Groups of Mixed Type

JOURNAL OF ALGEBRA 192, 524᎐571Ž. 1997 ARTICLE NO. JA966950

Tuna AltınelU

Uni¨ersite´ Claude Bernard Lyon 1, 43 bl¨ddu11no¨embre 1918, 69622 Villeurbanne Cedex, France

View metadata, citation and similar papers at core.ac.uk brought to you by CORE Alexandre Borovik† provided by Elsevier - Publisher Connector

Department of Mathematics, Uni¨ersity of Manchester, Institute of Science and Technology, Manchester M60 1QD, England

and

Gregory Cherlin

Department of Mathematics, Rutgers Uni¨ersity, New Brunswick, New Jersey 08903

Communicated by Gernot Stroth

Received July 4, 1995

1. INTRODUCTION

The Cherlin᎐Zil’ber conjecture states that an infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. The proofs of the following three conjectures would provide a proof of the Cherlin᎐Zil’ber conjecture: 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, и , 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.

* E-mail: altinel@desargues. univ-lyon1.fr. † E-mail: [email protected].

524

0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. GROUPS OF MIXED TYPE 525

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. The first conjecture appears to be very difficult, and the state of the second conjecture is unclear, but in the last decade it has become clear that ideas from finite group theory are very helpful for the classification of the simple groups of finite Morley rank satisfying the properties stated in Conjecture 3, namely, tame groups. Indeed, the first step toward the solution of Conjecture 3 is an idea from finite group theory. One analyzes a minimal counterexample, which in the context of groups of finite Morley rank means an example ‘‘of minimal rank.’’ Infinite, proper, definable, simple sections of such a counterexample are then algebraic groups over algebraically closed fields. We make the following definitions for our convenience:

DEFINITION 1.1. A group of finite Morley rank whose infinite, definable, simple sections are algebraic groups over algebraically closed fields is called a K-group.

DEFINITION 1.2. A group of finite Morley rank whose proper, definable subgroups are K-groups is called a K U-group. Conjecture 3 would follow from the following conjecture: Conjecture 3X. An infinite, simple, tame K U-group of finite Morley rank is an algebraic group over an algebraically closed field. Another reason why finite group theoretic ideas seem to be relevant in the context of tame groups of finite Morley rank is the impact of involutions on the structure of these groups. The following facts illustrate this principle: Fact 1.3wx 10, 13, 12, 18 . Bad groups do not have involutions. Fact 1.4wx 9, Theorem B.1, p. 353 . If G is a connected tame group of finite Morley rank, then any connected definable section of G which does not have involutions is nilpotent. In order to attack Conjecture 3X, one expects to follow the lead of finite group theory in analyzing the centralizers of involutions. As part of this analysis it is important to understand the structure of the Sylow 2-subgroups of a group of finite Morley rank. The main result in this direction is the following theorem of Borovik and PoizatŽ for the notation S 0 below cf. Definition 2.6. : Fact 1.5wx 11 . If S is a Sylow 2-subgroup of a group of finite Morley rank then the following hold: Ž.i S is nilpotent-by-finite. 526 ALTINEL, BOROVIK, AND CHERLIN

0 Ž.ii S s B)T is a central product of a definable, connected, nilpotent subgroup B of bounded exponent and a divisible, abelian 2-group T. Moreover, B and T are uniquely determined. It is helpful to compare the situation in Fact 1.5 to that of algebraic groups over algebraically closed fields. An algebraic group is defined over an algebraically closed base field K whose characteristic has a strong impact on the structure of the Sylow 2-subgroups. If charŽ.K s 2 then the Sylow 2-subgroups are nilpotent and of bounded exponent; if charŽ.K / 2 then the Sylow 2-subgroups areŽ. divisible abelian -by-finite. Consider, for Ž. Ž. example, SL2 K , where K is an algebraically closed field. If char K s 2 then any Sylow 2-subgroup of the group is isomorphic to the additive group Kq, e.g., 1 u : u K . ½5ž/01 g q On the other hand, if charŽ.K / 2 then the Sylow 2-subgroups are U isomorphic to a subgroup of K i ޚr2ޚ with ޚr2ޚ acting by inversion, e.g.,

␭ 0 U 2 n 0 y1 : ␭ g K , ␭ s 1 i . ½5ž/0␭y1 ¦;ž/10 Using the terminology in Fact 1.5, one can say that, in algebraic groups, either B s 1orTs1. In the context of groups of finite Morley rank, the general picture is somewhat different. For example, if K1 is an alge- Ž. braically closed field with char K12s 2 and K is an algebraically closed field with charŽ.K22/ 2 then SLŽ. K1= SL2 Ž. K2is a group of finite Morley rankŽ. although not an algebraic group each of whose Sylow 2-subgroups is isomorphic to the direct sum of a Sylow 2-subgroup of

SL21Ž. K and a Sylow 2-subgroup of SL22 Ž. K . Therefore, in a group of finite Morley rank, the Sylow 2-subgroups are in general a mix of the possibilities which arise in algebraic groups. Accordingly, the following definitions are natural:

DEFINITION 1.6. A group of finite Morley rank is said to be of e¨en type if its Sylow 2-subgroups are of bounded exponent. A group of finite Morley rank is said to be of odd type if the connected component S 0 of any Sylow 2-subgroup S is divisible abelian. A group of finite Morley rank is said to be of mixed type if it is not of one of the above types. If the Cherlin᎐Zilber conjecture is true, then there should be no simple group of finite Morley rank of mixed type: Conjecture 3A. A simple group of finite Morley rank cannot be of mixed type. GROUPS OF MIXED TYPE 527

We prove the following special case, which is the version that will actually be needed in the context of Conjecture 3X.

U THEOREM 1.7. There exists no simple, tame K -group of finite Morley rank of mixed type. Thus Conjecture 3X splits into two parts, which can be considered independently: Conjecture 3X E. An infinite, simple, tame K U-group of finite Morley rank of even type is an algebraic group over an algebraically closed field of even characteristic. Conjecture 3X O. An infinite, simple, tame K U-group of finite Morley rank of odd type is an algebraic group over an algebraically closed field of characteristic different from 2. There is already a considerable body of work on these two conjectures. The proof of Theorem 1.7 is by contradiction. We analyze a simple, tame K U-group G of mixed type. There are two main steps in this analysis. We first show that G has a weakly embedded subgroup. The second main step of our analysis allows us to conclude that this weakly embedded subgroup is a strongly embedded subgroup. This yields a contradiction because groups of finite Morley rank with strongly embedded subgroups cannot be of mixed type. The notions of weakly and strongly embedded subgroups are defined in Sections 3 and 7, respectively. The notion of a weakly embedded subgroup will also play a crucial role in the further treatment of Conjecture 3X, as we will show elsewhere, functioning as adequate replacement for such ideas from finite group theory as Glauber- man’s ZU theorem, in the context of groups of even type. Both the construction of a weakly embedded subgroup and the proof of the fact that this is a strongly embedded subgroup require a detailed analysis of some special proper, definable subgroups of G. These subgroups, which are necessarily K-groups, constitute the parts of G that are of ‘‘even type’’ or ‘‘odd type.’’ While analyzing these subgroups we will encounter central extensions of quasisimple algebraic groups which are perfect tame groups H of finite Morley rank. We prove that in such cases H is itself an algebraic group. It seems this result will be useful in the analysis of K-groups arising in other situations as well. The organization of the paper is as follows: in the next section we review the background results which we need. In Section 3 we begin the discussion of groups of finite Morley rank with weakly embedded subgroups. Section 4 is devoted to central extensions. In Section 5 we analyze the ‘‘even’’ and ‘‘odd’’ parts of G and the interactions between these parts. In Section 6 we find a weakly embedded subgroup in G. The final section, 528 ALTINEL, BOROVIK, AND CHERLIN

Section 7, contains the proof of the result that the weakly embedded subgroup of Section 6 is strongly embedded. Section 7 also contains the results about strongly embedded subgroups which imply that this second main step of the proof yields a contradiction.

2. BACKGROUND

In this section we list various facts needed in the sequel. For any subset Xof any group G, IXŽ.will be used to denote the set of involutions in X. Ž. If t g G then CtGs Ct.

DEFINITION 2.1. Let G be a group of finite Morley rank. A subgroup H of G is definably characteristic if it is invariant under definable group automorphisms. In the sequel, characteristic will mean definably characteristic.

Fact 2.2wx 9, Exercise 10, p. 78 . Let G be a group of finite Morley rank. G0 contains all connected definable subgroups of G.

Proof. See the hint on p. 380 ofwx 9 . Fact 2.3wx 26 . 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.

Fact 2.4wx 26 . Let H F G be a definable connected subgroup. Let X:Gbe any subset. Then the subgroup wxH, X is definable and connected.

Fact 2.5wx 26 . Let G be a group of finite Morley rank. Then G nand GŽ n. are definable. If G is connected, then G n and GŽn. are connected.

DEFINITION 2.6. Let G be a group of finite Morley rank. If X : G, then the definable closure of X, denoted by dXŽ., is the intersection of all the definable subgroups of G which contain X. Note that, as groups of finite Morley rank satisfy the descending chain condition on definable subgroups, this intersection is finite and therefore definable. If X is a subgroup of G then the connected component of X, denoted by X 0,is 0 defined to be X l dXŽ.. Fact 2.7wx 9, Exercise 2, p. 92 . Let G be a group of finite Morley rank. Ž. ŽŽ.. Assume X : G. Then CXGGsCdX. Ž. Ž. Ž. Proof. Let g g CXGGG. Then X : Cg. But Cgis a definable Ž. Ž. ŽŽ.. subgroup of G. Therefore, dX FCgGGand g g CdX. GROUPS OF MIXED TYPE 529

Fact 2.8wx 9, Lemma 5.36, p. 89 . Let G be a group of finite Morley rank. If B is a definable normal subgroup of G and B : X : G, then dXŽ.Ž.rBsdXrB. Fact 2.9wx 9, Corollary 5.38, p. 90 . Let G be a group of finite Morley rank and let H be a subgroup of G.If His solvableŽ. resp. nilpotent of class n, then dHŽ.is solvable Ž resp. nilotent . of class exactly n.

Fact 2.10wx 17 . Let G be an abelian group of finite Morley rank. Then the following hold:

Ž.i GsD[B, where D is a divisible subgroup and B is a subgroup of bounded exponent. Ž.ii D Žޚ . ޑ, where the index sets I are ( [[p prime IIppϱ [[ p finite. Ž.iii G s DC, where D and C are definable characteristic subgroups of G, D is divisible, C has bounded exponent, and their intersection is finite. The subgroup D is connected. If G is connected, then C can be taken to be connected.

Fact 2.11wx 22 . Let G be a nilpotent group of finite Morley rank. Then G s D)C, where D and C are definable characteristic subgroups of G, D is divisible, and C is of bounded exponent.

Fact 2.12wx 22 . Let G be a divisible nilpotent group of finite Morley rank. Let T be the torsion part of G. Then T is central in G and G s T [ N for some torsion-free divisible nilpotent subgroup N. Fact 2.13wx 9, Lemma 6.2, p. 96 . If G is an infinite nilpotent group of finite Morley rank then ZGŽ.is infinite.

Fact 2.14wx 9 , Lemma 6.3, p. 96x . Let G be a nilpotent group of finite Ž. Morley rank. If H - G definable subgroup of infinite indes then NHG rH is infinite.

DEFINITION 2.15. Let G be a group of finite Morley rank. Let ␴ Ž.G be the subgroup generated by all the normal solvable subgroups of G, and let FGŽ.be the subgroup generated by all the normal nilpotent subgroups of G.␴Ž.Gis called the solvable radical of G. FG Ž.is called the Fitting subgroup of G.

Fact 2.16wx 4, 21 . Let G be a group of finite Morley rank. Then FGŽ. and ␴ Ž.G are definable, and they are nilpotent and solvable, respectively.

Fact 2.17wx 27 . 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 530 ALTINEL, BOROVIK, AND CHERLIN

Ž. H-minimal. Assume CAH s1. Then the following hold:

Ž.i The subring K ޚwxH annޚ Ž. A of End Ž.A is a definable s r w H x algebraically closed field; in fact, there is an integer l such that every Ýl element of K can be represented as the endomorphism is1 hi, where Ž. higH. U Ž.ii A ( Kq, H is isomorphic to a subgroup T of K , and H acts on A by multiplication, in other words:

ta GAiH :tT,aK. s (½5ž/01 g g

Ž.iii In particular, H acts freely on A, K s T q иии qTl Žtimes . and ÄÝl 4 U with the additive notation A s is1 haii:hgH for any a g A . Fact 2.18wx 27 . Let G be a connected nonnilpotent solvable group of finite Morley rank. Then G interprets an algebraically closed field K. More precisely, a definable section of FGŽ.is isomorphic to Kqand a U definable section of GrFGŽ.is isomorphic to an infinite subgroup of K . Fact 2.19wx 20 . Let G be a connected solvable group of finite Morley 0 rank. Then GrFGŽ.ŽŽ.hence, GrFG.is divisible abelian group. Fact 2.20wx 14 . Let G be a nilpotent group. For a given 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. Fact 2.21wx 9, Exercises 11, 12, pp. 13, 14 . If G is a nilpotent-by-finite p-group, then the following hold: Ž.i ZG Ž ./1. Ž.ii If H is a nontrivial normal subgroup of G then ZGŽ.lH/1.

Ž.iii For any X - G, X - NXGŽ.; that is, G satisfies the normalizer condition. Proof. Seewx 1 . Fact 2.22wx 7 . Let G be a group of finite Morley rank and let H be a definable normal subgroup of G.If xis an element of G such that x is a p-element of G s GrH then the coset xH contains a p-element. In particular, if G is torsion-free, then GrH is torsion-free. DEFINITION 2.23. Let G be a group. For any prime number p,a pH-element g of G is an element whose order is either infinite or relatively prime to p. G is said to be a pH -group if every g in G is a pH -element. A pX-element is an element of finite order relatively prime to p. G is said to be a pX-group if all of its elements are pX-elements. GROUPS OF MIXED TYPE 531

Fact 2.24wx 9, Exercise 11, p. 72 . Let G be a pH-group of finite Morley rank, where p is a prime number. Then G is p-divisible. Proof. Seewx 1 . Fact 2.25wx 9, Exercise 12, p. 72 . Let G a pH-group of finite Morley rank. Then every element of G has a unique pth root.

p p Proof. Let x and y be two elements of G such that x s y . The Ž Ž .. Ž Ž .. H group ZCGG x lZC y is a definable abelian p -group which con- pp ŽŽ..ŽŽ.. tains x . By Fact 2.24, x has a pth root z in ZCGG Z lZC Y . But ŽŽ.. ZCG x is abelian and does not have p-torsion. Therefore, x s z. Simi- larly, y s z. Fact 2.26wx 8 . Let K and L be definable pH-subgroups of a group G of finite Morley rank. Assume K normalizes L. Then KL is also a pH -subgroup. Fact 2.27wx 11 . Let G be a group of finite Morley rank. Let D be a divisible abelian subgroup of G. Then, for every prime p, D has finitely many elements of order p. Proof. Seewx 1 . Fact 2.28wx 9, Exercise 1, p. 97 . An infinite nilpotent p-group of finite Morley rank and of bounded exponent has infinitely many central elements of order p. Proof. Seewx 1 .

DEFINITION 2.29. A divisible abelian p-subgroup of a group of a finite Morley rank is called a p-torus. Fact 2.30wx 11 . Let T be a p-torus in a group of finite Morley rank. Ž. Ž. Then w NTGG:CTx-ϱ. Moreover, there exists c g ގ such that Ž. Ž. wNTGG:CTxFcfor any torus in G. Fact 2.31wx 11 . The Sylow 2-subgroups of a group of finite Morley rank are conjugate. Fact 2.32wx 9, Lemma 10.22, p. 188 . Let S and T be as in Fact 1.5. If 0 g Ž. X,Y:S and X s Y, where g g G, then there exists h g NTG such h ŽŽ. 0. that X s Y that is, NG T controls fusion in S . Fact 2.33wx 3 . Let G be a group of finite Morley rankŽ or, more generally, ␻-stable. and solvable, N e G, and let H be a Hall ␲-subgroup of G for some set ␲ of primes. Then:

Ž.i HlN is a Hall ␲-subgroup of N, and all Hall ␲-subgroups of N are of this form. 532 ALTINEL, BOROVIK, AND CHERLIN

Ž.ii If N is definable then HNrN is a Hall ␲-subgroup of GrN, and all Hall ␲-subgroups of GrN are of this form. Fact 2.34wx 7 . Let G be a connected solvable group of finite Morley rank. Then the Hall ␲-subgroups of G are connected.

