. Let G be a connected tame group of finite Morley rank. Then any connected definable 2 H -section of G is nilpotent. F ACT 2.51 w1x. Let G be a connected nonsol¨ able K-group of finite Morley rank. Then Grs Ž G . is isomorphic to a direct sum of simple algebraic groups o¨ er algebraically closed fields. In particular the definable, connected, 2 H sections are sol¨ able. In order to prove Fact 2.51 we need some definition and results. 790 TUNA ALTINEL D EFINITION 2.52. A group G is called semisimple if G s G9 and GrZŽ G . is a direct sum of finitely many nonabelian simple groups. G is called quasisimple if this direct sum is actually a single group. D EFINITION 2.53. Let G be a group of finite Morley rank. The socle SŽ G . of G is the subgroup generated by all the minimal normal subgroups of G. SŽ G . is not necessarily definable. For example, SŽC*. s [p ZrpZ. F ACT 2.54 w20x. Let G be a connected group of finite Morley rank such that s Ž G . s 1. Then SŽ G . s S1 [ ??? [ Sm , where the Si are nonabelian, infinite, simple, definable subgroups. F ACT 2.55 w8, Theorem 8.4x. Let G s G i H be a group of finite Morley rank where G and H are definable, G is a infinite simple algebraic group o¨ er an algebraically closed field, and CH Ž G . s 1. Then, ¨ iewing H as a subgroup of Aut Ž G ., we ha¨ e H F Inn Ž G . G, where Inn Ž G . is the group of inner automorphisms of G and G is the group of graph automorphisms of G. Proof of Fact 2.51. By considering Grs Ž G . we may assume that s Ž G . s 1. By Fact 2.54, SŽ G . s S1 [ ??? [ Sm , where the Si are nonabelian definable simple subgroups. The subgroup CG Ž SŽ G .. intersects SŽ G . trivially. If CG Ž SŽ G .. / 1, then as it intersects SŽ G . trivially, it is solvable. But CG Ž S Ž G ..eG, and s Ž G . s 1. Therefore, CG Ž S Ž G .. s 1. Now, we consider the group SŽ G . i G. Since G is a K-group, the above paragraph implies that we are in the situation described in Fact 2.55. As G is connected, we can see G as a subgroup of Inn Ž SŽ G ... Thus, for any g g G there exists x g S Ž G . such that x and g induce the same automorphism on SŽ G .. This forces gxy1 g CG Ž SŽ G .. s 1. Hence, g s x. Therefore, G s SŽ G .. The proof of Theorem 1.5 is based on Fact 2.57 about Z assenhaus groups of finite Morley rank. A doubly transitive group G is said to be a Zassenhaus group if the stabilizer of any set of three distinct points is trivial. Let Gx denote a one-point stabilizer and Gx, y denote a two-point stabilizer. G is said to be a split Zassenhaus group if Gx, y has a normal complement in Gx . If, in addition, this normal complement is a 2-group then G is said to be of characteristic 2. F ACT 2.56 w7x. Let G be an infinite split Zassenhaus group of finite Morley rank of characteristic 2. Then G ( SL 2 Ž K . for some algebraically closed field K of characteristic 2. It is worth noting that in a setting where the strongly embedded subgroup is assumed to be connected, the rank computations of Section 6 GR OUPS OF FINITE MOR LEY R ANK 791 could be omitted as Theorem 1.5 would follow in this case from the following result: F ACT 2.57 w11x. Let G be an infinite, nonsol¨ able group of finite Morley rank such that O 2 Ž G . s 1. Suppose there exists a proper definable, nilpotent subgroup U of G such that U contains infinitely many in¨ olutions and for any u g U _  14 , CG Ž u. F U. Then G is isomorphic to PSL2 Ž K ., where K is an algebraically closed field of characteristic 2. In fact, the proof of Fact 2.57 includes a reduction to Fact 2.56, and we carry out a similar reduction when Fact 2.57 does not apply directly. 3. PR ELIMINAR Y R ESULTS ON GR OUPS OF FINITE MOR LEY R ANK WITH STR ONGLY EMBEDDED SUBGR OUPS This section contains results on groups of finite Morley rank with strongly embedded subgroups which do not depend on the tameness assumption. The proofs of the following two facts are direct translations of arguments in finite group theory. F ACT 3.1. Let G be a group of finite Morley rank with a proper definable subgroup M. Then the following are equi¨ alent: Ži. M is a strongly embedded subgroup. Žii. I Ž M . / B, CG Ž i . F M for any i g I Ž M ., and NG Ž S . F M for any Sylow 2-subgroup S of M. Žiii. I Ž M . / B, and NG Ž S . F M for any nontri¨ ial 2-subgroup S of M. D EFINITION 3.2. Let G be a group. For any x g G, CGU Ž x . is  g g G : x g s x or xy14 . A nontrivial group element is called strongly real if it can be written as a product of two involutions. F ACT 3.3 w8, Theorem 10.19; 12, Theorem 9.2.1x. Let G be a group of finite Morley rank with a strongly embedded subgroup M. Then the following hold: Ži. A Sylow 2-subgroup of M is a Sylow 2-subgroup of G. Žii. I Ž G . is a single conjugacy class in G. I Ž M . is a single conjugacy class in M. Žiii. If i g I Ž M ., and x is a nontri¨ ial strongly real element of CG Ž i ., then CGU Ž x . F M. 792 TUNA ALTINEL For the rest of this section, except for Proposition 3.5, we assume that G is a group of finite Morley rank with a strongly embedded subgroup M. Another general property of groups of finite Morley rank with a strongly embedded subgroup is the following: P R OPOSITION 3.4. If N is a definable subgroup of G such that M F N - G then N is a strongly embedded subgroup of G. Proof. We may assume that M - N. We only have to check N l N g is a 2 H -group for g g G _ N. Suppose i g I Ž N l N g .. Using the fact that N is a group of finite Morley rank with a strongly embedded subgroup Žnamely M . and taking conjugates if necessary, we may assume i g M. For j g I Ž M g ., i x s j for some x g N g . This implies M x s M g and therefore y1 M s M g x , gxy1 g M, implying xy1 g g M g and g g N g . Therefore, g g N, a contradiction. P R OPOSITION 3.5. Let G be a group of finite Morley rank. If the Sylow 2-subgroups of G contain infinitely many commuting in¨ olutions then the connected component of any Sylow 2-subgroup contains infinitely many in¨ olutions. Proof. The simple argument which proves this proposition will be used later also; it will be referred to as a commuting in¨ olutions argument. Suppose I Ž S 0 . is a finite set. Then there is a coset of S 0 in S which contains an infinite set J of commuting involutions. If we fix an element u in J, then  ¨ u : ¨ g J 4 will be infinite set of involutions in S 0 , a contradiction. Facts 2.48 and 3.3 and Proposition 3.5 yield the following corollary: C OR OLLAR Y 3.6. If the Sylow 2-subgroups of G contain infinitely many commuting in¨ olutions then the Sylow 2-subgroups are of bounded exponent. The following proposition is a consequence of the conjugacy of involutions in M. P R OPOSITION 3.7. I Ž M 0 .. If the Sylow 2-subgroups of G are infinite then I Ž M . s Our immediate goal is a useful factorization of M 0 , which will be given in Proposition 3.10 below. L EMMA 3.8. There exists a coset uM / M in G such that rk I Ž uM . G rk I Ž M .. Proof. We define the map m : I Ž G _ M . ª GrM u ¬ uM. GR OUPS OF FINITE MOR LEY R ANK 793 First note that rk I Ž G . s rk I Ž G _ M .. By Fact 2.4, rk I Ž G . s rk I Ž G _ M . F rkŽ GrM . q sup ug IŽG _ M . rk my1 Ž uM .. But rk I Ž G . s rkŽ GrM . q r k Ž M r C G Ž i .. s r k Ž G r M . q r k I Ž M . . H e n ce , r k I Ž M . F sup ug IŽG _ M . rk my1 Ž uM .. It follows that there exists uM / M such that rk I Ž uM . G rk I Ž M .. P R OPOSITION 3.9. If i g I Ž M . then rkŽ M . s rkŽ CG Ž i . K ., where K is a definable 2 H -subgroup of M. Proof. As I Ž M . is a single conjugacy class in M, we can take K to be the trivial group if I Ž M . is finite. Therefore, we will assume that I Ž M . is infinite. In what follows, C denotes CG Ž i .. We first construct the group K. We fix a coset uM as provided by Lemma 3.8. Fix w g I Ž uM .. Let H s ² wy : y g I Ž uM .: and K s dŽ H .. As M is definable, K F M. Define L s dŽ H, w .. Since w normalizes H, H e² H, w :. Therefore, K e L. Ž1. K does not contain involutions. Proof of Ž1.. Suppose I Ž K . is nonempty. Let R be a Sylow 2-subgroup of L containing w. Then R l K / 1, by Fact 2.42. As R is nilpotent-byfinite ŽFact 2.43Ži.., it has a nontrivial center ŽFact 2.20.. Now, if ¨ g I Ž R l K . then ZŽ R . F CG Ž ¨ . F M as M is strongly embedded. But if j g I Ž ZŽ R .. then w g CG Ž j . F M and this is a contradiction. Ž2. Let x 1 , x 2 g  wy : y g I Ž wM .4 . Then x 1C s x 2 C if and only if x 1 s x2 . Proof of Ž2.. Suppose x 1C s x 2 C although x 1 / x 2 . Let y 1 , y 2 g I Ž wM . such that x 1 s wy 1 and x 2 s wy 2 . Then x s xy1 2 x 1 s y 2 y 1 g C. Therefore, i g CG Ž x .. On the other hand, y 1 g CGU Ž x . _ CG Ž x . because x g K and K does not contain involutions. Hence, i and y 1 are not conjugate in CGU Ž x .. Then, by Fact 2.41, they commute with an involution z g dŽ i, y 1 .. This forces z g CG Ž i . F M first and then y 1 g M, a contradiction. Ž3. rkŽ M . s rkŽ KC .. Proof of Ž3.. Let Y s  wy : y g I Ž wM .4 and S s "¨ g Y ¨ C. Y is definable and rkŽ Y . s rk I Ž wM .. Define s: Y=CªG Ž ¨ , c . ¬ ¨ c. s is injective by Ž2. and s Ž Y = C . s S. Hence, rkŽ S . s rkŽ Y . q rkŽ C . G rk I Ž M . q rkŽ C . s rkŽ M .. As K s dŽ² Y :., KC > S. Hence, rkŽ KC . s rkŽ M .. This completes the proof of Proposition 3.9. 794 P R OPOSITION 3.10. TUNA ALTINEL M 0 s C 0 K 0 , where K is as in Proposition 3.9. Proof. By Proposition 3.9, we know that rkŽ M . s rkŽ KC .. Let w be the involution in the proof of Proposition 3.9. R ecall that rkŽ I Ž wM .. G rkŽ I Ž M ... Ž1. For m g M, u1 , u 2 g I Ž wM ., u1 mC s u 2 mC if and only if u1 s u 2 . Proof of Ž1.. Suppose for some u1 , u 2 g I Ž wM ., u1 / u 2 but u1 mC s y1 u 2 mC. Then i m g C Ž u 2 u1 .. u1 g C*Ž u 2 u1 . _ C Ž u 2 u1 . as u 2 u1 g K and y1 K is a 2 H -group. Hence, i m and u1 cannot be conjugate in C*Ž u 2 u1 .. But this will eventually force u1 g M ŽFacts 2.41 and 3.1Žii.., a contradiction. Ž2. For any m g M _ KC, rkŽ KmC . s rkŽ KC .. Proof of Ž2.. Let Y s  wu : u g I Ž wM .4 . rkŽ Y . s rk I Ž wM .. Define s : Y = C ª KmC Ž wu, c . ¬ wumc. s is well-defined, and by Ž1., it is injective. Hence, rkŽ KmC . G rkŽ Y . q rkŽ C . G rk I Ž M . q rkŽ C . s rkŽ M .. Therefore, rkŽ KmC . s rkŽ M .. We therefore have rkŽ KmC . s rkŽ KC . for any m g M. On the other hand, rkŽ KC . s rkŽ K 0 C 0 . and rkŽ KmC . s rkŽ K 0 mC 0 ., for any m g M. In particular, if m g M 0 then rkŽ M 0 . s rkŽ M . s rkŽ K 0 C 0 . s rk Ž K 0 mC 0 .. Now, if M 0 ) K 0 C 0 then for any m g M 0 _ K 0 C 0 , K 0 C 0 l K 0 mC 0 s B. This, together with the rank equalities which we have obtained, contradicts the connectedness of M 0 . 