DEFINITION 2.35. A definable, connected, 2-subgroup of bounded exponent of a group of finite Morley rank is called a unipotent 2-subgroup.

Note that a unipotent 2-subgroup is nilpotent by Fact 1.5Ž. i . A special case of the following fact was proven inwx 1 . Fact 2.36. Let Y be a connected solvable group of finite Morley rank and let S be a Sylow 2-subgroup of Y.If Bis the unique largest unipotent 2-subgroup of S as in Fact 1.5 then B F FYŽ.. In particular, B is a characteristic subgroup of Y.

Proof. By Fact 2.19, YrFYŽ.is a divisible abelian group. Thus, by Fact 2.27, YrFYŽ.has finitely many involutions. On the other hand a nontrivial unipotent 2-subgroup has infinitely many involutions by Fact 2.28. There- fore, YrFYŽ.has no nontrivial unipotent 2-subgroups and this implies BFFYŽ.. By Fact 2.20, B is contained in the unique Sylow 2-subgroup of FYŽ.. Therefore, by Fact 1.5Ž. ii , B is characteristic in Y.

The following fact follows from Fact 2.36.

Fact 2.37wx 9, Exercise 2, p. 175 . Let Q and E be subgroups of a group of finite Morley rank such that Q is normal, connected, solvable, definable, and does not contain involutions, and E is a definable connected 2-group of bounded exponent. Then wxQ, E s 1. Fact 2.38wx 19 . Let ␣ be a definable involutive automorphism of a group of finite Morley rank G.If ␣ has finitely many fixed points then G has a definable normal subgroup of finite index which is abelian and inverted by ␣.

Fact 2.39wx 19 . Let ␣ be a definable involutive automorphism of a group of finite Morley rank G.If ␣ has no nontrivial fixed points then G is abelian and inverted by ␣.

Fact 2.40wx 9, Theorem 8.4, p. 124 . Let G s G i H be a group of finite Morley rank, where G and H are definable, G is an infinite simple Ž. algebraic group over an algebraically closed field, and CGH s1. Then, viewing H as a subgroup of AutŽ. G , we have H F InnŽ.G ⌫, where Inn Ž.G is the group of inner automorphisms of G and ⌫ is the group of graph automorphisms of G. GROUPS OF MIXED TYPE 533

Fact 2.41wx 1 . Let G be a connected nonsolvable K-group of finite Morley rank. Then Gr␴ Ž.G is isomorphic to a direct sum of simple algebraic groups over algebraically closed fields. In Section 5, we will need the following facts about algebraic groups over algebraically closed fields. Apart from Fact 2.46, these facts are found inwxwx 25 , 16 , or wx 15 , which are our main references for the theory of algebraic groups and related subjects.

DEFINITION 2.42wx 16, Sect. 7.5 . Let G be an algebraic group over an algebraically closed field and let M be an arbitrary subset of G. AŽ.M Ž.the group closure of M is the intersection of all closed subgroups of G containing M. Fact 2.43wx 16, Proposition 7.5 . Let G be an algebraic group over an Ž. algebraically closed field, I an index set, and fii: X ª GigIa family of Ž. morphisms from irreducible varieties Xii, such that each Y s fXicon- tains the identity element of G. Set M s Dig IiY . Then: Ž.a A ŽM .is a connected subgroup of G. Ž.b For some finite sequence a s ŽŽ.a 1 ,...,an Ž..in I,

e1 en AŽ.M s YaŽ1.иии YaŽn. Ž.eis "1. Fact 2.44wx 16, Theorem 17.6 . Let G be a connected solvable subgroup of GLŽ. V ,0/V finite dimensional. Then G has a common eigenvector.

DEFINITION 2.45. Let G be a group of finite Morley rank. OGŽ.is the largest, definable, connected, normal 2 H -subgroup of G. Fact 2.46wx 6 . If H is a simple algebraic group over an algebraically closed field K of characteristic different from 2, and t is an involutive Ž. Ž0. automorphism of H, then FCttsZC is a finite extension of an Ž. algebraic torus over K and ECt sL1 иии Ln is the central product of quasisimple subgroups Li which are algebraic groups over K. In particular, Ž. 0 Ž.0Ž. OCtttts1 and C s FC EC . We will also need the more detailed description of the structure of simple algebraic groups in terms of root subgroups, as follows: Fact 2.47wx 25, Corollary to Lemma 15 . Let ␣, ␤ be roots with ␣ q ␤ / 0. Let k be a field. Then ij Ž.xt␣Ž.,xt␤ Ž. sŁxcti␣qj␤Ž.ij u, 11 where Ž.A, B s ABAy By, where the product on the right is taken over all roots i␣ q j ␤ Ž.i, j g ޚ arranged in some fixed order, and where the cij are integers depending on ␣, ␤, and the chosen ordering, but not on t and u. 534 ALTINEL, BOROVIK, AND CHERLIN

Fact 2.48w 25, ExampleŽ. a , p. 24x . If ␣ q ␤ is not a root, then the right-side of the commutator formula in Fact 2.47 reduces to 1.

Fact 2.49wx 15, Lemma 10.1 . Let ⌽ be a root system. If ⌬ is a base of ⌽, then Ž.␣, ␤ F 0 for ␣ / ␤ in ⌬, and ␣ y ␤ is not a root.

Fact 2.50wx 16, Theorem 27.3 . Let G be a reductive algebraic group, T Ž. a fixed maximal torus, and ⌽ s ⌽ G, T . Let ⌬ be a base of ⌽. Let Z␣ Ž. Ž .0 denote CTG ␣␣, where T s ker ␣ and ␣ g ⌽. Then G is generated by Ž. Ž. the Z␣ ␣ g ⌬ , or equivalently by T along with all U␣ "␣ g ⌬ .

Fact 2.51wx 16, Corollary 32.3 . A semisimple algebraic group of Lie rank 1 is isomorphic to SL22Ž. K or PGL Ž. K .

We conclude with some useful technical results.

PROPOSITION 2.52. Let Q be a group and i an inölution acting on Q. Suppose Q1 is a 2-diïsible, i-inäriant, normal subgroup of Q. If in Q1 eëry element has a unique square root or if i in erts Q then CŽ. i ¨ 1 Qr Q1 s Ž. CiQQ 11rQ.

y1 i Proof. We have to show that, for every x g Q, x x g Q1 implies that Ž. there exists y g Q1such that xy g CiQ . y1 i First, assume that i inverts Q11. Let x g Q such that x x g Q .AsQ1 y1 i 2 is 2-divisible, there is y g Q1 such that x x s y . This implies that i 1 i xy s xyy sŽ.xy . We are done. Now assume that in Q1 every element has a unique square root. y Ä i y14 y Consider Q11s z g Q : z s z . Let z g Q11.AsQ is 2-divisible, iŽ2.iŽi.2y1Ž2.y1 zhas a square root z11g Q . z s z 1s z 1and z s z1s Žy1.2 y1Ž.i y z11. By the uniqueness of the square roots, z s zi. Hence, Q1is a y1 i y y1 i 2 2-divisible set. Let x be as above. Then x x g Q1 . Hence, x x s y y for some y g Qi . Now, by the same argument as in the previous paragraph, we are done.

PROPOSITION 2.53. Let G s dŽ. S Q be a group of finite Morley rank, where S is Sylow 2-subgroup of G and Q is a normal, definable 2 H -subgroup.

Assume that Q has a definable, normal subgroup Q1 which is normalized by dŽ. S also. Assume also that S dSŽ..Then C ŽŽ.. d S Q Q CdSŽŽ... e Q 11r s QrQ1 Proof. Let L Q CdSŽŽ... It is sufficient to show that L r 1 s Qr Q1 F ŽŽ.. x Ž. CdSQQ 1. Let x g L. The assumptions imply that S F QdS1 . By the conjugacy of Sylow 2-subgroups and the fact that S e dSŽ.we can find xy y1 Ž. Ž. ŽŽ..Ž ygQ1 such that S s S . Thus xy g NSQQQsCSsCdS Fact .ŽŽ.. 2.7 . This implies L F CdSQQ 1. GROUPS OF MIXED TYPE 535

We need a definition before the following proposition.

DEFINITION 2.54. Let G be a group of finite Morley rank. The Prufer¨ 2-rank of a 2-torus of G is the number of copies of ޚ2ϱ whose direct product form this 2-torus. By rank considerations, this number is always finite in a group of finite Morley rank. If H is a subgroupŽ not necessarily definable.Ž. of G then the Prufer¨ 2-rank of H, denoted by pr H is the maximum of the Prufer¨ 2-ranks of the 2-tori in H.

PROPOSITION 2.55. Let Q i T be a group of finite Morley rank, where Q is a definable, connected, sol¨able 2-H group and T s dSŽ.,where S is a Sylow 2-subgroup of T. Assume that S is a 2-torus. If the Prufer¨ 2-rank of T is at least 2 then

²: QsCSQŽ.00:SFS and prŽ.S0s pr Ž.S y 1.

Proof. If the action of T on Q is trivial then there is nothing to do, so we may assume that T does not centralize Q. Let Q be a counterexample Ž. of minimal Morley rank. Consider the following action of TrCQT on Q: Ž. tt for any x g Q and t g TrCQT ,xis defined to be x . Let A be a Ž. TrCQT -minimal normal subgroup of Q.AsQis solvable, A is abelian. ²Ž. : By induction on Morley rank, QrA s CSQr A 00:Sis as above . But by Ž. Ž. ² Ž. Proposition 2.53, CSQrA 0sCSAQ 0 rA. This implies Q s CSAQ 0 : :²Ž.: S0 is as above . So it suffices to show that A F CSQ 00:Sis as above , Ž. Ž. and we show in fact that CAT has Prufer¨ rank at least pr T y 1. We Ž. may assume that A is not centralized by TrCQT , equivalently A is not centralized by T. We consider the action of T CQŽ.CA Ž.Ž r TTr rCTŽQ.s Ž. Ž. Ž.. TrCQTTTrCArCQ on A. By Fact 2.17, A ( Kq, where K is an Ž. Ž. Ž. U algebraically closed field and TrCQTTTrCArCQFK. Since Ž. Ž. Ž. Ž. Ž Ž.. TrCQTTTrCArCQ(TrCA T, we conclude that pr CATG prŽ.T y 1, as claimed.

3. WEAKLY EMBEDDED SUBGROUPS

DEFINITION 3.1. Let G be a group of finite Morley rank. A proper definable subgroup M of G is said to be weakly embedded if it satisfies the following conditions: Ž.i Any Sylow 2-subgroup of M is infinite. g Ž.ii For any g g G_ M, M l M has finite Sylow 2-subgroups. We first prove a few basic properties of weakly embedded subgroups. 536 ALTINEL, BOROVIK, AND CHERLIN

PROPOSITION 3.2. Let G be a group of finite Morley rank with a weakly embedded subgroup M. Then the following hold: Ž. Ž. i For any Sylow 2-subgroup S of M, NSG FM. Ž.ii If S is a Sylow 2-subgroup of M then S is a Sylow 2-subgroup of G. Proof. Ž.i This is immediate. Ž.ii Let S be a Sylow 2-subgroup of M and suppose T ) S is a Sylow Ž. Ž. Ž. Ž. 2-subgroup of G. By Facts 1.5 i and 2.21 iii , NSTT)S. But NSlM sSas S is Sylow 2-subgroup of M. This contradicts partŽ. i . Now we give a characterization of weakly embedded subgroups.

PROPOSITION 3.3. Let G be a group of finite Morley rank. A proper definable subgroup M of G is a weakly embedded subgroup if and only if the following hold: Ž.i M has infinite Sylow 2-subgroups. Ž. Ž . ii For any unipotent 2-subgroup U and 2-torus T in M, NUG FM, and NGŽ. T - M. Proof. It is clear that if M is a weakly embedded subgroup then it satisfies the above conditions. Therefore, we assume that M is a proper definable subgroup with the above properties and show that it is a weakly embedded subgroup. Let g g G and suppose S is an infinite Sylow g 2-subgroup of M l M . We will show that g g M. We claim that S is Sylow 2-subgroup of G. Suppose toward a contradiction that this is not the case. Let S1 be a Sylow 2-subgroup of G such that S-S. By Fact 2.21Ž. iii , S - NS Ž..IfS0 B Tas in Fact 1.5, then 1 S1 s ) NSŽ.NB Ž .NT Ž. M Mg, which contradicts S is a Sylow 2- SGG1F l F l g gy1 subgroup of M l M . Hence, S and S are Sylow 2-subgroups of M and xgy1 Ž. there exists x g M such that S s S . Therefore, gx g NSG FM, forcing g g M.

4. CENTRAL EXTENSIONS

It is possible to develop a theory of central extensions for tame groups. In this section, we show how to achieve this. Our proofs make use of the theory of central extensions of linear algebraic groups as explained inwx 25 and also of the ‘‘no bad fields’’ hypothesis. We will show inwx 2 that this hypothesis can be eliminated using K-theory and additional model theory. For the tame case, our present treatment is more straightforward. We prove the following theorem. GROUPS OF MIXED TYPE 537

THEOREM 4.1. Let G be a perfect tame group of finite Morley rank and let C be a definable central subgroup of G such that GrC is a uniërsal linear algebraic group oër an algebraically closed field; G is a central extension of finite Morley rank of a uniërsal linear algebraic group. Then C s Ž.1. In the proof of this theorem our notation and terminology will be the same as inwx 25 unless otherwise stated. We differ from wx 25 , and indeed from the theory of linear algebraic groups in general, in the use of the word simple, which we apply only to groups which are simple as abstract groups. Theorem 4.1 will be used in proving the following theorem:

THEOREM 4.2. Let G be a perfect tame group of finite Morley rank such that GrZŽ. G is a quasisimple algebraic group. Then G is an algebraic group. In particular, ZŽ. G is finite wx16, Sect. 27.5 . Before we start our argument, we would like to refer the reader towx 23 for results that are in a related vein. Throughout this section, for any group G and x, y g G, Ž.x, y will denote xyxy1 yy1 as inwx 25 . We start with an overview of some results from wx25 . In wx 25 , for a field k and a root system ⌺, the following relations over Ä Ž. 4 the set of symbols xt␣ :␣g⌺,tgkare defined:

Ž.A xt␣ Ž.is additive. Ž. Ž Ž. Ž.. BIf␣and ␤ are roots and ␣ q ␤ / 0, then xt␣ ,xt␤ s Ł Ž ij. xctui␣qj ␤ ij , where i and j are positive integers and the cij are integers depending on ␣, ␤, and the chosen ordering of the roots, but not on t or u. Ž X .Ž.Ž.Ž. Žy2. U Bwtxuw␣ ␣␣ytsxy␣ytufor t g k , where y1 wt␣Ž.sxtx␣ Ž.y␣␣ Žyt .xt Ž. U for t g k . Ž . Ž. Ž. Ž. Ž . C ht␣ is multiplicative in t, where ht␣␣␣swtwy1 for t g kU.

Let Xu denote the group presented byŽ. A and Ž. B if rank ⌺ ) 1 and by X Ž.A and Ž B.Ž if rank ⌺ s 1. If the the relation C. is added, then we get the universal Chevalley groupŽ seewx 25 ; the notation Xu is different from the one used inwx 25. . We quote the following lemmas and theorems:

LEMMA 4.3wx 25, Lemma 39, p. 70 . Let ␣ be a root and let Xu be as Ž. Ž.Ž.Ž.y1 abo¨e. In Xu, set f t, u s hthuhtu␣␣ ␣ .Then: 2 2 Ž.a ft Ž,u¨.Žsft,uft.Ž,¨ .. U Ž.bIf t, u generate a cyclic subgroup of k then fŽ. t, u s fu Ž.,t. 2 Ž.c If f Ž t, u .s fu Ž,t .,then f Ž t, u .s 1. Ž.d If t, u / 0 and t q u s 1, then fŽ. t, u s 1. 538 ALTINEL, BOROVIK, AND CHERLIN

THEOREM 4.4wx 25, Theorem 9, p. 72 . Assume that ⌺ is indecomposable and that k is an algebraic estension of a finite field. Then relations Ž.A and Ž.B X ŽŽor B ..if rank ⌺ s 1 suffice to define the corresponding uni¨ersal Che¨alley group, i.e., they imply relation Ž.C.

The following theorems and corollaries about central extensions are proven inwx 25 .

THEOREM 4.5wx 25, Theorem 10, p. 78 . Let ⌺ be an indecomposable root system and k a field such that<< k ) 4, and if rank ⌺ s 1, assume further that <

COROLLARY 4.6wx 25 . Xu is centrally closed. Each of its central extensions splits, i.e., its Schur multiplier is triïal. It yields the uniërsal coëring extension of all the Cheälley groups of the giën type.

THEOREM 4.7Žwx 25, Theorem 12Ž. Matsumoro, Moore. . Assume that ⌺ is an indecomposable root system and k a field with<< k ) 4. If X is the Ž. uni¨ersal Che¨alley group based on ⌺ and k, if Xu is the group defined by A, Ž. ŽX. B,and B,and if ␲ is the natural map from Xu to X with C s ker ␲ , the Schur multiplier of X, then C is isomorphic to the abstract group generated by U the symbolsÄ4 t, utŽ,ugk. subject to the relations:

Ž.aÄ4Ä4Ät,utu,¨st,u¨ 4Ä4Ä4Äu,¨;1,usu,1 4s1; 1 Ž.bÄ4Ät,ut,yuy4Äst,y1; 4 1 Ž.cÄ4Ät,usuy,t4; Ž.dÄ4Ät,ust,ytu 4; Ž.eÄ4Ät,ust,1Ž.ytu4; Ž. and in the case ⌺ is not of the type Cn n G 1 the additional relation X Žab . Ä4, is bimultiplicati¨e. In this case relationsŽ.Ž. a ᎐ e may be replaced by Ž abX. and ŽcX . Ä4,is skew. X Ždt. Ä4,yts1. X Žet. Ä4,1yt s1. Ä4 Ž. Ž . Ž .y1 The isomorphism is gi¨en by ␾: t, u ¬ hthuhtu␣ ␣␣ ,␣a fixed long root. GROUPS OF MIXED TYPE 539

For a proof of Theorem 4.1 we have to consider a perfect tame central extension G of finite Morley rank of a universal linear algebraic group X over an algebraically closed field K. Let Xu be the universal covering extension of X. Let C0 s ker ␺ , where ␺ is the covering map from G onto X. By the universality of Ž.␲ , Xuu, there exists a map ␪ from X into G Ž. Ž .ŽŽ.Ž.. such that ␺␪ s ␲ . Then ␪ XCu 0 sG. But G s G, G s ␪ Xuu, ␪ X Ž.Ž. s␪Xuu,Xs␪X u. Hence, ␪ is surjective. Then C s ker ␲ s ker ␺␪ s y1 y1Ž. ␪ ker ␺ s ␪ C0 . If f is the function defined in Lemma 4.3, we will need to show that ␪(f:K=KªZGŽ.is interpretable in G. The following fact is proved in wx24 : Fact 4.8wx 24, Corollaire 4.16 . In a simple algebraic group over an algebraically closed field, definability from the field and definability from the pure group coincide. We prove a generalization of this fact. The proof makes use of the following results fromwx 24 : Fact 4.9wx 24, Corollaire 4.2 . There are no constructibleŽ in the sense of algebraic geometry. bad groups. Fact 4.10wx 24, Theoreme´` 4.15 . An infinite field definable in a pure algebraically closed field F is definablyŽ. in F isomorphic to F.

PROPOSITION 4.11. In a quasisimple algebraic group o¨er an algebraically closed field, definability from the field and definability from the pure group coincide.

Proof. Let X s XFŽ.be a quasisimple algebraic group over an algebraically closed field F. We will useᎏ -notation to denote quotients by ZXŽ.. By Fact 4.9, no bad groups are definable in X. Therefore, we can interpret an algebraically closed field in XrZXŽ.using its Borel subgroups. Let us denote this field by K.As K is interpretable in F, these two fields are definably isomorphic in F by Fact 4.10. Let us denote this isomorphism by ␪. Let XKŽ.be a linear algebraic group over X isomorphic to XFŽ.by an isomorphism ␺ induced by ␪. The isomorphisms ␺␣ : Ž.Ž. Ž Ž .. Ž Ž .. XF␣ ªXK␣␣defined by ␺ xt␣sx␣␪t are definable in X since they can be written as the composition of the isomorphism XF␣Ž. ªXF␣Ž.induced by the canonical homomorphism X ª X with the Ž. definable isomorphism XF␣Ž.ªXK␣ given by

xt␣Ž.¬t¬␪ Ž.t¬x␣Ž.␪ Ž.t, Ž. Ž Ž.. where the maps xt␣Ž.¬tand ␪ t ¬ x␣ ␪ t are F-algebraic, hence definable in X by Fact 4.8. 540 ALTINEL, BOROVIK, AND CHERLIN

Ž. Ž . The isomorphisms ␺␣are induced by the isomorphism ␺ : XFªXK and conversely ␺ is definable from the ␺␣ since every element of X is a product of a bounded number of elements from the root groups X␣ . Therefore, ␺ is definable. Now, let A be a subset of XFŽ.ndefinable from F. ␺ Ž.A is definable in XKŽ.n from K and hence is definable in X. But ␺ and K are definable in 1 X. Hence, A s ␺␺y ŽŽ..Ais definable in X. PROPOSITION 4.12. Let G be a group of finite Morley rank. Assume that G is a perfect central extension of a uniërsal linear algebraic group X, such that the kernel of the coëring map from G onto X is a definable central subgroup of G. If Xuu is the uniërsal coëring of X and ␪: X ª G is the Ž. unique induced map as in Theorem 4.5, and f: K = K ª Xu is the function defined in Lemma 4.3, then the function ␪ ( f is interpretable in G. Proof. In order to prove Theorem 4.5, Steinberg proves that relations Ž.Ž.A , B , and Ž BX. can be lifted from a universal linear algebraic group X to any of its central extensions. To do so he starts with a central extension Ž.␺,Gof X and he constructs a map ␾ from the root subgroups of X into G. We will make use of this map in order to show that the function ␪ ( f is interpretable in G. The first step in the proof is to show that ␾ is interpretable in G.Todo so, we need to look at the at the definition of ␾. First, an element a of K U 2 ŽŽ.Ž..Ž. is chosen so that c s a y 1 / 0. In GrC, ha␣ ,xt␣␣sxctfor all ŽŽ.. ␣g⌺,tgK. Then ␾ xt␣ is defined so that: Ž . Ž Ž Ž ... Ž . i ␺␾xt␣ sxt␣ ; Ž . Ž Ž Ž .. Ž Ž ... Ž Ž .. ii ␾ ha␣ ,␾xt␣␣s␾xct. Steinberg observes that this determines ␾ as a map from the root group ²Ž. : X␣ sxt␣␣:tgK into G. The x are definable from the field over which X is defined. Therefore, by Proposition 4.11, they are definable from the pure group G. On the other hand, the following formula defines ␾:

␾ Ž.x s y if and only if

᭚ x11, y Ž.␺ Ž.x 1s y 1& Žg 01, y .s y & ᭚txŽ.1sxt␣␣Ž.&xsxct Ž., where g0 is the group element defined by ha␣ Ž.. As a result, we conclude that ␾ is an interpretable map from X␣ into G. One can do the same thing for all roots and get a map ␾ which liftsŽ. interpretably in G the X␣ from X to G. GROUPS OF MIXED TYPE 541

Now we define the following functions from K into G:

␾ ␾ y1 ␾ wt␣Ž.sŽ.xt␣ Ž.Ž.xy␣␣ Žyt .Ž.xt Ž.;

ht␣Ž.swtw␣␣ Ž. Žy1. .

As ␾ and the x␣ are interpretable in G,soisw␣␣and therefore, h . Hence, using Proposition 4.11, the following function also is interpretable in G:

UU f:K=KªG y1 Ž.t,u¬hthuhtu␣ Ž.Ž.Ž.␣␣ . Ž. Ž . Ž. But f s ␪ ( f since i and ii hold in Xu for xt␣ and are preserved by homomorphisms. This finishes the proof. Now we can prove Theorem 4.1: U Ä4 Ä Proof of Theorem 4.1. Let t g K _ 1 . We consider the set Bt s u g U K :ftŽ.,us1.As4 K is an algebraically closed field, by Lemma 4.3Ž. a , f is multiplicative in the second component. Therefore, Bt is a subgroup containing t by Lemma 4.3Ž. b and Ž. c . If t is of infinite order, then Bt is infinite. Hence, using our assumption about the nonexistence of bad fields U in the environment, we conclude that Bt s K . On the other hand, if t is U 1 of finite order, then fix u g K of infinite order. Then fuŽ ,ty . s1by 1 the first part of the argument. But fuŽ ,ty .Ž.sft,us1 by Theorem 4.7Ž. c . Hence, again Bt contains elements of infinite order, and thus U Bt s K . This finishes the argument. Finally, we can prove Theorem 4.2. Proof of Theorem 4.2. We start with a perfect tame group G of finite Morley rank such that GrZGŽ.is a quasisimple algebraic group. Let X be the universal group of the same type as GrZGŽ.. Then we have the following diagram:

G ␲ 1 6 6 GrZGŽ. X ␲2

We form the pullback fo this diagram:

␪1 G ␲ 1 6

6 YG6 rZGŽ. ␲ ␪2 X 2 542 ALTINEL, BOROVIK, AND CHERLIN

ÄŽ. Ž.Ž.4 Then Y ( g, x g G = X: ␲ 12g s ␲ x . In this diagram, ␲ 1and GrZGŽ.are interpretable in G. On the other hand, as X is an algebraic group, it is interpretable in GrZGŽ.and hence in G. Moreover, the triple ŽŽ.. X,GrZG,␲2 is algebraic and hence interpretable in G, say as UUU U U Ž.X,G,␲, where G ( GrZG Ž.definably; hence we may take G s UU GrZGŽ.and ␲ : X ª GrZGŽ.. U The pullback Y˜of ␲ 1 and ␲ is interpretable in G. Hence it is of finite Morley rank. Since Y˜( Y, Y also has finite Morley rank. Moreover, Y is a definable central extension of X. Therefore, we can apply Theorem 4.1 to Yand X and conclude, usingw 25,Ž. iii , p. 75x that Y ( X ) A, where A is Ž X .ŽŽ..XX X abelian. Note that ␪11Y s ␪ Y s G s G. But Y ( X is an algebraic group. Therefore, G is a quotient of an algebraic group by a finite group. We conclude that G also is an algebraic group.

5. B-D

The proof of Theorem 1.7 is based on the analysis of the interaction between the unipotent 2-subgroups and the 2-tori of a counterexample to the theorem. In this section, we introduce concepts which will be used in this analysis and obtain some basic results.

DEFINITION 5.1. Let G be a group of finite Morley rank. Then UŽ.G is the set of unipotent 2-subgroups in G and TŽ.G is the set of 2-tori in G; BGŽ.s ²U:UgU Ž.:G and DG Ž.s ²Ž.dT:TgT Ž.:G . A group G of finite Morley rank is said to be of D-type if G s DGŽ.and of B-type if GsBGŽ.. Note that, for any group of finite Morley rank G, BGŽ.and DG Ž.are definable and connected by Fact 2.3. In the sequel, we will need information about BHŽ.and DH Ž., where H is a proper definable subgroup of a counterexample to Theorem 1.7. Such a subgroup is a tame K-group of finite Morley rank. Therefore, we prove some lemmas about tame K-groups. Nevertheless, we will try to prove statements that are as general as allowed by our context.

LEMMA 5.2.Ž. i Let X be a B-type group. Assume that X1 is a normal definable subgroup of X and X1 contains a maximal unipotent 2-subgroup. Then X s X1.

Ž.ii Let Y be a D-type group. Assume that Y1 is a normal definable subgroup of Y and Y11 contains a maximal 2-torus. Then Y s Y .

Proof. Ž.iAsX1 is normal and definable in X, by Facts 1.5Ž. ii and 2.31, X1 contains all the unipotent 2-subgroups of X. But X is a B-type group, hence X s X1. GROUPS OF MIXED TYPE 543

Ž.ii The same argument as in the first part works for a D-type group after one replaces unipotent 2-subgroups with the definable closures of 2-tori.

PROPOSITION 5.3. Let R be a connected group of finite Morley rank such 0 0 that FŽ. R is di¨isible. Thenw R, FŽ. Rx is torsion-free. Ž.0 Proof. Let F s FR . By Fact 2.12, F s R12[ R , where R 1is torsion, X R21is torsion-free, and R is central in F. Therefore, F F R2and, in particular FX is torsion-free. We mimic the proof of Theorem 3 inwx 22 . Let g g R. We define the following mapping:

X ␥g:FrdRŽ.1ªwxR,FrF, X adŽ. R1¬ wxg, aF.

We first check that ␥g is well-defined. Assume a, b g F such that y1 Ž. Ž. Ž. bagdR11.R and therefore dR1is abelian. By Fact 2.10 ii R1is the direct sum of p-tori for some prime numbers p.As R1 is normal in R,by Fact 2.30, it is centralized by R. This forces dRŽ.1 to be central in R also. 1 1a 1 1 1 Therefore, we have 1 s w g, bay xwsg,ag xw,byxswxg,a ay gy bgby a. This last expression is equal to wxg, aay1 by1 agy1 bg because bay1is g Ž. y1y1y1 central in R, b g F, and dR1 is central in F.Aswxg,aa b ag bgs X wxwxwxwxg,aa,bb,g,g,b, and wxg, a are in the same coset of F . Therefore, ␥g is well-defined. Next, we check that ␥g is a group homomorphism. Let a, b g F. Then b X wxwxwxg, ab s g, bg,aswg,bg xw,ag xww,a x,b x' wg,ag xw,b x modulo F . Ž. Ž As FrdR1 is torsion-free, its image under ␥g is also torsion-free Fact 2.22. . Ž. Now, for any g1,..., gn gR, let g s g1,..., gn and define the following mapping:

n X ␥g:Ž.FrdRŽ.1 ªwxR,FrF, n X Ž.ad11Ž. R ,...,adRn Ž.1ªŁwxgkk,aF. ks1

X This mapping is a homomorphism because wxR, F rF is abelian. As Ž. Ž. FrdR1 is torsion-free, so is the image of ␥g Fact 2.22 . This implies that X X wxR,FrFis torsion-free as every element of wxR, F rF is contained in the X image of some ␥g . But F is a torsion-free definable subgroup. Therefore, we conclude that wxR, F is also torsion-free. 544 ALTINEL, BOROVIK, AND CHERLIN

LEMMA 5.4. If R is a sol¨able D-type group of finite Morley rank then 0 RsdTŽ.Ž. F R ,where T is a maximal 2-torus of R. Proof. By Fact 2.19, RrFRŽ.is a divisible abelian group,

RrFRŽ.s²:dT Ž.Ž. F RrFR Ž.:TgT Ž.R .