4. SOLVABILITY OF M In this section, unless otherwise stated, G will denote a simple, K *-group of finite Morley rank with a strongly embedded subgroup M whose Sylow 2-subgroups contain infinitely many commuting involutions. Under these assumptions, the definable connected 2 H -sections of G are solvable ŽFact 2.51.. In this section, we prove that M is solvable. The main effort is spent on proving the solvability of M 0 . This will yield information about the structure of M 0 , which will be used throughout the paper. To pass from the solvability of M 0 to the solvability of M will require the use of the Feit ]Thompson theorem. We will use Ci to denote CG Ž i .. K denotes the subgroup constructed in Proposition 3.9. GR OUPS OF FINITE MOR LEY R ANK 795 We start with a lemma about linear algebraic groups: L EMMA 4.1. Let G be a linear algebraic group, S a sol¨ able subgroup of G such that S s S s Ž X s denotes the set of semisimple elements of X .. Then S˜ s Ž S˜ .s , where S˜ is the Zariski closure of S. Proof. The proof is by induction on the solvable length of S. Let Ae S be abelian so that the solvable length of SrA is less than that of S. A˜ is also abelian and A˜ s Ž A˜ .s . Let N s NG Ž A˜ .. S F N. Moreover, as N is a closed subgroup, S˜ F N. ˜ Then p Ž S . s SArA ˜ ˜ and p Ž S˜ . s SrA. ˜ ˜ ˜ SArA Consider p : N ª NrA. w x does not contain unipotent elements because p is a morphism 15 and & S s S s . By induction, p S does & not contain unipotent elements. MoreŽ . & ˜ ˜ ˜ ˜ ˜ does not contain Ž . over, p S : p Ž S . , i.e., SrA : p Ž S . . Hence, SrA unipotent elements. This implies that all the unipotent elements of S˜ are ˜ But the only unipotent element in A˜ is the identity. contained in A. Therefore, S˜ s S˜ s . T HEOR EM 4.2. M 0 F s Ž M 0 .Ci , where i is any in¨ olution in M. Proof. As there is nothing to prove in the case M 0 is solvable, we may assume that s Ž M 0 . - M 0 . By Fact 2.51, M 0rs Ž M 0 . is isomorphic to a direct sum of algebraic groups over algebraically closed fields. Note that these fields are of characteristic 2 by Corollary 3.6. We first prove M 0 F s Ž M 0 .Ci for some i g I Ž M . when M 0rs Ž M 0 . is simple. Let M 0s M 0rs Ž M 0 .. By Proposition 3.10, M 0s Ci0 K 0. K 0 is a 2 H -group and, hence, does not contain unipotent elements and it is & solvable by Fact 2.51. By Lemma 4.1, Ž K 0 . is a group whose elements are semisimple. As M 0 is a simple group, it is connected as an algebraic group. Therefore, if we denote the connected component Žin the Z ariski topology. & of K 0 by K 1, then we get M 0s Ci0 K 1. K 1 lies in a maximal torus T. As a result, we have the following factorization: M 0s Ci0 T. Let U be a unipotent subgroup of M 0 such that U i T is a Borel subgroup. We first argue that Ci0 contains a conjugate of U. As U is nilpotent and s Ž M 0 . is solvable, U is solvable. Now, let U1 be a Sylow 2-subgroup of U. As U is solvable, U1 is a Sylow 2-subgroup of U, by Fact 2.30Žii., i.e., U s U1. On the other hand, U10 F B, where B is the connected component of a Sylow 2-subgroup of M. B is a 2-group containing U1 , which forces B s U1 ŽU1 is a Sylow 2-subgroup of M 0.. Hence, we have B s U s U1. Now, by the conjugacy of convolutions in M, there exists m g M such that B m F Ci0 . B m s U m F Ci0. By replacing i with i m , we may assume that U F Ci0. 796 TUNA ALTINEL As M 0 is simple algebraic and 1 / ²U T C i:eM 0, we get M 0s ²U T C i: s ²U C i: : Ci. Hence, M 0s Ci and this implies M 0 F s Ž M 0 .Ci for some i g I Ž M .. Now, we assume that M 0s M1 [ ??? [ Mk, where the M j are simple algebraic groups. Moreover, we have M 0s Ci0 K 0, where Ci , K are as usual. From this decomposition, we obtain Mn s Ž Ci . n K n by taking projections. As in the case k s 1, we can replace K n with a maximal torus Tn such that Mns Ž Ci . nTn. Now, for each n let Un be a maximal unipotent group corresponding to Tn. We get Mns ²Un Ž C i . n: s ²Ui C i:. Hence, M 0s k ²UnC i : 1 F n F k :. [ns1 Un is a 2-subgroup of M 0. As in the case k s 1, k we may assume that [ns1 Un F Ci0. We then get M 0s Ci. Hence in this general case also, M 0 F s Ž M 0 .Ci for some i g I Ž M .. By the conjugacy of involutions in M, M 0 F s Ž M 0 .Ci is true for any i g I Ž M .. T HEOR EM 4.3. M 0 is sol¨ able. Proof. We assume toward a contradiction that M 0 is not solvable. Then M 0rs Ž M 0 . is the direct sum of simple algebraic groups over algebraically closed fields. I Ž M . ; s Ž M 0 . because otherwise by Theorem 4.2, M 0rs Ž M 0 . has a nontrivial center, which is impossible. Now a commuting involutions argument shows that I Ž M . ; s Ž M 0 . 