Therefore, RrFRŽ.also is a D-type group and since it is abelian, we have Ž. Ž Ž.. Ž. Ž. RrFRsdT11rFR , where T rFR is a maximal 2-torus of RrFR. Ž. Ž. Ž. Ž. But by Fact 2.33 ii , T1rFRsTF R rFR, where T is a maximal 2-torus of R. Therefore, RrFRŽ .sdTFR Ž Ž .rFR Ž ..sdTFR Ž Ž ..rFR Ž . s 0 dTŽ.Ž. F RrFR Ž., and the connectedness of R implies R s dTŽ.Ž. F R . LEMMA 5.5. Let Y be a tame D-type K-group. Then UŽ.Y s л. Proof. We carry out a case-by-case analysis: Y is nilpotent. In this case, Y s dTŽ., where T is the maximal 2-torus of Y. By Fact 2.10, dTŽ.is a divisible abelian group. Fact 2.27 implies Y has finitely many involutions. Therefore, using Fact 2.28, we conclude that UŽ.Ysл. Y is sol¨able nonnilpotent. We argue by induction on the rank of Y. Suppose Y is a counterexample of minimal rank. Let S be a Sylow 2-subgroup of Y. By Fact 2.34, S is connected. Hence, S s B)T with B / 1 and T / 1. By Fact 2.36, B e Y. Hence, B is subgroup of every Sylow 2-subgroup of G and it is centralized by every 2-torus in G. Therefore, B is central in Y. 0 0 We claim that FYŽ.sB. By Fact 2.11, FYŽ.sC)D, where C is definable, connected, and of bounded exponent, and D is definable and divisible. If D / 1, then rkŽ.YrD - rk Ž.Y . This yields a contradiction because BDrD is a nontrivial unipotent 2-subgroup of YrD and YrD is a 0 D-type group. Thus, FYŽ.sC. Fact 2.20 implies FYŽ.0is the direct sum of its Sylow p-subgroups, which are definable. If P is such a Sylow p-subgroup, where p / 2, then Ps1 because otherwise, P is infiniteŽ. it is connected by Fact 2.34 and rkŽ.YrP - rk Ž.Y , and arguing as above yields a contradiction. Therefore, 0 0 FYŽ.sB. In particular, FYŽ. is central in Y. By Fact 2.19, Y is nilpotent, a contradiction. Y is nonsol¨able. We will useᎏ -notation to denote quotients by ␴ Ž.Y . By Fact 2.41, Ys Y1 [ иии [ Ymi, where the Y are simple algebraic groups over algebraically closed fields of odd characteristic. 0 Let S be a Sylow 2-subgroup of Y. Then S s B)T, where B is definable, connected, and of bounded exponent and T is divisible abelian. 0 If UŽ.Y / л, then both B and T are nontrivial. We claim ␴ Ž.Y s 00 FYŽ.sZY Ž.sB.AsY is the direct sum of simple algebraic groups 0 over algebraically closed fields, BYŽ.s1. Therefore, B F ␴ Ž.Y . By Fact GROUPS OF MIXED TYPE 545

2.36, B is characteristic in ␴ Ž.Y , and thus B e Y.As Bis the largest unipotent 2-subgroup of every Sylow 2-subgroup of Y, it is centralized by the 2-tori of Y. But Y is a D-type group, therefore B is central in Y. Now the same argument as in the case in which Y is solvable implies that 0 0 0 FYŽ.sB. But by Fact 2.19, ␴ Ž.Y rFYŽ.is an abelian group and B is 0 0 central in Y. This forces ␴ Ž.Y to be nilpotent and in particular ␴ Ž.Y s FYŽ.0. Ž.0 We have B s ZY and YrB s Y1rB) иии )YmirB, where the Y rB are finite central extensions of simple algebraic groups over algebraically closed fields of odd characteristic. We will do component analysis in the X style ofwx 21 . If i / j, then wYij, Y xF B. For every i, Yiis connected and X X X X YiiBrBeYrB. Therefore, we have Yiis Y B and Y s Y1иии YmB. The X X subgroup Y1 иии Ym contains a maximal 2-torus of Y. Therefore, by Lemma X X X 5.2, Y s Y s Y1 иии Ym. By continuing to take commutators we may assume that Y s Y1) иии )Ymi, where the Y are perfect groups. We also have Ž. Ž . Ž . Ž. ZY sZY1 иии ZYmii. Therefore, each Y rZY is a perfect group which is a central extension of a simple algebraic group by a finite center. By Ž. Theorem 4.2 each YiirZY is a quasisimple algebraic group. By Theorem 4.2, Yiiis an algebraic group. Therefore, ZYŽ.is finite. This last conclusion proves that Y cannot have unipotent 2-subgroups.

LEMMA 5.6. Let G s K i H be a group of finite Morley rank, where H and K are connected definable subgroups of G, and K is a K-group of B-type. Assume that H centralizes all definable simple and B-type sol¨able sections of K normalized by H. Then H centralizes K. Ž A similar statement can be made for D-type groups..

Proof. We prove the statement for B-type groups. The same proof works for D-type groups mutatis mutandis. By Fact 2.41, Kr␴ Ž.K is isomorphic to the direct sum of simple algebraic groups over algebraically closed fields of characteristic 2. Since H is connected, H normalizes each of these simple algebraic groups. Thus by the hypotheses, H centralizes Kr␴Ž.K. Therefore, H centralizes U␴ Ž.K r␴ Ž.K , where U is a unipotent 2-subgroup of K. This implies that H normalizes U␴ Ž.K . U␴ Ž.K is a definable solvable subgroup of K. Therefore, H acts trivially on BUŽŽ..␴ X , and in particular, U.As K is a B-type group, H centralizes K.

LEMMA 5.7. If B is a unipotent 2-subgroup acting on a tame D-type K-group Y, then the action is tri¨ial. Proof. We carry out a case-by-case analysis: Y is nilpotent. Y is nilpotent implies that Y s dTŽ., where T is the 2-torus of Y. By Fact 2.30, B centralizes T. Therefore, it also centralizes dTŽ.. 546 ALTINEL, BOROVIK, AND CHERLIN

0 Y is sol¨able and nonnilpotent. By Lemma 5.4, Y s dTŽ.Ž. FY , where T is maximal 2-torus of Y. The group B acts on FYŽ.0. We claim that the group BFŽ. Y 0is nilpotent. Suppose this is not the case. As BFŽ. Y 0is solvable and connected, by Fact 2.18, an algebraically closed field K can be interpreted in BFŽ. Y 0, and a definable section of B is isomorphic to an infinite definable subgroup of K U. But then, by Fact 2.28, K U has infinitely many involutions, a contradiction. Therefore, BFŽ. Y 0is nilpotent. Then, by Fact 2.11 and Lemma 5.5, B centralizes FYŽ.0. By replacing T by one of its conjugates in Y, we may assume that B and T are in the same Sylow 2-subgroup of BY. But this implies that B centralizes T and therefore dTŽ.. By the above paragraph, B centralizes 0 0 FYŽ.. Thus, we conclude that B centralizes Y s dTŽ.Ž. FY . Y is nonsol¨able. This follows from Fact 2.40, the result for Y solvable, and Lemma 5.6.

COROLLARY 5.8. If H is a tame K-group then BŽ. H and D Ž. H commute.

LEMMA 5.9. If X is a tame B-type K-group then TŽ.X s л. Proof. We carry out a case-by-case analysis. X is sol¨able. By Fact 2.36, X is a unipotent 2-group and we are done in this case. X is nonsol¨able. We assume that X is a counterexample of minimal rank to the statement. Therefore, DXŽ./1. Xr␴ Ž.X is the direct sum of simple algebraic groups over algebraically closed fields of characteristic 2. 0 0 0 We claim that ␴ Ž.X s FXŽ.sZXŽ.. By Lemma 5.7, DXŽ.is central in X and DXŽ.sdT Ž., where T is the maximal 2-torus of X.By 0 Fact 2.11, FXŽ.sC)D, where C is definable, connected, and of bounded exponent and D is definable and divisible. By induction on rank, 0 0 we may assume C s 1. If FXŽ.-␴Ž.X then, by Fact 2.18, an alge- 0 0X 0 braically closed field K is interpretable in ␴ Ž.X . Since ␴ Ž.X : FXŽ., Ž.0 Kq is isomorphic to a definable section of FX , and a definable section 0 0 U of ␴ Ž.X rFXŽ.is isomorphic to an infinite subgroup of K . The ‘‘no bad fields’’ assumption implies that this definable section is isomorphic to K U. The characteristic of K is 0 because FXŽ.0is divisible. Therefore, K U 0 0 contains a nontrivial 2-torus. Fact 2.33Ž. ii implies that ␴ ŽX .rFXŽ. contains a nontrivial 2-torus. This implies that ␴ Ž.X 0contains a noncen- 0 0 tral 2-torus, a contradiction. Therefore, FXŽ.s␴Ž.X . Let U be a unipotent 2-subgroup. We claim that the group FXŽ.0 Uis nilpotent. If this is not the case then, as FXŽ.0 Uis connected and solvable, by Fact 2.18, an algebraically closed field K can be interpreted in FXŽ.0 Uand a definable section of U is isomorphic to an infinite subgroup of K U. But then Fact 2.28 implies that K U has infinitely many involutions, GROUPS OF MIXED TYPE 547 a contradiction. Therefore, FXŽ.0 Uis nilpotent and Fact 2.11 implies that FXŽ.0centralizes U. Since X is a B-type group, we conclude that FXŽ.0 0 0 0 is central in X; ␴ Ž.X s FXŽ.sZXŽ.. Now we can finish the argument as in the proof of Lemma 5.5.

LEMMA 5.10. If X is a B-type K-group and C is a nilpotent, connected, 2X-group of bounded exponent of finite Morley rank acting definably on X, then the action is triïal. Proof. We carry out a case-by-case analysis: X is soläble. By Facts 2.36 and 1.5Ž. i , X is nilpotent. The rest of the argument is an application of Fact 2.18. X is nonsoläble. This follows from Fact 2.40, the result for X solvable and Lemma 5.6.

LEMMA 5.11. If X is a tame B-type K-group and T is a torsion-free nilpotent group of finite Morley rank acting on X definably then this action is triïal. Proof. We carry out a case-by-case analysis: X is soläble. By Fact 2.36, X is nilpotent. We claim that the group XT is nilpotent. If this is not the case then as XT is a connected solvable group of finite Morley rank, by Fact 2.18, an algebraically closed field K is interpretable in XT. The ‘‘no bad fields’’ hypothesis implies that a definable section of T is isomorphic K U. But T is torsion-free, a contradiction. Therefore, XT is nilpotent. Therefore X is centralized by T. X is nonsoläble. This follows from Fact 2.40, the result for X solvable, and Lemma 5.6.

U LEMMA 5.12. Let T be a group such that T ( F , where F is an algebraically closed field of characteristic 2. If T acts on a tame D-type K-group Y, then the action is triïal. Proof. The argument is again a case-by-case analysis: Y is nilpotent. Y is the definable closure of its sole 2-torus and an application of Fact 2.30 proves the statement in this case. Y is soläble and nonnilpotent. Suppose the action of T is nontrivial. By Fact 2.18, we can interpret an algebraically closed field of characteristic 2 in the group H s YT. This forces Y to have a definable section which is a unipotent 2-group. But then BYŽ./1, a contradiction to Lemma 5.5. Y is nonsoläbe. This follows from Fact 2.40, the result for Y solvable, and Lemma 5.6.

LEMMA 5.13. If an inölution i acts on a tame D-type K-group Y, and Ž. CY i does not contain nontriïal 2-tori, then Y is soläble. If, in addition, Y is nilpotent, then i inërts Y. 548 ALTINEL, BOROVIK, AND CHERLIN

Proof. We first assume that Y is nilpotent and prove the second statement. As Y s DYŽ.,YsdT Ž., where T is the maximal 2-torus in Y. In particular, Y is abelian. By Fact 2.38, it is enough to show that under the assumptions of the statement CiY Ž.is finite. 1 i We let K s Ä ggy :ggY4. Note that the elements of K are inverted by i.AsY is abelian, K is a definable subgroup and the following map is a homomorphism from Y onto K:

␪ : Y ª K , 1 i g ¬ gy g .

As the fibers of this map correspond to the cosets of CiYŽ.in Y,we Ž. Ž. Ž Ž.. have rk Y s rk K q rk CiY . The assumptions on Y imply that rkŽŽ..Ž.CiY -rk Y . Therefore, K is an infinite divisible subgroup of Y. 1 i 2 i 1 Let g g Y. Then gy g s k for some k g K. This implies g ky s gk. 1 i i 1 But ky s k . Therefore, Ž.gk s gk.As gsgkky, we conclude that Ž. Ž.0 YsCiKYY.AsY is connected, we have Y s CiK. The assumption on 0 H CiYYŽ.implies that CiŽ. is 2 . But then K contains the Sylow 2-sub- Ž.0 group of Y. This forces K s Y, thus CiY s1. Now we will show that Y is solvable. Assume toward a contradiction that 0 Yis nonsolvable. We will first show that ␴ Ž.Y s 1. Suppose this is not the case. We first analyze FYŽ.0.If FYŽ.0is a 2H -group then, by Fact 0 0 0Ž. Ž. Ž . Ž. 2.25 and Proposition 2.52, CiY r FŽY . sCiFYY rFY . Therefore, 0Ž. CiY r FŽY . does not contain nontrivial 2-tori. By induction on rank, 0 YrFYŽ.is solvable, forcing Y to be solvable, a contradiction. Therefore, 0 0 FYŽ.has a nontrivial 2-torus. Let T11be the maximal 2-torus of FYŽ..T is i-invariant. By the assumption on CiY Ž., and using Fact 2.38, i inverts T. It follows that i inverts dTŽ.. By Proposition 2.52, CiŽ. 11YrdŽT1.s Ž.Ž.Ž. Ž. CidTY 11rdT . By induction on rank, we conclude that YrdT1is solvable, which forces Y to be solvable, a contradiction. As a result, we 0 0 have FYŽ.s1, and therefore, ␴ Ž.Y s 1. The conclusion of the above paragraph and Fact 2.41 imply that Y s Y1 иии Yni, where the Y are central extensions of simple algebraic groups over algebraically closed fields of odd characteristic. Moreover, these can be taken to be perfect subgroups. Then the results on central extensions imply that the Yi are quasisimple algebraic groups over algebraically closed fields of odd characteristic. Now we will argue that we may take n s 1. The involution i is contained 0 in a Sylow 2-subgroup S of Y i ²:i . T s S is a 2-torus. The hypotheses Ž. i on CiY imply that i inverts T. But this implies that T l Y111F Y l Y i and, in particular, this intersection is infinite. This forces Y11s Y . Hence, it suffices to deal with the case in which Y is a quasisimple algebraic group over an algebraically closed field of odd characteristic. Now the conclusion follows from Facts 2.38 and 2.46. GROUPS OF MIXED TYPE 549

DEFINITION 5.14. Let G be a group of finite Morley rank. Define a graph on UŽ.G as follows. The vertices of the graph are the elements of

UŽ.G. Two vertices U12and U are connected by an edge if U12and U normalize each other; this will be denoted by U12; U . We will investigate some basic properties of this graph. Note that two vertices U and V are in the same connected component of UŽ.G if and only if there exists a sequence A0 ,..., An of unipotent 2-subgroups such that U s A0 , V s Anii, and, for 0 F i F n y 1, A and A q1normalize each other. U Ž. PROPOSITION 5.15. Let G be a tame K -group, and let U12, U g U G . If U12;U and neither of U1 and U 2 is a central subgroup of G, then Ž Ž .. Ž Ž .. DCG U1 sDCG U2 .

Proof. As U12normalizes U , U1normalizes DCŽŽ..G U22. Since U is not central in G, DCŽŽ..G U21is a K-group. Then, by Lemma 5.7, U centralizes ŽŽ.. ŽŽ.. Ž. ŽŽ.. DCG U2 . Hence, DCG U2 FCUG 1 and we get DCG U2 F DCŽŽ..G U1 . By symmetry, we obtain equality. U COROLLARY 5.16. Let G be a tame K -group that has no central unipo- Ž. tent 2-subgroups, and U12, U g U G . If U1 and U 2 are in the same con- Ž. ŽŽ..ŽŽ.. nected component of U G , then D CG U1 s DCG U2 . U COROLLARY 5.17. Let G be a tame K -group that has no central unipotent 2-subgroups, and U g UŽ.G . If U Ž.G is a connected graph then ŽŽ.. DCG U eG,for any unipotent 2-subgroup U of G.

PROPOSITION 5.18. Let G be a group of finite Morley rank. Let W denote Ž . Ž² :. Ž . a connected component of U G , and M s NG W . Then U M s W . Proof. We need to show that UŽ.M : W . Let U, V g UŽ.M such that U g W . We will show that V g W . First we argue that U and V can be replaced by maximal unipotent 2-subgroups. U F S121and V F S , where S Ž. 0 and S2 are two Sylow 2-subgroups of M. By Fact 1.5 ii , S111s B )T , and 0 Ž.0 S222sB)T, and U F B1and V F B2. Note that B1; ZB 1;U 0 0 ŽŽZB12 ./1 by Facts 1.5Ž. i and 2.28 . and B ; ZBŽ.2;V. Hence, we can replace U and V by B12and B , which are maximal unipotent 2-subgroups. g By the Sylow theorem, there exists g g M such that B12s B . Hence, ²: ²: B2 FW and g can be taken to be in W . Proceeding inductively, it g suffices to treat the case in which g g W g W . In this case, W s W,so gg WsWand B21s B g W .

COROLLARY 5.19. Let G be a group of finite Morley rank. Let W be a Ž . Ž² :. Ž . Ž . connected component of U G . NG W s Stab W . In particular, Stab W is a definable subgroup. 550 ALTINEL, BOROVIK, AND CHERLIN

In the sequel we will encounter groups which are isomorphic to either

PSL22Ž. K or SL Ž. K over an algebraically closed field K. We will denote this situation by Ž.PSLK2 Ž ..

LEMMA 5.20. If H is a nonsol¨able, tame, connected K-group with a Ž. Ž. Ž. weakly embedded subgroup, then HrOH ( PSLK2 ,where K is an algebraically closed field.

Proof. Let M be a weakly embedded subgroup of H. Note that 0 0 FHŽ.sOHŽ.because if this is not the case, then FHŽ.)OHŽ.ŽFact 1.4.Ž. and FH0has an infinite Sylow 2-subgroupŽ. Fact 2.34 which is unique in FHŽ.0Ž.Fact 2.20 . Therefore this Sylow 2-subgroup is normal in H as well. This contradicts the assumption that H has a weakly embedded subgroup. We will show that either MOŽ. H rOH Ž.is a weakly embedded subgroup of HrOHŽ.or H s MO Ž. H . We will use Proposition 3.3. We use ᎏ-notation to denote quotients by OHŽ.. Clearly, M has infinite Sylow 2-subgroups. Let U be a subgroup of H, which contains OHŽ., such that x U is a unipotent 2-subgroup of M. Suppose U s U for some x g H. This x ŽŽ.. Ž. implies that U s U, and therefore x g NBUH . Clearly, U F MO H and we have U s OHŽ.Ž MlU ..AsUis solvable and its Sylow 2-subgroups are of bounded exponent, by Fact 2.36, U has a unique Sylow 2-subgroup, which is a subgroup of M l U by Fact 2.33. Hence, BUŽ.FM. ŽŽ.. But x g NBUH and M is a weakly embedded subgroup. This implies x g M, forcing x g M. Now let T be a subgroup of H such that T is a xx 2-torus of M and x g H such that T s T. Hence, T s T, and therefore Ž Ž Ž ... Ž . Ž . x g NDdTH .If MO H - H then MO H is a weakly embedded subgroup of H. Therefore, replacing M by MOŽ. H , we may assume that Ž . Ž Ž Ž ... M contains OH and therefore T. But then x g NDdTH forces x g M, and x g M.If M-Hthen M is a weakly embedded subgroup of H, otherwise H s MOŽ. H . We first analyze the possibility M - H. In this case, MOŽ. H - H and the discussion in the foregoing paragraph implies that M is a weakly embedded subgroup in H.If␴Ž.H0 )OH Ž., then M has a normal infinite 2-subgroupŽ. Fact 2.34 , which contradicts that H has a weakly 0 embedded subgroup. It follows that ␴ Ž.H s OHŽ.and therefore, by Fact 2.41, H s HrOHŽ.is the central product of quasisimple groups of finite Morley rank. As H has a weakly embedded subgroup, we conclude that H is a quasisimple group. By the results on central extensions, H is a quasisimple algebraic group with a weakly embedded subgroup. Ž. Ž . We claim that H ( PSLK2 , where K is an algebraically closed field. Let X s H. We will show that X has Lie rank 1. Then the result will follow from Fact 2.51. Let M denote a weakly embedded subgroup of X. GROUPS OF MIXED TYPE 551

By Proposition 3.2, M has a Sylow 2-subgroup S which is also a Sylow 2-subgroup of X. We first assume that charŽ.K s 2. In this case S is a maximal unipotent ² q: y ² subgroup of X. We may assume that S s X␣ : ␣ g ⌽ . Let S s X␣ : y:Ž. ␣g⌽. By Proposition 3.3, NUX FMfor any unipotent 2-subgroup in Ž. q M. In particular, NUX ␣␣FMfor every ␣ g ⌽ , where U is the root subgroup corresponding to ␣ Ž.by Proposition 4.11, U␣ is definable in X . Suppose X has Lie rank greater than 1. Then ⌽q has a base with at least two distinct roots ␣ and ␤. By Fact 2.49, ␣ y ␤ is not a root. Fact 2.47 implies that the root subgroups U␣ and Uy␤ commute. Therefore, Uy␤ centralizes a unipotent 2-subgroup of M and Uy␤ F M. This forces ZSŽ y.ŽFM. But then, as ZSy.is a nontrivial unipotent 2-subgroup, SyF M. Hence, by Fact 2.50, M s G, a contradiction. Therefore, if charŽ.K s 2 then the Lie rank of X is 1. Now we assume charŽ.K / 2. Then the Sylow 2-subgroups of X are contained in maximal tori. Let T␣ and Z␣ be as in Fact 2.50. Suppose the rank is greater than 1. Then T␣ / 1 for any ␣ g ⌬. Moreover, Z␣ F M by Proposition 3.3. This forces M s X, a contradiction. Therefore, in this case also the rank of X is 1. Now we analyze the remaining possibility, H s MOŽ. H . Note that under this assumption BHŽ.s1 because otherwise H has a nontrivial unipotent 2-subgroup U which normalizes OHŽ.. But then Fact 2.37 implies that U centralizes OHŽ., forcing OH Ž.FMand thus H s M,a contradiction. We now show similarly that the Sylow 2-subgroups of H have Prufer¨¨ 2-rank 1. If the Prufer 2-rank of the Sylow 2-subgroups of H is bigger than 1 then Proposition 2.55 implies that OHŽ.is generated by the centralizers of nontrivial 2-tori in M. This forces OHŽ.FMand H s M. By Fact 2.33Ž. ii , the Sylow 2-subgroups of HrOHŽ.are images of the Sylow 2-subgroups of H. Thus the Prufer¨ 2-rank of HrOHŽ.is 1. 0 0 0 We claim that ␴ Ž.H s OHŽ.Žequivalently, ␴ Ž.H s FHŽ...If 0 0 ␴Ž.H)OHŽ., then ␴ Ž.H rOHŽ.has a nontrivial connected Sylow 2-subgroupŽ. Fact 2.34 . This nontrivial 2-torus is contained in the connected component of every Sylow 2-subgroup of HrOHŽ.. But the Prufer¨ 0 2-rank of HrOHŽ.is 1. Therefore, the Sylow 2-subgroup of ␴ Ž.H rOHŽ. is the connected component of every Sylow 2-subgroup of HrOHŽ.and H is solvable, a contradiction. Fact 2.41 together with the facts that the Prufer¨ 2-rank is 1 and that 0 ␴Ž.HsOHŽ.imply that HrOHŽ.is isomorphic to a quasisimple algebraic group. This group is over an algebraically closed field of characteristic different from 2 as BHŽ.s1. The fact that its Prufer¨ 2-rank is 1 implies that its Lie rank is 1 as well and, by Fact 2.51, HrOHŽ.( Ž.PSLK2 Ž .. 552 ALTINEL, BOROVIK, AND CHERLIN

PROPOSITION 5.21. If X is a tame B-type K-group and UŽ.X is not Ž. connected, then X ( PSL2 K , where K is an algebraically closed field of characteristic 2.

Proof. As UŽ.X is not connected, Fact 2.36 implies that X is not solvable. Similarly, BŽŽ␴ X ..s 1. Therefore, using Lemma 5.9, we conclude that ␴ Ž.X 0is a 2H -subgroup. By Fact 2.37, the action of any 0 0 0 unipotent 2-subgroup on ␴ Ž.X is trivial. Therefore, ␴ Ž.X s ZXŽ.. 0 We claim that UŽŽ.XrZX .Žis isomorphic to U X .. We will use 0 ᎏ-notation to denote quotients by ZXŽ.. Let U, V g U Ž.X be such that 0 Uand V normalize each other and ZXŽ.FU,V. Let x g U. Then x V s V. V is a definable nilpotent subgroup of X. Therefore, by Fact 2.20, it has a unique Sylow 2-subgroup normalized by x. This argument applies to U also. Therefore, U and V have a single unipotent 2-subgroup each, and these subgroups normalize each other. Thus UŽ.X is isomorphic to UŽ.Xand, in particular, UŽ.X is not connected. By Fact 2.41, it follows that Xr␴ Ž.X is a simple algebraic group, over X 0 an algebraically closed field of characteristic 2. As X s XZŽ. X and X is X a B-type group, we have X s X . Using Theorem 4.2, we conclude that X is a quasisimple algebraic group over an algebraically closed field of characteristic 2. We claim that the stabilizer of a connected component of UŽ.X is a weakly embedded subgroup of X, and hence Lemma 5.20 applies to complete the argument. Let W be such a connected component and let MsStabŽ.W . M is definable by Corollary 5.19. As U Ž.X is not connected, by Proposition 5.18, M - X. M has infinite Sylow 2-subgroups. Let g g g X _ M. Then M l M does not contain a nontrivial unipotent 2-sub- g group because otherwise W s W by Proposition 5.18 and g g M,a g contradiction. Using Lemma 5.9, we conclude that M l M has finite Sylow 2-subgroups. Therefore M is a weakly embedded subgroup.

DEFINITION 5.22. Let G be a group of finite Morley rank. Define a graph on TŽ.G as follows. The vertices of the graph are the elements of

TŽ.G. Two vertices T12and T are connected by an edge if and only if they nromalizeŽ. equivalently, by Fact 2.30, centralize each other; this will be denoted by T12f T .

Thus two vertices T12and T of TŽ.G are in the same connected Ž. Ž. component of T G if and only if there exists R0 ,...,Rn gT G such that T10sRand T 2s Rnii, and for 1 F i F n y 1, wxR , R q1s 1. U PROPOSITION 5.23. Let G be a tame, K -group of finite Morley rank. Let Ž. T12,TgTG such that T1f T 2. If neither of T1 and T 2 is a central ŽŽ..ŽŽ.. subgroup of G, then B CG T1 s BCG T2 . GROUPS OF MIXED TYPE 553

Proof. Let T12and T be as in the statement of the proposition. Then T1 normalizes BCŽG Ž T2 ...As BC ŽG Ž T21 .. is a K-group, by Corollary 5.8, T ŽŽ.. ŽŽ..ŽŽ.. centralizes BCG T2 . This forces BCG T2 FBCG T1 . We get equality by symmetry.

U COROLLARY 5.24. Let G be a tame, K -group of finite Morley rank that has no central 2-tori. If T12 and T are in the same connected component of Ž. ŽŽ..ŽŽ.. TG,then B CG T1 s BCG T2 .

U COROLLARY 5.25. Let G be a tame, K -group of finite Morley rank. If Ž. Ž Ž.. TG is connected and G has no central 2-tori then B CG T e G for e¨ery TgTŽ.G. The next lemma is an analog of Proposition 5.18.

LEMMA 5.26. Let G be a group of finite Morley rank. Let W denote a Ž . Ž² Ž .:. Ž . connected component of T G and M s NdG W.Then T M s W .

Proof. Let T12and T be two 2-tori in M such that T1g W . We will show that T21g W . We may assume that T and T2are maximal in G since any 2-torus is in the same connected component of TŽ.M as a maximal 2-torus which contains it. By the Sylow theorem, there exists g g M such g Ž² :. that T12s T . Therefore, T2F d W and g can be taken to be in dŽ²W :.. Since d Ž²W :.s ²dT Ž .: TgW :, proceeding inductively it suf- Ž. fices to treat the case in which g g dT33, where T g W . In this case as ggŽŽ . . T33sTdT 3is abelian , W s W .

COROLLARY 5.27. Let G be a group of finite Morley rank. Let W be a Ž . Ž² Ž .:. Ž . connected component of T G . Then NG d W s Stab W . In particular, StabŽ.W is definable.

COROLLARY 5.28. If Y is a nonsol¨able tame D-type K-group and TŽ.Yis Ž. Ž. Ž. not connected then YrOY (SL22 K or PSL K , where K is an algebraically closed field of characteristic different from 2.

Proof. Let W be a connected component of TŽ.Y and M s Stab ŽW .. By Corollary 5.27, M is definable. We claim that M is a weakly embedded g subgroup. M has infinite Sylow 2-subgroups. If g g X _ M, then M l M g does not have infinite Sylow 2-subgroups because otherwise W s W and g g M. Now the result follows from Lemma 5.20. Before we finish this section, we would like to mention an interesting property of B-type and D-type subgroups of a tame K U-group of finite Morley rank. Let G be a group of finite Morley rank. Let B and D denote the posets of B-type and D-type subgroups of G, respectively, 554 ALTINEL, BOROVIK, AND CHERLIN ordered by inclusion. One can define the following mappings:

BC: D ª B DC: B ª D

Y ¬ BCŽ.GGŽ. Y X¬DCŽ.Ž. X

These mappings define a Galois connection Žseewx 5. between B and D because they satisfy the following properties:

PROPOSITION 5.29.Ž. i The mappings BC and DC are order-re¨ersing. Ž.ii If X g B then X F BCDCX, and if Y g D then Y F DCBCY. The following properties of BC and DC Ž.or any Galois connection can be checked easily:

PROPOSITION 5.30. Let X g B and Y g D. Then DCX s DCBCDCX and BCY s BCDCBCY. COROLLARY 5.31. BCDC and DCBC are closure operations on B and D, respectiëly, i.e.: Ž.i For any X g B and Y g D, X F BCDCX and Y F DCBCY. Ž.ii For any X g B and Y g D, BCDCX s BCDCBCDCX and DCBCY s DCBCDCBCY. Ž.iii Both BCDC and DCBC are order-preserïng. The closed elements of B and D are of the form BCY and DCX, respectiëly. In the sequel we will use these properties several times and refer to them under the rubric of Galois connection.

6. CONSTRUCTION OF A WEAKLY EMBEDDED SUBGROUP

In this section, we assume that G is a simple, tame K U-group of mixed type. We will show that G has a weakly embedded subgroup. Let S be a 0 Sylow 2-subgroup of G. Then S s B) D, where B is a maximal unipotent 2-subgroup and D is a maximal 2-torus, with B / 1 and D / 1. Observe ŽŽ.. ŽŽ.. that D F DCGG B and B F BC D , so these groups are nontrivial. As we have assumed that G is simple, UŽ.G is not a connected graph by Corollary 5.17. Let W be the connected component of UŽ.G containing B Ž. and M11s Stab W . By Corollary 5.19, M is definable. Also note that Ž. WsUM1 by Proposition 5.18. We prove that M1 is a reasonable candidate for a weakly embedded subgroup of G. GROUPS OF MIXED TYPE 555

g PROPOSITION 6.1. For g g G_ M11, M l M 1does not contain nontri¨ial Ž. Ž. unipotent 2-subgroups. In particular, for any U g U M1, NUG FM1. g Proof. Suppose U is a unipotent 2-subgroup of M11l M . Then U g W gg lW, and thus W s W , a contradiction. This proposition has useful corollaries: Ž. COROLLARY 6.2. NBG FM1.

COROLLARY 6.3. S F M1. Propositions 3.3 and 6.1 show that if the normalizer of every 2-torus of

M111were in M then M would be a weakly embedded subgroup of G. Therefore, for the remainder of this section, we analyze the case in which this does not happen. We assume that M1 has a 2-torus R such that Ž. Ž Ž .. Ž Ž .. NRG gM1. Note that, by Fact 2.30, BNGG R sBC R .

PROPOSITION 6.4. UŽŽ..NG R is not connected.

Proof. Suppose UŽŽ..NRG is connected. Corollary 6.3 implies that R F S11F M , where S 1is some Sylow 2-subgroup of G. Therefore, ŽŽ.. ŽŽ.. Ž. Ž. UNRGGlW/л. This forces U NR:W. Thus NRGFStab W sM1, which contradicts our assumption. ŽŽ.. Ž. By Proposition 5.21, BCG R (PSL2 K , where K is an algebraically closed field of characteristic 2. Note that BCŽŽ..G R is not contained in M1.