0 . By Fact 2.46, s Ž M 0 . 0 has a unique Sylow 2-subgroup, O 2 Ž s Ž M 0 . 0 .. As O 2 Ž s Ž M 0 . 0 . has central involutions, I Ž M . ; ZŽ O 2 Ž s Ž M 0 . 0 ... Hence, ² I Ž M .: is an elementary abelian 2-group, normal in M. Fact 2.27 implies that s Ž M 0 . 0 s O 2 Ž s Ž M 0 . 0 . i T, where T is a definable, 2 H -group, and any complement of O 2 Ž s Ž M 0 . 0 . in s Ž M 0 . 0 is conjugate to T. By the Frattini argument, M 0 s O 2 Ž s Ž M 0 . 0 . NM 0 ŽT . and in particular, M 0 s s Ž M 0 . 0 NM 0 ŽT .. We claim that NM 0 ŽT . does not contain involutions. If i g I Ž NM 0 ŽT .. then T F Ci , so s Ž M 0 . 0 F Ci . By Theorem 4.2, M 0 F Ci and thus M contains only finitely many involutions. This contradicts our assumption on the structure of Sylow 2-subgroups. M 0rs Ž M 0 . 0 is a connected 2 H -group, and hence solvable. Thus, M 0 is solvable. C OR OLLAR Y 4.4. M 0 s O 2 Ž M 0 . i ŽT ., where T is a 2 H , connected, definable subgroup, and any complement to O 2 Ž M 0 . in M 0 is conjugate to T in M 0 . C OR OLLAR Y 4.5. A s ² I Ž M .: is a definable, connected, elementary abelian 2-subgroup, and A F ZŽ O 2 Ž M 0 ... C OR OLLAR Y 4.6. Let a g G _  14 . Assume that there exist i, j g I Ž G . such that a i s a and a j s ay1. Then a g I Ž G .. GR OUPS OF FINITE MOR LEY R ANK 797 Proof. Let a, i, j be as in the statement of the proposition. Let M denote the strongly embedded subgroup which contains i. By Fact 3.1Žii., a g M. As a is inverted by j, it is a strongly real element. Thus, by Fact 3.3Žiii., j g I Ž M . also. This implies ja g I Ž M .. But ² I Ž M .: is an elementary abelian 2-group. Hence, a g I Ž M .. In the remainder, A will denote the elementary abelian subgroup in Corollary 4.5. C OR OLLAR Y 4.7. I Ž NM ŽT .. s B. C OR OLLAR Y 4.8. The in¨ olutions in M are conjugate in M 0 . Proof. Let i g I Ž M .. The following rank equalities show that I Ž M . and i have the same rank: M0 rk Ž M . s rk CM Ž i . q rk I Ž M . 0 rk Ž M 0 . s rk Ž CM 0 Ž i . . q rk i M . As A is a connected group, I Ž M . s A _  14 is an irreducible set by Fact 0 2.9. Since rk I Ž M . s rkŽ i M . for any involution in M, there cannot be 0 more than one M 0-conjugacy class in I Ž M .. Hence, i M s I Ž M .. It remains to prove: P R OPOSITION 4.9. MrM 0 is a finite group of odd order. Proof. Let x g M be such that x 2 g M 0 . By Fact 2.36, we may assume that x is a 2-element. As M 0 e M and the complements of O 2 Ž M 0 . in M 0 are conjugate in M 0 , T x s T u for some u g O 2 Ž M 0 .. xuy1 is a 2-element in M normalizing T. By Corollary 4.7, xuy1 s 1. Thus, x g M 0 . C OR OLLAR Y 4.10. M is sol¨ able, and O 2 Ž M . s O 2 Ž M 0 .. 5. O Ž M . s 1 In this section we assume that G is a simple, tame, K *-group of finite Morley rank with a strongly embedded subgroup M. L EMMA 5.1. For m g M, if m centralizes one in¨ olution in M, then it centralizes all the in¨ olutions in M. Proof. We consider the action of MrCM Ž A. on A, i.e., the group A i MrCM Ž A.. -notation will be used to denote the quotients by CM Ž A.. As T acts transitively on A _  14 ŽCorollary 4.8., A is T-minimal. Therefore, by Fact 2.24, T 9 F CM Ž A., and T is abelian. The transitivity of the 798 TUNA ALTINEL action of T on A _  14 forces T to act transitively on A _  14 . In particular, A is T-minimal. This is the situation described in Fact 2.23. Therefore, A is a finite dimensional vector space over an algebraically closed field K of characteristic 2, M F GLŽ A. and T F ZŽ M .. This means that for any m g M and t g T, w m, t x g CM Ž A.. Now, assume m g M and i g A are such that i m s i. Consider i t , where t g T. i t m s i m t c, where c g CM Ž A.. But i m t c s i t c and i t g A, hence i t c s i t. Therefore, we get i t m s i t. As T acts transitively on A _  14 , we conclude that m commutes with A. We need the following notation from group theory for the next proposition: V 1Ž G . s ² g g G : g 2 s 1:. P R OPOSITION 5.2. C Ž A. 0 s O 2 Ž M . = B, where B is a 2 H -group. Proof. By Fact 2.27, we have C Ž A. 0 s O 2 Ž M . i T0 , for some connected subgroup T0 of T. As there is nothing to prove when T0 s 1, we may assume that T0 is an infinite definable connected subgroup of T. In order to simplify the notation, we will let S s O 2 Ž M . and S0 s CS ŽT0 .. We have A F S0 F S. Our aim is to prove that S s S0 . Suppose toward a contradiction that S0 - S. Let X s NS Ž S0 .. As S0 - S, S0 - X ŽFacts 2.43Ži., 2.20.. As S is connected ŽFact 2.28., w X : S0 x s ` by Fact 2.18Žii.. Let S1rS0 s V 1Ž ZŽ XrS0 ... S1 ) S0 by Fact 2.18Ži.. We analyze some definable subgroups of S1. The first of these is w T0 , S1 x. By Z il’ber’s Indecomposability Theorem, it is definable and connected. Note that w T0 , S1 x F w C Ž S0 ., N Ž S0 .x F C Ž S0 .. We claim that for s g S1 _ S0 , w T0 , s x g S0 . Suppose w T0 , s x F S0 