PROPOSITION 6.5. R canŽ. and will be taken to be a maximal 2-torus in G.

Proof. R is contained in a maximal 2-torus R11of G. Since R is ŽŽ..ŽŽ.. abelian, R f R1. By Proposition 5.23, BCG R1 sBCG R gM1.

Rcentralizes a unipotent 2-subgroupŽ. say A of M1. This implies ŽŽ.. Ž. ŽŽ..ŽŽ.. RFDCG A . But then, as U M1 s W , R F DCGG A sDC B . ŽŽ.. This implies that B F BCG R .As Bis a maximal unipotent 2-subgroup of G, we conclude that B is a Sylow 2-subgroup of BCŽŽ..G R .As ŽŽ.. Ž. BCG R (PSL2 K , B is an elementary abelian 2-group and NBŽ. BT, where T is a maximal torus of BCŽ Ž.. R . Let w be an BŽCGŽ R.. s G w involution in BCŽŽ..G R which inverts T. T normalizes B and B , and w w therefore normalizes CBGGŽ.,CB Ž .ŽŽ..,DC G B , and DC ŽŽG B ... Lemma w 5.12 implies that T centralizes DCŽŽ..GG B and DC ŽŽ B ... Hence, ²Ž Ž..Ž Žw..: Ž Ž .. DCGG B ,DC B FDC G T . ŽŽ.. ŽŽ.. We let L s DCGG T and Q s BC R . Then B F Q and the discussion in the foregoing paragraph can be summarizes as follows:

DCŽ.Ž.Ž.GGGŽ. Q FDC Ž. B FDC Ž. T sL. 556 ALTINEL, BOROVIK, AND CHERLIN

Before we state the next lemma, we remark that the group QDŽŽ.. CG Q ŽŽ.. ŽŽ.. sQ=DCGG Q . Indeed, Q and DC Q centralize each other, and Ž. Ž. Ž. QlCQG sZQ s1asQsPSL2 K .

Ž Ž .. Ž Ž .. Ž Ž Ž ... LEMMA 6.6. If D CGG Q s D C B then N GG Q = DC Q s NGŽ. Q is a weakly embedded subgroup.

Ž Ž Ž ... Proof. Let X s NQGG=DC Q . Ž Ž Ž ... Ž Ž Ž ... We claim that BQ=DCGG Q sQ. Let U g U Q = DC Q . Ž Ž .. Ž Ž .. Ž Ž .. Then as DCGG Q eQ=DC Q ,Uacts on DC G Q . But Lemma 5.7 implies that this action is trivial. Hence, U centralizes DCŽŽ..G Q and, ŽŽ.. in particular, it centralizes R, forcing U F BCG R sQ. This also im- Ž. plies that X s NQG . On the other hand, Q acts on DQŽ=DC ŽG Ž Q ... trivially by Lemma 5.7. Ž Ž Ž ... Ž Ž .. This forces DQ=DCGG Q sDC Q . Next, we claim that if U g UŽ.X then U F Q. Let U g UŽ.X . Then U Ž Ž .. Ž Ž Ž ... Ž Ž .. acts on Q = DCGG Q , and hence, on DQ=DC Q sDCG Q . But by Lemma 5.7, this action is trivial. In particular, U centralizes R forcing U F Q. Similarly, as Q acts on DXŽ.trivially by Lemma 5.7, if T g T Ž.X , then ŽŽ.. TFDCG Q . Now, we can prove the lemma. We will make use of the characterization Ž. obtained in Proposition 3.3. Let B1 g U X . It follows from the above Ž.. Ž. x discussion that B1 g U Q . Let g g NBG 11. By Fact 2.31, B F B , Ž Ž Ž ... x where x can be taken to be in Q as BQ=DCG Q sQ.As B and B1 Ž.ŽŽ..ŽŽ..x are in the same connected component of U G , DCG B1 sDCG B ŽŽ..x Ž Ž .. Ž Ž .. s DCGG Q sDC Q . Hence g normalizes DCG Q . But Q s BCŽGG Ž DC Ž Ž Q ....,so g normalizes Q. Ž. Ž Ž Ž ... Ž. Let R11g T X . By the above, R g T DCGG Q . Let g g NR1. ŽŽ..g ŽŽ.. This implies BCG R1 sBCG R1 . By Proposition 6.5, R is maximal. x Thus using Fact 2.31 we conclude R1 F R , where x can be taken to be in Ž Ž .. Ž Ž Ž ... Ž Ž .. x ŽŽ.. DCGG Q as DQ=DC Q sDCG Q .As R1fR,BCG R1s ŽŽ..xx Ž BCG R sQ sQ. Hence, g normalizes Q and thus g g NQG = DCŽG Ž Q .... This finishes the proof.

LEMMA 6.7. Let B be a maximal unipotent 2-subgroup of G which is in Q.

Suppose B1 is another unipotent 2-subgroup of G such that B and B1 are in Ž. the same connected component of U G . Then B1 F B.

ŽŽ..ŽŽ.. Proof. By Corollary 5.16, DCGG B sDC B1and this implies that B1 centralizes DCŽŽ..G B ; in particular, B1 centralizes dRŽ.. Therefore, Ž. B121FQ. But Q ( PSL K . This forces B F B. GROUPS OF MIXED TYPE 557

In the remainder of our argument we consider various possibilities for the structure of L and come back to Lemma 6.6 in all cases. After some preliminary analysis, the main argument begins with Proposition 6.23 below. The following lemma will help clarify a frequently used assumption in the sequel:

LEMMA 6.8. If DŽ CGG Ž Q ..- DC Ž Ž B ..,then D Ž C B Ž B .. - L. ŽŽ.. ŽŽw.. Proof. Suppose DCGG B sL. Then L s DC B also, where w is an involution in Q inverting T. This implies that L centralizes Q as ² w :ŽŽ.. QsB,B. Therefore, L s DCG Q . This contradicts our assumption that DCŽŽ..ŽŽ..GG Q -DC B .

PROPOSITION 6.9. If TŽ.L is connected then G has a weakly embedded subgroup. ŽŽ..ŽŽ.. Proof. Let R1 be any 2-torus in L. Then, Q s BCGG R sBC R1 by Corollary 5.24 since TŽ.L is connected. In particular, R1 centralizes Q. Therefore, dRŽ.1 centralizes Q. But L is a D-type group. Hence, L ŽŽ..ŽŽ.. centralizes Q. Therefore, L F DCGG Q FDC B FL, and we have equality. Now, by Lemma 6.6, NQGGŽ=DC Ž Ž Q ... is a weakly embedded subgroup of G. ŽŽ..ŽŽ.. Ž. LEMMA 6.10. If L is sol¨able, DCGG Q -DC B -L and T L Ž.Ž. Ž Ž.. Ž.ŽŽ. is not connected, then L s dROL, DCG B sdR OL l 0 DCŽGG Ž B ... ,and OŽ. L / 1. L and DŽ C Ž.. B are not nilpotent. Proof. Since TŽ.L is not connected, L is not nilpotent. Since R is 0 maximalŽ. Proposition 6.5 , Lemma 5.4 implies L s dRFL Ž.Ž..AsTŽ.L is 0 not connected, FLŽ.sOLŽ.. Hence, we have L s dROLŽ.Ž.. This im- Ž Ž .. Ž .Ž Ž . Ž Ž ...0 Ž Ž . Ž Ž ...0 plies DCGG B sdR OLlDC B . Also OLlDC G B Ž. ŽŽ..ŽŽ.. ŽŽ.. /1as dRFDCGG Q -DC B . We also claim that DCG B is ŽŽ..Ž. ŽŽ.. not nilpotent because otherwise DCGG B sdR and DC Q s DCŽŽ..G B , a contradiction.

PROPOSITION 6.11. If L is nonsol¨able, TŽ.L is not connected and ŽŽ..ŽŽ.. ŽŽ..Ž. ŽŽ..Ž. DCGG Q -DC B -L,then L s DCGG B OL sDC Q OL. Ž. Ž. Ž . Proof. By Corollary 5.28, LrOL( PSLK2 , where K is an algebraically closed field of characteristic different from 2. We will use ᎏ-notation to denote quotients by OLŽ.. Ž. Ž . As R ( R, R is a 2-torus of L. Moreover, as the 2-tori of PSLK2 , where K is an algebraically closed field of characteristic different from 2, are all one-dimensional, R is a maximal 2-torus of L. R F DCŽ.GŽ. Q . 558 ALTINEL, BOROVIK, AND CHERLIN

We claim that DCŽ.GŽ. Q is a Zariski closed subgroup of L. Let R1be Ž Ž.. Ž. Ž. a 2-torus of DCG Q . We claim dR11sdRŽ.. Clearly dR 11FdRŽ.. Ž. For the other inclusion we argue as follows. dR11sdROLŽ.Ž.by Fact ŽŽ..Ž.Ž. Ž. 2.8. But dROL11sdR OL. Therefore, dR11sdRŽ.. Therefore, ²Ž . : DCŽ.GŽ. Q s dR11:R is a 2-torus centralizing Q . Thus, by Fact 2.43, it is sufficient to show that dRŽ.11, where R is a 2-torus centralizing Q,isa U closed subgroup of L. The Zariski closure of R1 in L is isomorphic to K and thus the ‘‘no bad fields’’ hypothesis implies that dRŽ.1 and the Zariski closure of R1 are the same group. Therefore, DCŽ.GŽ. Q is of one of the following three forms: dRŽ.,a Borel subgroup UdŽ. R ,orL. We will show that the first two possibilities cannot occur, which will complete the proof. Ž. Ž.Ž. Case i DCŽ.GŽ. Q sUd R . Let ¨ g ILsuch that ¨ inverts the torus dRŽ.. The element ¨ acts on ROŽ. L . By Fact 2.31, there exists g g OLŽ. ¨ g y1 Ž. y1 such that R s R and ¨g g NRL . This implies that ¨g normalizes Ž Ž .. y1 Ž Ž .. BCGG R sQ. Hence, ¨g normalizes DC Q , and so ¨ normalizes Ž. DCŽ.GŽ. Q . But ¨ f Ud R , which contradicts the fact that Borel subgroups are self-normalizing. Hence, DCŽ.GŽ. Q /UdŽ. R . Ž. Ž. Ž. Case ii DCŽ.GŽ. Q sdR. Let ¨ be as in Case i . Then we also get Ž. y1 ŽŽ.. ggOL such that ¨g normalizes R, Q, and DCG Q . ²: In L, we can find W - R ¨ , such that W ( Q822or ޚ = ޚ , depending Ž. Ž. on whether L ( SL22 K or PSL K , respectively. We take an element x Ž. ²Ž.Ž.:x of order 3 from NWL . We claim that L s dR,dR . To illustrate we give an example in the SL2Ž. K situation:

i 00y10yi Wsus ,¨s ,ws , ¦;ž/ž/ž0yi 10 yi0 / 1 y1qiy1yi xs , 2ž/1yiy1yi

2 where i sy1. One can choose x so that

xx x us¨,¨sw,wsu. ²Ž . Ž .:x ŽŽ .. Ž . ²: Let L1s dR,dR . We have NdRL sdR i ¨ FL1. Therefore, L11cannot be a Borel subgroup of L.As L is connected, L1sL. In L, we may assume that WOŽ. L s OL Ž.iW, where W ( W.As x normalizes W, x normalizes OLŽ.iW. By Fact 2.31, we can find h g Ž. y1 Ž. OL such that xh g NWL . GROUPS OF MIXED TYPE 559

By taking conjugates, we may assume W - R²:¨ . Then W normalizes R. This implies W normalizes Q. Therefore, using Fact 2.40, we may assume W is a group of inner automorphisms of Q.AsWcentralizes T, W can be seen as a subgroup of T Žmaximal tori are self-centralizing in reductive algebraic groups. . But Q is an algebraic group over an algebraically closed field of characteristic 2, thus T is a 2 H -group. Therefore, the action of W ŽŽ.. on Q is trivial. Hence, we get Q F BCG W . ŽŽ.. ŽŽ.. We claim that Q s BCGG W . Suppose Q - BC W . Then there is a unipotent 2-subgroup B1 in BCŽŽ..G W but not in Q. By Lemma 6.7, B1 cannot be in the same connected component of UŽ.G as any unipotent ŽŽ.. Ž. 2-subgroup of Q. Therefore by Proposition 5.21, BCG W (PSL2 L , where L is an algebraically closed field of characteristic 2. But Q contains maximal unipotent 2-subgroups of G. In particular, these are maximal ŽŽ.. ŽŽ.. Ž. unipotent 2-subgroups of BCGG W .As BC W (PSL2 L , where L is an algebraically closed field of characteristic 2, these subgroups correspond to root subgroups of BCŽŽ..G W . But then Q contains two opposite ŽŽ.. ŽŽ.. root subgroups of BCGG W . This forces Q s BC W , a contradiction. 11 1 1 Since xhynormalizes W and ¨gynormalizes R, ²ŽdR .,xhy, ¨gy: ²Ž .y1 y1 : normalizes Q. Let L22s dR,xh , ¨g . Then L s L.As L2normal- izes Q, L2normalizes DCŽŽ..G Q . Therefore, L2normalizes DCŽ.GŽ. Q . Ž. Ž. Ž. But DCŽ.GŽ. Q sdR and L22s L ( PSLK. This is a contradiction because Ž.PSLK2 Ž .cannot have a large normal subgroup. Ž Ž .. Ž . Ž Ž .. Ž Ž .. PROPOSITION 6.12. If L s DCGG Q OL andDC Q -DCG B Ž. -L,then CZŽOŽ L.. B s 1. Ž. Ž Ž .. Proof. Let A1s CBZŽOŽL.. . Suppose A1 / 1. Then BCG A1 -G. The subgroup DCŽG Ž B .. centralizes B and normalizes ZOLŽ Ž ... Hence, ŽŽ.. ŽŽ..Ž. DCG B normalizes A1. But L s DCG B OL. Hence, L normalizes A1. Therefore, L normalizes BCŽŽ..G A1 . By Lemma 5.8, L centralizes Ž Ž .. Ž Ž .. BCG A1 . In particular, L centralizes B as B F BCG A1 . But this ŽŽ.. forces L s DCG B , which contradicts our assumptions. Ž Ž .. Ž . Ž Ž .. Ž Ž .. LEMMA 6.13. If L s DCGG Q OL andDC Q -DCG B -L, Ž Ž Ž Ž .... then B s BCGG DC B . Ž Ž Ž Ž .... Ž Ž Ž Ž .... Proof. Clearly, BCGG DC B FBCGG DC Q . By the Galois Ž Ž Ž Ž .... Ž Ž .. Ž Ž .. connection, BCGG DC Q sQ.As DCGG Q -DC B , we have the following strict inequality: BCŽGG Ž DC Ž Ž B ....-BC ŽGG Ž DC Ž Ž Q ..... Ž Ž Ž Ž .... This forces BCGG DC B sB. Ž Ž .. Ž . Ž Ž .. Ž Ž .. PROPOSITION 6.14. If L s DCGG Q OL andDC Q -DCG B Ž.0 -L,then COŽ L. B is di¨isible. 560 ALTINEL, BOROVIK, AND CHERLIN

Proof. By Fact 2.11, OLŽ.sE)D, where E is of bounded exponent Ž.0 and D is divisible. We may assume that E is connected. Let A1s CBE . Ž.0 ŽŽ.. Suppose A12/ 1. Let A s NAE1.DCG B normalizes A1. There- fore, DCŽŽ..GG B normalizes BCŽŽ.. A1. By Lemma 5.8, this action is Ž Ž .. Ž Ž Ž Ž .... trivial. Hence BCG A1 FBCGG DC B . By lemma 6.13, Ž Ž Ž Ž .... Ž Ž .. BCGG DC B sB. But B F BCG A1. Therefore, we conclude B s Ž Ž .. Ž Ž Ž Ž .... BCG A1 sBCGG DC B . A consequence of the last paragraph is that A22normalizes B. B i A is a nilpotent group, because otherwise, using Fact 2.18 we can interpret in

B i A2 an algebraically closed field K and the ‘‘no bad fields’’ hypothesis U U implies that a definable section of A2 is isomorphic to K . But K and hence A2 is not of bounded exponent, a contradiction. Hence, B i A2 is nilpotent and, by Fact 2.20, A22centralizes B. Therefore, A s A1.As Eis nilpotent of finite Morley rank, we conclude that E s A21s A . This Ž. Ž. forces ZEFCBZŽOŽ L.. . But this last group is trivial by Proposition Ž.0 6.12, a contradiction. As a result, A1 s 1 and therefore CBOŽ L. is divisible. Ž Ž .. Ž . Ž Ž .. Ž Ž .. COROLLARY 6.15. If L s DCGG Q OL andDC Q -DCG B Ž Ž . Ž Ž ...0 - L, then O L l DCG B isdi¨isible and torsion-free. Ž Ž . Ž Ž ...0 Proof. By Fact 2.11, OLlDCG B sE)D, where E is a definable, connected group of bounded exponent and D is a definable, divisible 0 group. If E / 1 then CBOŽ L.Ž. has a nontrivial definable connected subgroup of bounded exponent. But this contradicts Proposition 6.14. Ž Ž . Ž Ž ...0 ŽŽ. Therefore, OLlDCG B is divisible. The torsion part of OLl 0 DCŽG Ž B ... is also contained in the torsion divisible part of OLŽ.. But, by Fact 2.12, this last subgroup of OLŽ.is central in OLŽ.. Now, Proposition Ž Ž . Ž Ž ...0 6.12 forces the torsion part of OLlDCG B to be trivial. Ž Ž .. Ž . Ž Ž .. Ž Ž .. PROPOSITION 6.16. If L s DCGG Q OL andDC Q -DCG B Ž Ž Ž ... - L, then B s BCGO CŽL. B .

Proof. As DCŽŽ..GO B normalizes CBŽL.Ž.,itnormalizes BCŽŽGO CŽL. Ž... B . Thus, by Lemma 5.8, DCŽŽ..G B centralizes

BCŽGO Ž CŽL. Ž B .... Therefore,

BCŽ.GOŽ. CŽL.Ž. B FBCŽ.GGŽ. DCŽ.Ž. B .

But by Lemma 6.13, this last subgroup is equal to B. ŽŽ..Ž.ŽŽ..ŽŽ.. PROPOSITION 6.17. If L s DCGGG Q OL, DC Q -DC B - 0 0 L,and the group NOŽL.ŽŽ.. COŽL. B is torsion-free, then B i NCOŽL.ŽŽ..OŽL. B is nilpotent. GROUPS OF MIXED TYPE 561

ŽŽ..0 Proof. Let A s NCOŽ L. OŽ L. B. By Proposition 6.16 A normalizes B. Clearly B i A is a solvable group. Suppose it is not nilpotent. Then we can interpret in B i A an algebraically closed field K using Fact 2.18. As there are no bad fields in the environment, we conclude that a definable section of A is isomorphic to the multiplicative group of this field, in particular A is not torsion-free by Fact 2.22. This contradiction shows that B i A is nilpotent.

PROPOSITION 6.18. If L is sol¨able, TŽ.L is not connected, and ŽŽ..ŽŽ.. Ž. DCGG Q -DC B -L,then d R has p-torsion for e¨ery prime p. Ž Ž . Ž Ž ...0 Proof. By Lemma 6.10 and Corollary 6.15, OLlDCG B is divisible and torsion-free. As DCŽŽ..G B is nonnilpotentŽ. Lemma 6.10 and solvable, by Fact 2.18 an algebraically closed field K is interpretable Ž Ž . Ž Ž ...0 ŽŽ. in it. Kq is a definable section of OLlDCG B . Since OLl 0 DCŽG Ž B ... is torsion-free, K is of characteristic 0. The ‘‘no bad fields’’ hypothesis implies that a definable section of dRŽ.is isomorphic to K U and, using Fact 2.22, we conclude that dRŽ.has p-torsion for every prime p.

LEMMA 6.19. LetGbeaD-typeŽ. resp. B-type group of finite Morley X rank. Assume G s G C, where C is a definable 2 H -subgroup of G. Then G is perfect. X Proof. GrG is a 2H -group. Hence, all the 2-toriŽ resp. unipotent XX 2-subgroups. are in G , forcing G s G . PROPOSITION 6.20. Let G be a tame group of finite Morley rank, Ta definable di¨isible abelian subgroup of G, and O a definable, connected, nonnilpotent, torsion-free subgroup of G. Assume T normalizes O. Assume also that T does not ha¨ep-torsion for some prime p. Then H s TO is nilpotent. Proof. Suppose that H is not nilpotent. Then we can interpret in H an algebraically closed field K, using Fact 2.18. As no bad fields are interpretable in the environment, a definable section of T will be isomorphic to U Ž. K. This forces char K s p, where p is a prime missed by T. Kq is isomorphic to a definable section of O. Hence, it is torsion-free by Fact 2.22. This yields a contradiction. Therefore, H is nilpotent.

PROPOSITION 6.21. If L is nonsol¨able, TŽ.L is not connected, and ŽŽ..ŽŽ.. Ž.Ž.Ž. DCGG Q -DC B -L, then LrOL( PSLK2,where K is an algebraically closed field of characteristic 0. Proof. We will useᎏ -notation to denote quotients by OLŽ..By Ž Ž .. Ž Ž Ž ... Proposition 6.11, L s DCŽ.GGŽ. B (DC B rODC G B . By Corol- Ž. Ž. Ž . lary 5.28 we know that LrOL( PSLK2 , where the characteristic of 562 ALTINEL, BOROVIK, AND CHERLIN

K is different from 2. Suppose the characteristic of K is not 0. Then it is at Ž Ž .. Ž Ž Ž ... ² Ž . Ž Ž Ž ... Ž Ž Ž ... least 3. DCGG B rODC B s dSOCCGG B rODC B :S is a 2-torus of DCŽG Ž B ..:. Proposition 6.20 and Corollary 6.15 imply that Ž . Ž Ž Ž ... Ž Ž Ž ... Ž Ž Ž ... dSODCGG B is nilpotent. Thus SO D C B s S [ ODCG B . As S is an arbitrary 2-torus of DCŽŽ..GG B , and DC ŽŽ.. B is a D- type group, ODCŽ ŽGG Ž B ... is a central subgroup of DCŽ Ž B ...As Ž Ž .. Ž Ž Ž ... Ž . Ž . Ž Ž Ž ... DCGG B rODC B ( PSLK2, we conclude that ODCG B Ž Ž Ž ...0 s ZDCG B . Therefore,

0 DCŽ.Ž.GGŽ. B rZDCŽ. Ž. B ( Ž.PSLK2 Ž., where K is an algebraically closed field of characteristic at least 3. By

Lemma 6.19, DCŽŽ..G B is a perfect group. Theorem 4.2 implies that Ž Ž Ž ... Ž Ž .. Ž Ž .. ODCGG B s1. But this forces DC B sDCG Q , which contradicts our assumptions. This finishes the proof of the claim.

LEMMA 6.22. Let G s dŽ. T E be a connected sol¨able group of finite Morley rank, where T is a p-torus and E is a normal nilpotent group of bounded exponent a power of p. Then G is nilpotent. Proof. Suppose G is nonnilpotent. Then, by Fact 2.18, we can interpret an algebraically closed field K in G so that a definable section of E is isomorphic to Kq. This implies that the characteristic of K is p. On the other hand, a definable section of dTŽ.is isomorphic to an infinite subgroup of K U. As the characteristic of K is p, T centralizes E. Therefore, dTŽ.centralizes E. This forces G to be nilpotent, a contradiction.

PROPOSITION 6.23. If TŽ.L is not connected and D ŽŽ..ŽŽ.. CGG Q - DC B -L,then OŽ. L is di¨isible. Proof. We carry out a case-by-case analysis. L is sol¨able. By Lemma 6.10, L is not nilpotent. By Theorem 2.11, OLŽ.sE)D, where E is of bounded exponent and D is divisible. dREŽ. is a connected solvable group. Suppose E / 1. dRŽ.has p-torsion for every prime p by Proposition 6.18. Suppose p is an odd prime such that E has a nontrivial Sylow p-subgroup, say Epp. By Lemma 6.22, E centralizes the p-torus Tpppof dRŽ..As E centralizes T , it normalizes BCŽ G Ž Tp ...

By Lemma 5.10, EpGcentralizes BCŽŽ.. Tp. In particular, E pcentralizes B. Ž. But CBZŽOŽ L.. s1 by Proposition 6.12 and Lemma 6.10. This is a contradiction. Therefore E s 1. Ž. L is nonsol¨able. Let D0 be the divisible part of OL. The quotient Ž.Ž.Ž.ŽŽ.. Ž.Ž.Ž. LrD00rOLrD s LrD 0rOLrD 0(LrOL ( PSLK2, where K is an algebraically closed field of characteristic 0, by Proposition Ž. 6.21. LrD00is generated by dSDrD0, where S ranges over all the 2-tori GROUPS OF MIXED TYPE 563

Ž. Ž . of L. Consider the action of dSD00rD on OLrD0. The group ŽŽ. . Ž . dSD00rDOLrD 0is nilpotent, because otherwise, using Fact 2.18 and Proposition 6.21, we can interpret an algebraically closed field of Ž. characteristic 0 in it in such a way that a definable section of OLrD0 is Ž. isomorphic to the additive group of this field. But OLrD0 is of bounded Ž. Ž. exponent. Therefore, the action dSD00rD on OLrD0is trivial. As S Ž. is an arbitrary 2-torus in L, we conclude that OLrD0 is central in LrD00. By Lemma 6.19, LrD is perfect. Therefore, by Theorem 4.2, Ž. Ž. OLrD00s1. This implies that OLsD.

PROPOSITION 6.24. If TŽ.L is not connected and DŽŽ..ŽŽ.. CGG Q - DC B -L,then OŽ. L is torsion-free. Proof. We carry out a case-by-case analysis. L is sol¨able. By Lemma 6.10, L is not nilpotent. By Proposition 6.23, OLŽ.is divisible. By Proposition 5.3 wL, OLŽ.xis torsion-free. Clearly, OLŽ.rwL,OLŽ.xwis central in Lr L, OLŽ.x. Also ŽLr wL, OLŽ.x.ŽŽ.rOLr wL,OLŽ.x.Ž.(LrOL and this last group is abelian because L s dROLŽ.Ž.by Lemma 6.10. Therefore, Lr wL, OLŽ.xis a nilpotent group. Ž. Ž. Ž Since L is a D-type group, so is Lr w L, OLxwand Lr L, OLxsdR1r Ž.. Ž. Ž. wL,OLx, where R1r wL, OLxwis a maximal 2-torus of Lr L, OLx.As Lis solvable, Fact 2.33Ž. ii implies that LrwL, OLŽ.xsdRŽ.w L,OLŽ.xr 0 wL,OLŽ.x. Thus L s dRŽ.w L,OLŽ.xand OLŽ.s ŽŽ.OLldR Ž..w L, 0 OLŽ.x. By Corollary 6.15, ŽŽ.OLldR Ž..is torsion-free. Thus Fact 2.26 implies that OLŽ.is torsion-free. Ž. L is nonsol¨able. Proposition 6.23 and Fact 2.12 imply that OLsA1= A212, where A and A are the torsion and torsion-free parts of OLŽ., respectively. By Proposition 5.3, wL, OLŽ.xis torsion free. We have, by Proposition 6.21,

Ž.Lr L, OLŽ.rŽ.OL Ž.rL,OL Ž.(LrOL Ž.( Ž.PSLK2 Ž ., where K is an algebraically closed field of characteristic 0. Clearly, OLŽ.rwL,OLŽ.xwis central in Lr L, OLŽ.x. We have a central extension of finite Morley rank of a simple algebraic group by a group of finite Morley rank. By Proposition 6.19, Lrw L, OLŽ.xis a perfect group. By Theorem 4.2, we conclude that OLŽ.rwL,OLŽ.xs1. Hence, OLŽ.swL,OLŽ.xis torsion-free. U THEOREM 6.25. A simple tame, K -group G of mixed type contains a weakly embedded subgroup.

Proof. Let M1 denote the stabilizer of a connected component of Ž. Ž. UG. By Proposition 6.1, if NTG FM111for any 2-torus of M , then M is a weakly embedded subgroup. If this is not the case then there is a 564 ALTINEL, BOROVIK, AND CHERLIN

Ž . Ž Ž Ž ... 2-torus R in M1such that NRG gM1. By Proposition 6.4, U BNG R is not connected. By Proposition 6.5, R can be assumed to be maximal in ŽŽ.. Ž. G. Fact 5.21 implies that Q s BCG R (PSL2 K , where K is an algebraically closed field of characteristic 2. Then, as discussed after Ž. Proposition 6.4, NBQ sBT, where T is a maximal torus in Q and

DCŽ.Ž.GGŽ. Q FDC Ž. B FLsDC Ž. GŽ. T .

By Proposition 6.9, if TŽ.L is a connected graph then G has a weakly embedded subgroup. Therefore, we may assume that TŽ.L is not a connected graph. ŽŽ..ŽŽ.. If DCGG Q sDC B then by Lemma 6.6, G has a weakly embedded subgroup. Therefore, we may assume that DCŽŽ..ŽŽ..GG Q -DC B . Then, by lemma 6.8, DCŽŽ..ŽŽ..GG Q -DC B -L. We will show that DCŽŽ..ŽŽ..GG Q -DC B -L cannot happen. Let ŽŽ.. Ž. AsNCOŽ L. OŽ L. B.Ais connected since, by Proposition 6.24, OL is Ž. Ž. Ž Ž.. torsion-free. We have CBOŽL. -OLbecause otherwise L s DCG B , Ž. Ž. Ž. contrary to our assumptions. In particular, wOL:CBOŽL. xsϱ.AsOL ŽŽ..Ž. is nilpotent, by Fact 2.14, wNCOŽ L. OŽ L. B:CBOŽ L. xsϱ. By Proposition 6.24, A is torsion-free. This, together with Proposition 6.17, implies that ŽŽ..Ž. BiAsB[A. Hence, w NCOŽ L. OŽ L. B:CBOŽ L. xs1, a contradiction.

7. FROM WEAK TO STRONG EMBEDDING

In this section we will finish the proof of Theorem 1.7. We will continue working in the hypothetical simple tame K U-group G of mixed type. In the previous section we showed that such a group has a weakly embedded subgroup. The main result in this section will show that this weakly embedded subgroup is a strongly embedded subgroup. This will yield a contradiction for reasons we will mention shortly.

DEFINITION 7.1. 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:

Ž.i Mcontains involutions. g Ž.ii For every g g G_ M, M l M does not contain involutions. GROUPS OF MIXED TYPE 565

The following are two characterizations of strongly embedded subgroups:

Fact 7.2wx 9, Theorem 10.19, p. 186 . Let G be a group of finite Morley rank with a proper definable subgroup M. Then the following are equiva- lent:

Ž.i M is a strongly embedded subgroup. Ž.Ž. Ž. Ž. Ž. ii IM /л,CiGGFMfor any i g IM, and NSFMfor any Sylow 2-subgroup S of M. Ž. Ž . Ž. iii IM /л, and NSG FMfor any nontrivial 2-subgroup S of M.

The following fact is crucial for our argument:

Fact 7.3wx 9, Theorem 10.19, p. 186 . Let G be a group of finite Morley rank with a strongly embedded subgroup M. Then IGŽ.is a single conjugacy class in G, and IMŽ.is a single conjugacy class in M.

The following proposition shows that the proof of Theorem 1.7 will be over once we have shown that G has a strongly embedded subgroup.

PROPOSITION 7.4wx 1 . 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.

0 Proof. Let S be a Sylow 2-subgroup of G. S s B)T by Fact 1.5Ž. ii . We will show that we cannot have B / 1 and T / 1. Suppose toward a contradiction that B / 1 and T / 1. By Fact 2.27, T contains finitely many 0 involutions. As IBŽ.is infinite, we can find u g IS Ž _T.Ž.and ¨ g IT. Ž. Since these are conjugate in G, by Fact 2.32, there exists g g NTG such g that u s ¨ . But this implies that u g T, contradicting the choice of u. U THEOREM 7.5. Let G be a simple tame K -group of mixed type with a weakly embedded subgroup M. Then M is a strongly embedded subgroup.