for some s g S1 _ S0 . Let t g T0 . Then w s, t x s sn 0 for some s0 g S0 , equiva2 n lently, s t snss0 . Then we get for any n, s t s ss02 . Hence, for n large t2 enough, s s s. As T0 does not contain involutions, by Fact 2.35, it is n 2-divisible. We may therefore replace t 2 by t and conclude that s t s s. As t was arbitrarily chosen, we conclude s g S0 , a contradiction. As w T0 , S1 x is connected, ww T0 , S1 x S0 : S0 x s `. Let S2 s w T0 , S1 x S0 . Let S3 be a subgroup of S2 such that S0 F S3 and S3rS0 is T0-minimal. By Fact 2.24, T0X centralizes S3rS0 . We let T0s T0rCT 0Ž S3rS0 .. Fact 2.22 and the tameness assumption imply that S3rS0 i T0( Kqi K *. In particular T0 and thus T0 acts on S3rS0 _  14 transitively. Now let a, b g w T0 , S1 x l S3 be such that aS0 / bS0 . The transitivity of the action of T0 on S3rS0 implies that there exist t 1 g T0 and s1 g S0 such that a t 1 s bs1. This implies a2 s Ž a2 . t 1 s Ž a t 1 . 2 s b 2 s12 . We also have Ž by1 a. 2 s w by1 , ax by2 a2 s w by1 , ax s12 . Therefore Ž by1 asy1 . 2 s w by1 , ax. 1 y1 2 y1 y1 y1 On the other hand, w b , ax s w b , axw b , ax s w b , axw by1 , ax a s w by1 , a2 x s 1, for any such a and b. This shows that w by1 , ax g A. Let us denote this element of A by i. Note that i / 1 because otherwise, by1 asy1 1 GR OUPS OF FINITE MOR LEY R ANK 799 would be an involution, contradicting the choice of a and b. Hence i is an . 2 is an involution in A. involution. Therefore, Ž by1 asy1 1 We first analyze the possibility A - S0 . In this case, in order to get a contradiction it is enough to find an element s2 g S0 whose square is . 2 , because then as a, b, and s1 g C Ž S0 ., we will get equal to Ž by1 asy1 1 y1 y1 y1 2 y1 Ž b as1 s2 . s 1, which will force by1 asy1 1 s 2 g A and therefore, y1 b a g S0 . This contradicts the choice of a and b. As A - S0 , we can find j g A _  14 which has a square root in S0 . As T acts on A _  14 transitively, i also has a square root. If S0 s A then a2 s Ž by1 a. 2 s w by1 , ax and we get a2 s bay1 by1a or equivalently a b s ay1 . Now, as S3rS0 is infinite and every coset of S0 in S3 contains an element of w T0 , S1 x l S3 , we can find c g w T0 , S1 x l S3 such that cS0 / aS0 , cS0 / bS0 , and cS0 / by1 aS0 . Then a c s ay1 and a b c s ay1 . But a b c s a. This forces a to be an involution, which contradicts the choice of a. P R OPOSITION 5.3. If O Ž M . / 1 then ZŽ M 0 . is infinite. Proof. If O Ž M . / 1 then it is infinite, and, as it is normal in M, it is contained in every complement of O 2 Ž M . in M 0 , in particular in T. Therefore, by Fact 2.18Žiii., it has infinite intersection with ZŽT .. On the other hand, O Ž M . is centralized by O 2 Ž M .. Hence, O Ž M . l ZŽT . centralizes both O 2 Ž M . and T. Thus, it centralizes M 0 . The proof that O Ž M . s 1 will make use of the following lemma. L EMMA 5.4. If a g O Ž M . _  14 then C Ž a. F M. We postpone the proof of this lemma and prove first that it implies O Ž M . s 1. We need the following lemma: L EMMA 5.5. There exist w g I Ž G _ M . and a g M 0 _  14 such that a is 2 H and aw s ay1 . Proof. Fix w as in the proof of Proposition 3.9. If M 0 has a 2 H -element inverted by w, then there is nothing to prove. Suppose now that all 2 H -elements of M inverted by w are in M _ M 0 . In particular, J s  wu : u g I Ž wM .4 : M _ M 0 . J is an infinite set. Therefore we can find distinct u1 , u 2 g I Ž wM . such that wu1 , wu 2 are in the same coset of M 0 in M. u 2 u1 s Ž wu 2 .y1 wu1 g M 0 . We know from the proof of Proposition 3.9 that u 2 u1 is a 2 H -element and is inverted by u1 Žor u 2 as well.. Therefore, we can replace w by u1. T HEOR EM 5.6. O Ž M . s 1. Proof. Suppose O Ž M . / 1. By Lemma 5.5, there exist w g I Ž G _ M . and a g M 0 _  14 such that a is 2 H and aw s ay1. 800 TUNA ALTINEL We claim that w inverts C Ž a. 0 . By Fact 2.49, it is enough to show that w fixes finitely many points of C Ž a. 0 . Suppose this is not the case. Then Ž C Ž w . l C Ž a. 0 . 0 is an infinite, definable, connected subgroup of M g , where w g M g . By Corollary 4.6, it is a 2 H -group. By Lemma 5.1 and Proposition 5.2, it is contained in O Ž M g .. Now, let x g Ž C Ž w . l C Ž a. 0 . 0 _  14 . Then a g CG Ž x .. Therefore, by Lemma 5.4, a g M g . aw s ay1 implies aw g I Ž M g .. But w g I Ž M g .. Hence a g I Ž M g ., a contradiction. We can now finish the proof of the theorem. Let b g ZŽ M 0 . 