Proof. Proposition 3.2Ž. i and Fact 7.2Ž. ii imply that it is sufficient to Ž. Ž. prove that, for any t g IM,CtG FM. Suppose toward a contradiction Ž. Ž. Ž. that there is an involution t g IM such that CtGGgM. Let H s Ct. Ž.1 HlMhas an infinite Sylow 2-subgroup.

Proof of Ž.1 . Let S be a Sylow 2-subgroup of M such that t g S. 0 S s B) D, where B / 1 and D / 1. The involution t acts on the definable subgroup B. As there are infinitely many involutions in B, t cannot fix finitely many elements of B Ž.Fact 2.38 . 566 ALTINEL, BOROVIK, AND CHERLIN

It follows fromŽ. 1 that M l H is a weakly embedded subgroup of H, 0 and in particular contains a Sylow 2-subgroup of H. Similarly, M l H is a weakly embedded subgroup of H 0. Note that it follows from the proof ofŽ. 1 that B l H and thus H contains an infinite elementary abelian 2-group. In particular, BHŽ./1. 0 Ž.2 H gM. Proof of Ž.2 . Ž. 1 and the remarks that follow its proof imply that H contains infinite Sylow 2-subgroups. This implies that H 0 contains infinite Sylow 2-subgroups as well. If A is a Sylow 2-subgroup of H 0, then by the Ž. 00 0 Frattini argument H s NAHH .If H FMthen A F M as well and, Ž. 0 since M is a weakly embedded subgroup, H s NAHH FM, a contradiction to the assumption that H g M. Ž.3 H0 is not solvable.

0 0 Proof of Ž.3 . Suppose H is solvable. As BHŽ.FH, we conclude using Fact 2.36 that BHŽ.is a unipotent 2-subgroup normal in H.As BHŽ.eH, BH Ž.is contained in every Sylow 2-subgroup of H and, hence, in H l M.As HlMis a weakly embedded subgroup of H, ŽŽ .. HFNBHH FHlM, a contradiction. Ž. 0 Ž. Ž. 4 HsL=OH, where L ( PSL2 K and the characteristic of 0 K is 2. In particular, H does not contain 2-tori and t g H _ H Proof of Ž.4 . Lemma 5.20,Ž. 3 , and the fact that H has nontrivial 0 Ž. Ž. unipotent 2-subgroups imply that H rOH (PSL2 K , where K is an 0 algebraically closed field of characteristic 2. Therefore, H s BHOHŽ.Ž.. By Fact 2.37, BHŽ.centralizes OH Ž.. Since H l M contains a nontrivial unipotent 2-subgroup, OHŽ.FM. Thus Ž. 2 implies that BHŽ.gM.We therefore conclude that BHŽ.lMis a weakly embedded subgroup of BHŽ.. In particular, UŽŽ..BH is not connected. Therefore, by Proposition Ž. Ž. 5.21, BH (PSL2 K , where K is an algebraically closed field of characteristic 2. This proves the claim.

Ž.5 MlLcontains a unique Sylow 2-subgroup of L. 0 Proof of Ž.5 . Ž. 1 and the fact that H s L = OHŽ., where L ( Ž. Ž. PSL2 K with char K s 2, imply that a Sylow 2-subgroup of L l M is an infinite elementary abelian 2-group. Let A be such a subgroup. If A F A1, where A11is a Sylow 2-subgroup of L, then A centralizes A as A1is abelian. But A is a subgroup of M and M is a weakly embedded subgroup. Therefore, A1 is a subgroup of M. But A is a Sylow 2-subgroup of L l M. Therefore, A1 s A. If M contains another Sylow 2-subgroup of L Then L F M because Ž. Ž. Ž. L(PSL2 K . But this is impossible by 2 and 4 . GROUPS OF MIXED TYPE 567

Ž. We fix A, a Sylow 2-subgroup of L in M l L. NAL sAiT, where Ž. Ž. Tis a torus. Let w g NTL be such that w inverts T. Let Y s DM.By ŽŽ.. w Lemma 5.7, DCG A GY. Moreover, as T normalizes both A and A , ²Ž Ž..Ž Žw..: Ž Ž .. we have, by Lemma 5.12, DCGG A ,DC A FDC G T . We will use R to denote DCŽŽ..G T . We can summarize the above discussion as follows:

DMŽ.sYsDCŽ.GG Ž. A FRsDC Ž.Ž. T .

Ž.6 Ris solvable. In particular, Y is solvable. Ž. Ž. Proof of 6. AsTFLFCtG and R is a characteristic subgroup of Ž.Ž. Ž. Ž. Ž. CTGG,tgNR. Moreover, since CtRGRFCt,Ctdoes not contain 2-tori. By Lemma 5.1, R is solvable.

Ž.7 RsY. Proof of Ž.7. By6, Ž. Rand Y are both solvable. By Lemma 5.4, R and Y can be factored as a product of their Fitting subgroups and the definable closure of their maximal 2-tori. As M is a weakly embedded subgroup,

Proposition 3.2Ž. ii implies that Y contains a maximal 2-torus T1 of G. Ž.Ž.0 Ž.Ž.0 Therefore, we have Y s dT11 FY and R s dT F R . Suppose that Y - R. We will reach a contradiction. If R is nilpotent, then we immediately conclude that R s Y. Therefore, R is nonnilpotent. 0 Note that R g M because otherwise R s Y. This forces FRŽ.to be a 2H -group because otherwise, by Lemma 5.5, FRŽ.0 would contain a nontrivial 2-torus Tcc.AsT eR,T cis centralized by R, and therefore by T1. This forces Tc F M and thus R F M, a contradiction to R g M. This Ž.Ž. Ž.Ž Ž.. implies R s dT11 O R and Y s dT YlOR .

Ž.7.1 If t inverts T11Žequivalently dT Ž.., then CtOŽR. Ž.is infinite and Ž.0 CtOŽR. FM. Proof of Ž.7.1 . The involution t centralizes T. Therefore, it acts on R.

As ORŽ.is characteristic in R, t acts on OR Ž.also. If CtOŽ R. Ž.is finite then t inverts ORŽ.by Fact 2.38. Then by Fact 2.25 and Proposition 2.52, Ž. Ž.Ž. Ž. Ž. Ž. CtRrOŽR. sCtORR rOR.Astinverts dT1, t inverts RrOR. Ž. Hence, CtR r OŽ R. is finite. But

CtORRRŽ. Ž .rOR Ž .(Ct Ž.rCtOŽR. Ž..

Ž. Ž. Ž. Ž. Hence, CtROrCtŽR.is finite. As CtOŽR.is finite, CtRis finite also, which forces R to be abelian. This contradicts our assumption that R is not nilpotent. Hence, CtOŽ R.Ž.is infinite. 568 ALTINEL, BOROVIK, AND CHERLIN

0 The connected group CtOŽ R.Ž. acts on L and centralizes T. The action 0 0 of CtOŽ R.Ž. on L is by inner automorphisms. Therefore, CtOŽ R.Ž. normal- Ž.0 izes A. This forces CtOŽ R. FMas M is a weakly embedded subgroup. Ž. Ž. 7.2 CAZŽOŽ R.. s1. Ž. Ž. ŽŽ.. Proof of 7.2 . Let C1s CAZŽOŽR.. . Suppose C1 / 1. Then BCG C1 ŽŽ.. Ž.Ž. Ž. Ž. -G. Note that A F BCG C11.As RsdT O R and dT 1FCAG, Rnormalizes C1. Hence, by Lemma 5.8, R centralizes BCŽŽ..G C1 , and in particular it centralizes A. But M is a weakly embedded subgroup, which forces R F M. This contradicts our assumption on R. Ž.7.3 OR Ž.is a divisible group.

Proof of Ž.7.3 . By Fact 2.11, OR Ž.sC)D, where C is definable, connected, and of bounded exponent and D is definable and divisible.

Suppose C / 1. Then the action of dTŽ.1 on C is nontrivial because, otherwise, dTŽ.1 Dis a definable subgroup R which contains all the 2-tori Ž. of R. But R is a D-type group and thus, by Lemma 5.2, R s dT1 D. Ž . Ž Ž . Ž .. Ž Therefore, ORs dT1 lOR D, which forces C s 1 Facts 2.11, 2.27, and 2.28. . This contradicts our assumption on C. Therefore, C has an infinite p-subgroup Cp on which dTŽ.1 acts nontrivially.

We claim that dTŽ.1 does not have p-torsion. Suppose Tp is the nontrivial p-torus of dTŽ.1 . By Lemma 6.22, Cpppcentralizes T . Thus C normalizes BCŽŽ..Gp T . This last subgroup contains A. By Lemma 5.10, Cp Ž Ž .. Ž Ž .. centralizes BCGp T . Hence, Cpcentralizes A. But Cpl ZOR /1. Ž. This contradicts 7.2 . Therefore Tp s 1. As dTŽ.1 has no p-torsion, an application of Facts 2.18 and 2.22 and the ‘‘no bad fields’’ assumption implies that dTŽ.11 Dis nilpotent. T is central in dTŽ.111 D. This implies dTŽ. Dcentralizes dTŽ.. Thus dT Ž.1 Ccontains all the 2-tori of R. But R is a D-type group. Therefore, by Lemma 5.2, Ž. Ž. RsdT11 C. In particular, this implies that D - dT . 0 0 We claim that Ž.C l M / 1. Suppose to the contrary that Ž.C l M s Ž.Ž Ž..Ž.Ž . 1. We already know that Y s dT11 YlOR sdT YlCD . D F Ž. Ž.Ž . Ž.Ž . dT11FYimplies that Y s dT DYlC sdT1 YlC.AsYlCF Ž. Ž. MlC,YlCis finite. But Y s DM is connected. Therefore, Y s dT1 . Ž. Ž. This implies that t acts on dT1 .As HsCtG does not contain nontrivial 2-tori, Lemma 5.13 implies that t inverts dTŽ.1 . By Ž 7.1 . , CtOŽ R. Ž.is Ž.0 Ž. infinite and CtOŽ R. FM.AsORsCD and C l M is finite, any Ž.0 element of CtOŽ R. can be written as dcii, where d g D and c g C l M Ä4 Ä 4 scim,...,c . There exists c g C l M such that dcj: jgގ is an Ž.0 Ž.Ž.y1 y1 infinite set of distinct elements of CtOŽR. .dc1 dcj sdd1 j g Ž.0 Ž.0 CtOŽ R. lDfor all j g ގ. This implies that CtOŽ R. lD/1, a con- 0 tradiction. Therefore, Ž.C l M / 1. GROUPS OF MIXED TYPE 569

0 The subgroup Ž.C l M centralizes A by Lemma 5.10. The subgroup ŽŽ.. ŽŽŽ... NCGC A normalizes BCGC C A . On the other hand, Y F DNŽGC Ž C Ž A ... because Y centralizes BMŽ. and Y normalizes C. This Ž Ž Ž ... Ž Ž Ž ... implies BCGC C A centralizes Y and, therefore, BCGC C A FM. ŽŽ.. ŽŽ..Ž. But then NCCC AFClM. This forces wNCCC A:CA C x-ϱ,so Ž. by Fact 2.14, CAC sC,C-M, and thus R - M, a contradiction. This final contradiction shows that ORŽ.is divisible.

Ž. Ž. 7.4 CAOŽ R. s1.

Ž. Ž. Ž Ž Ž... Proof of 7.4 . Suppose CAOŽR. /1. Then A F BCGO CŽR. A - G. By Corollary 5.8, dTŽ.11centralizes BMŽ.and in particular A.AsdTŽ. normalizes ORŽ., we conclude that dTŽ.1 normalizes CAOŽ R.Ž.. Therefore, dTŽ.1 normalizes BCŽGO Ž CŽR. Ž A .... By Corollary 5.8, dTŽ.1centralizes Ž Ž Ž ... Ž Ž Ž ... BCGO CŽR. A .As Mis weakly embedded, BCGO CŽR. A FM. Therefore, NCOŽR.ŽOŽR. Ž A .., which also normalizes BCŽGO Ž CŽR. Ž A ...,isa subgroup of M. An immediate consequence of the above paragraph is that ZORŽŽ..- X M. This also implies that ORŽ.is not abelian. Therefore, ŽŽ.OR l 0 ZORŽ Ž ... is an infinite, definable, connected subgroup of M l ORŽ..By X 0 Ž.7.3 and Fact 2.10, ŽŽ.OR lZORŽ Ž ... is divisible abelian. Moreover, X 0 ŽŽ.OR lZORŽ Ž ... is torsion-free, because w R, ORŽ.xis torsion-free by X 0 Proposition 5.3. By Lemma 5.11, ORŽ.lZORŽ Ž ... centralizes BMŽ. Ž. Ž. and in particular A. This contradicts 7.2 . Therefore, CAOŽ R. s1.

Ž. Ž. Ž. Ž. 7.4 implies that ORlYs1 because ORlYFCAOŽ R. . This Ž. Ž. Ž. implies that Y s dT11. Therefore, t acts on dT .As HsCtGdoes not contain nontrivial 2-tori, Lemma 5.13 implies that t inverts dTŽ.1 . By Ž 7.1 . , Ž. Ž.0 CtOŽR. is infinite and CtOŽR. FM. Ž. Ž.0 Ž.0 Being a subgroup of H s CtGO,CtŽR.normalizes L.AsCtOŽR.F Malso, it normalizes M l L.By5,Ž. MlLhas a unique Sylow 2-sub- 0 group, namely, A. Therefore, we conclude that CtOŽ R.Ž. normalizes A. 0 Another consequence ofŽ. 7.4 is that the action of CtOŽ R.Ž.on A is faithful. The facts that the action of T on A_Ä41 is transitive and that T 0 normalizes CtOŽ R.Ž.imply that the action of CtOŽ R. Ž. on A is fixed-point- 0 free. Therefore, we have A i CtOŽ R.Ž., a centerless, hence nonnilpotent solvable group. By Fact 2.18, we can interpret in this group an algebraically 0 closed field K of characteristic 2. A definable section of CtOŽ R.Ž. is U 0 isomorphic to K Ž.‘‘no bad fields’’ . This implies that CtOŽ R. Ž.has some torsion in itŽ. Fact 2.22 . We will show that this is not possible. Now, we consider the group w R, ORŽ.x. It is nontrivial as R is assumed to be nonnilpotent. By Proposition 5.3, w R, ORŽ.xis torsion-free. Clearly, 570 ALTINEL, BOROVIK, AND CHERLIN

ORŽ.rwR,ORŽ.xwis central in Rr R, ORŽ.x,

Ž.RrR,ORŽ.rŽ.OR Ž.rR,OR Ž.(RrOR Ž.(dT Ž1 ..

This forces RrwR, ORŽ.xto be a nilpotent group. But this is also a D-type group, so Rrw R, ORŽ.xis the definable closure of a 2-torus. Hence, using the solvability of R and Fact 2.33Ž. ii , we conclude RrwR, ORŽ.xs Ž. Ž. Ž. Ž. Ž. Ž. dT11w R,ORxwrR,ORx. This implies R s dTw R,ORx.AsORl Ž. Ž. Ž. Ž. Ž. Ys1, dT1 FY, and wR, ORxFOR,ORswR,ORx. Therefore, 0 ORŽ.is torsion-free, which contradicts the above conclusion on CtOŽR.Ž.. This last contradiction finishes the proof ofŽ. 7 . Ž. ŽŽ.. ŽŽw.. 7 implies that DCGG A sYsR. Therefore, DC A sR. Hence, wnormalizes Y, forcing w g M. This means L s ²:A, w F M. On the other hand, OHŽ.FMbecause OH Ž.centralizes L and in particular A, which is a unipotent 2-subgroup of M. Therefore, we conclude that 0 H FM, a contradiction toŽ. 2 , which finishes the proof of the theorem. Now Theorem 1.7 follows from Theorem 6.25, Proposition 7.4, and Theorem 7.5.

ACKNOWLEDGMENTS

The authors would like to thank Ali Nesin for carefully reading the paper and correcting some errors. They also thank the referee for crucial corrections and helpful suggestions about the organization of the paper.

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