0 _  14 . Then b g C Ž a. 0 and b w s by1. But b commutes with involutions, contradicting Corollary 4.6. Now we prove Lemma 5.4. Proof of Lemma 5.4. Let a g O Ž M . _  14 . Suppose C Ž a. g M. Then C Ž a. _ M contains involutions, as otherwise AeC Ž a., and C Ž a. F N Ž A. s M, a contradiction. Therefore, C Ž a. l M is a strongly embedded subgroup of C Ž a., so I Ž C Ž a.. is a single conjugacy class. As A is connected, C Ž a. 0 l M is a strongly embedded subgroup of C Ž a. 0 . From now till the end of the proof of Lemma 5.4, s will denote s Ž C Ž a. 0 .. I Ž s 0 . s B because otherwise by Fact 2.46, s 0 has a unique Sylow 2-subgroup which contradicts the fact that Ž C Ž a. 0 l M . is a strongly embedded subgroup of C Ž a. 0 . Now, a commuting involutions argument shows that I Ž s _ s 0 . s B. Hence, s does not contain involutions. In the rest of the proof of Lemma 5.4, -notation will be used to denote quotients by s . B 0 C LAIM 5.1. C Ž a . contains a strongly embedded subgroup. Proof of Claim 5.1. We will prove that Ž C Ž a. 0 l M . is a strongly 0 embedded subgroup of C Ž a . . As s does not contain involutions, Ž C Ž a. 0 l M . contains involutions. Now we show that Ž C Ž a. 0 l M . l g Ž C Ž a. 0 l M . Ž C Ž a.00 l M . . 0 does not contain involutions for any g g C Ž a . _ Suppose this group contains an involution. By Fact 2.36, Ž C Ž a. l M . s l Ž C Ž a. 0 l M . gs contains an involution. If Ž C Ž a. 0 l M . s - C Ž a. 0 then Proposition 3.4 implies that Ž C Ž a. 0 l M . s is a strongly embedded subgroup of C Ž a. 0 . But then Ž C Ž a. 0 l M . s l Ž C Ž a. 0 l M . gs does not contain an involution, a contradiction. On the other hand, if Ž C Ž a. 0 l M . s s C Ž a. 0 then, as both s and C Ž a. 0 l M are solvable and s eC Ž a. 0 , C Ž a. 0 is solvable. But C Ž a. 0 is not solvable Ž s does not contain involutions although C Ž a. 0 does... Thus in either case we have a contradiction. This finishes the proof. 801 GR OUPS OF FINITE MOR LEY R ANK 0 By Fact 2.51, C Ž a . is isomorphic to a direct sum of simple algebraic groups over algebraically closed fields. The conjugacy of involutions in 0 0 C Ž a. 0 implies that C Ž a . is actually a simple algebraic group. Since C Ž a . contains a strongly embedded subgroup by Claim 5.1, it is isomorphic to PSL2 Ž K ., where K is an algebraically closed field of characteristic 2. 0 0 As C Ž a . l M is a strongly embedded subgroup of C Ž a . , it is con0 0 0 nected. Hence, we get C Ž a . l M s Ž C Ž a . l M . . Moreover, Ž C Ž a. 0 l 0 0 0 M . 0 F C Ž a. 0 l M 0 and we get C Ž a . l M 0s C Ž a . l M. Z Ž M 0 . centralizes this group; hence it is trivial because PSL2 Ž K .’s strongly embedded subgroups are centerless. This implies ZŽ M 0 . 0 F s , actually ZŽ M 0 . 0 F s 0 . Fact 2.45 implies that s 0 , and in particular ZŽ M 0 ., commutes with all the involutions in C Ž a. 0 . Let S be the subgroup generated by I Ž C Ž a. 0 .. 0 0 S eC Ž a. 0 . Therefore, S eC Ž a . . This forces S s C Ž a . . Moreover, S F ž ž 0 // C Ž ZŽ M 0 . 0 .. We have C Ž ZŽ M 0 . 0 .rC Ž ZŽ M 0 . 0 . l s ( C Z Ž M 0 . G S ( PSL2 Ž K .. We conclude that C Ž ZŽ M 0 . 0 . is not solvable. M normalizes C Ž ZŽ M 0 . 0 .. Thus, MC Ž ZŽ M 0 . 0 . F N Ž C Ž ZŽ M 0 . 0 .. - G. By Proposition 3.4, we conclude that MC Ž ZŽ M 0 . 0 . is a strongly embedded subgroup of G. Then by Theorem 4.2, MC Ž ZŽ M 0 . 0 . is solvable. But the above paragraph shows that C Ž ZŽ M 0 . 0 . is not solvable. This contradiction finishes the proof of the lemma. 6. DOUBLE TR ANSITIVITY Proposition 5.2 and Theorem 5.6 of Section 5 have important consequences for the structure of the centralizers of involutions in M; since O Ž C Ž A.. F O Ž M . s 1, C Ž A. s O 2 Ž M . i B, where B is a finite group of odd order. In this section we finish the proof of Theorem 1.5. We will first prove that CM 0 Ž A. s O 2 Ž M .. This conclusion, together with Fact 2.57, would yield Theorem 1.5 in a setting where the strongly embedded subgroup is connected, using w11x. To prove Theorem 1.5 without assuming that M is connected, we use the ideas as well as results from w11x. We analyze the action of G on the left coset space of M 0 and show that G is a split Z assenhaus group of characteristic 2. Then Fact 2.56 implies that G ( PSL2 Ž K ., where K is an algebraically closed field of characteristic 2. In this section T denotes a complement to O 2 Ž M . in M 0 . The following lemma will be used to prove that CT Ž A. s 1 and will also be useful subsequently. 802 TUNA ALTINEL L EMMA 6.1. If a g M 0 _  14 and w g I Ž G _ M . is such that aw s ay1 then w in¨ erts C Ž a. 0 . Proof. By Fact 2.49, it is enough to show that w acts on C Ž a. 0 with finitely many fixed points. Suppose this is not the case. Then C Ž w . l C Ž a. 0 is an infinite definable subgroup of M g , where w g I Ž M g .. An application of Corollary 4.6 shows that C Ž a. is a 2 H -group. Therefore, so is Ž C Ž w . l C Ž a. 0 . 0 . Moreover, Ž C Ž w . l C Ž a. 0 . 0 F CM 0 Ž A g . 0 . This forces O Ž M g . / 1, a contradiction. C OR OLLAR Y 6.2. T is abelian. Proof. Let a and w be as in Lemma 5.5. As a is a 2 H -element in M 0 , it must be contained in a complement to O 2 Ž M .. Hence, after taking conjugates, we may assume that a g T. Then w inverts C Ž a. 0 , which contains ZŽT . 0 , and the latter subgroup is nontrivial as T is nilpotent. Hence, T has a nontrivial central element inverted by w. A second application of Lemma 6.1 shows that w inverts T, and in particular, T is abelian. P R OPOSITION 6.3. CT Ž A. s 1. Proof. Suppose x g CT Ž A.. Let a and w be as in Lemma 5.5. As in the roof of Corollary 6.2, we may assume a g T. Then x g C Ž a. 0 because T is proof of abelian and connected and x g T F C Ž a.. Lemma 6.1 implies that w inverts x. But x is a 2 H -element and x commutes with an involution. By Corollary 4.6, x s 1. C OR OLLAR Y 6.4. ZŽ M 0 . s 1. It follows that CM 0 Ž A. s O 2 Ž M .. In a setting where the strongly embedded subgroup is connected, no further argument is necessary to finish the proof of Theorem 1.5; apply Fact 2.57 with U s O 2 Ž M .. The remaining part of this section is needed to prove Theorem 1.5 in a more general setting where M is not necessarily connected. The following notation is taken from w11x: T w w x s  m g M : mw s my1 4 X1 s  w g I Ž G _ M . : T w w x l M 0 s 14 X2 s  w g I Ž G _ M . : T w w x l M 0 / 14 . Following w11x, we compare the ranks of X 1 , X 2 , and I Ž G .. L EMMA 6.5. rkŽ X 2 . s rkŽ I Ž G ... Proof. It is enough to show that rkŽ X 1 . - rk I Ž G .. For w 1 , w 2 g I Ž G ., w 1 M 0 s w 2 M 0 if and only if w 2 w 1 g T w w 1 x l M 0 Žor T w w 2 x l M 0 as well.. If w 1 , w 2 g X 1 , this is equivalent to w 1 s w 2 . Hence, rkŽ G . s GR OUPS OF FINITE MOR LEY R ANK 803 rkŽ I Ž G .. q rkŽ CG Ž i .. G rkŽ"w g X 1 wM 0 . s rkŽ X 1 . q rkŽ M 0 .. Thus, we have rkŽ I Ž G .. G rkŽ X 1 . q rkŽ M . y rkŽ CG Ž i ... But I Ž M . is infinite. Hence, rkŽ I Ž G .. ) rkŽ X 1 .. The following lemma will be useful in proving Proposition 6.8 and also in the remaining part of this section. L EMMA 6.6. For w g X 2 , there exists a unique conjugate of T which is contained in T w w x. Proof. Let w g X 2 and t i g T w w x l M 0 , where t g T _  14 and i g O 2 Ž M .. By Lemma 6.1, w inverts C Ž t i . 0 G T i. Hence, T w w x G T i. We will show that T i is the only conjugate of T in M 0 which is contained in T w w x. Suppose T w w x G T j, j g O 2 Ž M .. By taking conjugates, we may assume that j s 1. Hence, T w w x G T, T i. Let u g O 2 Ž M . and t g T _  14 such that ut g T i. Then u w t w s Ž ut . w s Ž ut .y1 s ty1 uy1 s t w uy1. This implies Ž ty1 ut . w s uy1, which forces u g M l M w and consequently, u s 1. Therefore, t g T lT i and we get t, t i g T i implying w t, i x g T i l O 2 Ž M . s 1. Hence, t g CT Ž i . s CT Ž A. s 1 ŽLemma 5.1, Corollary 4.7.. As t / 1, the only possibility is i s 1. C OR OLLAR Y 6.7. For w g X 2 , T w w x l M 0 is a conjugate of T. We examine the ranks of some relevant groups more closely. P R OPOSITION 6.8. rkŽ G . s rkŽ C ŽT .. q 2 rkŽ O 2 Ž M .., where T is a complement to O 2 Ž M . in M 0 . Proof. Let w g I Ž G _ M . be an involution inverting T. By Lemma 6.6, w 1 g X 2 if and only if T w w 1 x G T u for some unique u g O 2 Ž M .. For such a u, t g T _  14 implies t uw 1 s tyu s t w u. Therefore, y1 y1 w 1u w g C ŽT .. As a result, we get c g C ŽT . such that w 1u s cw. Note that such a c is unique. Define f : X 2 ª w CŽT .O 2 Ž M . w 1 ¬ w cu . f is well-defined as both u and c are unique for any given w 1. f is also a definable map. Ž1. f has finite fibers. Proof of Ž1.. If w cu s w c1 u1 , then this element inverts both T cu and Ž . T . Therefore, T u s T u1 , by Lemma 6.6. Thus, uuy1 1 g NM 0 T . But by Corollary 4.7, I Ž NM ŽT .. s B. Hence, u s u1. We therefore get w c s w c1 . Ž . Ž . This implies ccy1 1 g C w l C T , which is a finite group. This proves the claim. c1 u1 804 TUNA ALTINEL Claim Ž1. implies that rkŽ X 2 . F rkŽ w CŽT .O 2 Ž M . .. On the other hand, for any c g C ŽT . and u g O 2 Ž M ., T w w cu x G T u, in other words, w cu g X 2 . Hence, w CŽT .O 2 Ž M . : X 2 . We conclude that rkŽ X 2 . s rkŽ w CŽT .O 2 Ž M . .. 0 Now, we show that rkŽ w C ŽT .O 2 Ž M . . s rkŽ w CŽT . O 2 Ž M . .. It is enough to show that u : w C G ŽT . 0 k O 2Ž M . ª w C G ŽT . 0 O 2Ž M . w ck u ¬ w cu , where k g C ŽT ., is a bijection because w C ŽT .O 2 Ž M . is a union of finitely many sets of this form. For any c, c1 g C ŽT . 0 , u, u1 g O 2 Ž M ., w ck u s w c1 k u1 implies u s u1 , as above. Therefore, w c s w c1 . But c, c1 g C ŽT . 0 . Lemma 6.1 implies wc 2 s wc12 . We eventually get c s c1 , using Fact 2.35. This argument is reversible and shows that u is well-defined and bijective. Before we conclude, we define one more mapping as follows: 0 g : CG Ž T . O 2 Ž M . ª w C G ŽT . 0 O 2Ž M . cu ¬ w cu . g is well-defined because for any c, c1 g CG ŽT . 0 and u, u1 g O 2 Ž M ., y1 cu s c1 u1 if and only if cy1 g CG ŽT . 0 l O 2 Ž M . s 1. It has 1 c s u1 u finite fibers by the arguments in the proof of Ž1. and it is surjective. 0 We have rkŽ I Ž G .. s rkŽ X 2 . s rkŽ w CŽT . O 2 Ž M . .. This gives us rkŽ G . y 0 rkŽ O 2 Ž M .. s rkŽ X 2 . s rkŽ w C ŽT . O 2 Ž M . . s rkŽ C ŽT . 0 . q rkŽ O 2 Ž M ... The last equality holds because C ŽT . l O 2 Ž M . s 1. We therefore get rkŽ G . s rkŽ C ŽT .. q 2 rkŽ O 2 Ž M ... The proofs of Lemma 6.9 and Proposition 6.10 below follow the proofs of very similar facts in w11x. L EMMA 6.9. rkŽ I Ž G . M 0 . s rkŽ G .. Proof. We define ; on X 2 as follows: w 1 ; w 2 if and only if w 1 M 0 s w 2 M 0 . For w 1 , w 2 g X 2 , if w 1 ; w 2 , then w 2 w 1 g T w w 2 x, that is, w 1 g w 2 T w w 2 x. Therefore, w 1 ; w 2 if and only if w 2 w 1 g T w w 2 x l M 0 . Since, by Corollary 6.7, T w w 2 x l M 0 is a conjugate of T, w 2r;s w 2 T x , for some x g O 2 Ž M .. As a result, rkŽ wr;. s rkŽT . for w g X 2 and hence rkŽ I Ž G . M 0 . G rkŽ X 2 M 0 . s rkŽ X 2r;. q rkŽ M 0 . s rkŽ X 2 . yrkŽT . q rkŽ M 0 . s rkŽ G . y rk C Ž i . y rkŽT . q rkŽ M 0 . s rkŽ G .. P R OPOSITION 6.10. Let a g T _  14 . Then C Ž a. s CM Ž a.. Proof. Suppose CM Ž a. - C Ž a.. Ž1. If u g O2 (M) and c g C(a) _ T then I(ucM 0 ) s B. Proof of Ž1.. Suppose ucb g I Ž ucM 0 .. As b s t¨ , where ¨ g O 2 Ž M ., t g T, we may assume that uc¨ g I Ž ucM 0 . with u, ¨ g O 2 Ž M .. Since Ž uc¨ . u s c¨ u g cM 0 , we may assume cu g I Ž cM 0 .. Therefore, Ž cu. 2 s 1. 805 GR OUPS OF FINITE MOR LEY R ANK cu g I Ž cM 0 . : I Ž cM . s Ž cu.T w cu x. As Ž cu. a s cu a g I Ž cM ., we get Ž cu.y1 Ž cu. a s uy1 u a g T w cu x. But uy1 u a g O 2 Ž M . also. Therefore, u s u a and we conclude u s 1, which forces c 2 s 1. But Corollary 4.6 implies that C Ž a. does not contain involutions, thus c s 1. This contradicts the choice of c. Ž2. Let u 1 , u 2 g O2 (M), c1 , c 2 g C(a) _ M, with u 1 c1 M 0 s u 2 c 2 M 0 . Then u 1 s u 2 and c1T s c 2 T. Proof of Ž2.. Suppose u1 c1 s u 2 c 2¨ for some ¨ g M 0 . c1 s uc2¨ with u s uy1 ¨ g O 2 Ž M . i T, we may also assume ¨ g O 2 Ž M .. 1 u 2y1and as y1 y1 y1 c 1¨ ¨ Since a s a s a u c 2 , a¨ g M 0 l M 0 c 2 . Hence, ay1 a¨ s w a, ¨ y1 x is a 2 H -element. But it is also a 2-element. Therefore, w a, ¨ y1 x s 1, Ž . Ž . implying ¨ s 1. This forces c1 s uc2 and c1 cy1 2 s u g C a l O 2 M s 1. Hence, we get u s 1 and c1 s c 2 , which is what we wanted. Now, let Y s D ucM 0 : c g C Ž a. _ M, u g O 2 Ž M .4 . If C Ž a. g M then Ž2. implies rkŽ Y . s rkŽ O 2 Ž M .. q rkŽ C Ž a.. y rkŽT . q rkŽ M . s rkŽ C Ž a.. q 2 rkŽ O 2 Ž M .. G rkŽ C ŽT .. q 2 rkŽ O 2 Ž M .. s rkŽ G .. On the other hand, Ž1. implies that Y and I Ž G . M 0 ŽLemma 6.9. are disjoint. As a result, we get in G two disjoint sets having the same rank as G, which contradicts the connectedness of G. This finishes the proof of Proposition 6.10. C OR OLLAR Y 6.11. rkŽ G . s rkŽT . q 2 rkŽ O 2 Ž M ... Proof. Proposition 6.10 implies that C ŽT . F M. But I Ž C ŽT .. s B by Corollary 4.7. Therefore, C ŽT . 0 s T. The conclusion follows from Proposition 6.8. The following lemmas are crucial to concluding that G is a group of the type described by the hypotheses of Fact 2.56. L EMMA 6.12. For any g g G _ M, u, ¨ g O 2 Ž M ., and m1 , m 2 g M, ugm1 s ¨ gm 2 if and only if u s ¨ and m1 s m 2 . y1 Proof. ugm1 s ¨ gm 2 implies ¨ y1 u s gm 2 my1 gMg 1 g 1. Hence u s ¨ . L EMMA 6.13. y1 l O2 Ž M . s For any g g G _ M, rkŽ M 0 gO 2 Ž M .. s rkŽ G .. Proof. Let g g G _ M. We define the map u : O 2 Ž M . = T = O 2 Ž M . ª M 0 gO 2 Ž M . Ž x 1 , t , x 2 . ¬ x 1 tgx 2 . 806 TUNA ALTINEL By Lemma 6.12, x 1 tgx 2 s xX1 t9gxX2 if and only if x 1 t s xX1 t9 and x 2 s xX2 . The former identity is also equivalent to xXy1 x 1 s t9ty1 g O 2 Ž M . l T. 1 But this last group is trivial. Hence u is injective. As a result, using Corollary 6.11, we get rkŽ M 0 gO 2 Ž M .. G 2 rkŽ O 2 Ž M .. q rkŽT . s rkŽ G .. Now we can prove Theorem 1.5. Proof of Theorem 1.5. Lemma 6.13 and the connectedness of G imply that we have G s M 0 " O 2 Ž M . gM 0 for any g g G _ M. Therefore, the action of G on the left coset space of M 0 is doubly transitive. Now, we will show that G is a split Z assenhaus group of characteristic 2 with M 0 a one-point stabilizer and O 2 Ž M . the normal complement of any two-point stabilizer in M 0 . This, together with Fact 2.56, will finish the proof of Theorem 1.5. Write G s M 0 " M 0 wO 2 Ž M ., where w is an involution inverting a complement T of O 2 Ž M . in M 0 . M 0 and wM 0 are two points in the left coset space of M 0 whose pointwise stabilizer is T s M 0 l M 0 w . The pointwise stabilizer of M 0 is M 0 . We already know that M 0 s O 2 Ž M . i T. Thus, all we have to do is to check that a three-point stabilizer is trivial. Suppose t g M 0 l M 0 w and assume that t stabilizes a third point uwM 0 , where u g O 2 Ž M .. Then uwM 0 s tuwM 0 s y1 y1 y1 y1 u t twM 0 s u t wty1 M 0 s u t wM 0 . By Lemma 6.12, u s u t . Proposition 6.3 implies t s 1. This finishes the proof of Theorem 1.5. ACKNOWLEDGMENTS This paper is part of the author’s Ph.D. thesis. He thanks his advisor Gregory Cherlin, without whom this work would not exist. The author also thanks Alexander Borovik for many fruitful ideas he has generously offered. R EFER ENCES 1. T. Altinel, ‘‘Groups of Finite Morley R ank with Strongly Embedded Subgroups,’’ Ph.D. thesis, R utgers, the State University of New Jersey, 1994. 2. T. Altinel, G. Cherlin, L.-J. Corredor, and A. Nesin, A Hall theorem for v-stable groups, submitted for publication. 3. O. V. 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