Tame groups of odd and even type A. V. Borovik∗ Department of Mathematics UMIST PO Box 88 Manchester M60 1QD United Kingdom E-mail:
[email protected] 9 November 1997
Introduction This is a report on a joint work with Tuna Altınel, Gregory Cherlin and Luis Corredor which is a part of a programme of classification of simple tame ωstable groups of finite Morley rank; other people working on the programme are Christine Altseimer, Ay¸se Berkman, Mark DeBonis, Ali Nesin and Paul Watson and we mention some of their results. For a mathematician not involved with model theoretical stability theory the purpose of the programme can be best explained as an attempt to understand the logical structure of the classification of finite simple groups (CFSG). We share the (maybe heretical) belief that CFSG actually consists of several more or less independent theories. One theory, ‘sporadic’, deals with ‘sporadic objects’, most of which, we presume, are still unknown, but some happen to be finite simple groups. Another, ‘generic theory’, describes, in a more or less uniform way, structural properties of the majority of finite simple groups, namely, ‘large’ groups of Lie type. The theory of ω-stable groups of finite Morley rank captures the essence of this ‘generic’ part of CFSG, and it works approximately in the following way. Assume that we study a finite simple group G with the aim to identify it eventually with the group of points over a finite field of a simple algebraic group e Let us look how the standard arguments of CFSG applied to G are reflected G.
∗ The text is distributed as a background material for the participants of the Euroconference Groups of Finite Morley Rank, Crete, 22–26 June 1998, sponsored by the Traininig and Mobility of Researches Programme of the Commission of the European Union.
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e For in the structural properties of a hypothetical simple algebraic group G. example, we can do different tricks with the centralisers of involutions in G—but e we can also do the same manipulations with the centralisers of involutions in G. e Of course, we cannot use the concept of order in the infinite group G, but we can e q )| use dimension instead. By the Lang-Weyl theorem the logarithm logq |X(F e asymptotically of the number of Fq -points of an irreducible affine variety X e behaves as dim X. This to some extent explains why many properties of ‘large’ finite simple groups can be translated as ‘dimension’ properties of algebraic groups and why methods of CFSG can be successfully applied to those infinite groups which admit a reasonable concept of dimension. An explicit description of this class of infinite groups can found in model theory, as the class of ω-stable groups of finite Morley rank. Model theory provides a very efficient language for developing the theory of groups of finite Morley rank as an ‘asymptotic limit’ of CFSG. In an ω-stable structure M every definable (by a formula of first order language) set can be assigned an ordinal which is called its Morley rank. Simple algebraic groups over algebraically closed fields are ω-stable, and Morley rank on them is finite and coincides with dimension. Though of model theoretic origin, the concept of Morley rank can be introduced in a purely axiomatic (and very elementary) way, see Section 1 below. There is a conjecture, due to Gregory Cherlin and Boris Zil’ber, that Simple infinite groups of finite Morley rank are algebraic groups over algebraically closed fields. We are interested in a special case of this conjecture, classification of tame simple groups of finite Morley rank.
Tame Groups Every group G of finite Morley rank has a family of the definable subgroups, i.e. those subgroups which can be assigned rank. Further, G contains a unique minimal definable subgroup of finite index, denoted G◦ and called the connected component of identity. A group G of finite Morley rank is connected if G = G◦ . Let G be a group of finite Morley rank. A section of G is a quotient H/K with K H and H, K both definable in G. G is a K-group if every connected simple section of G is isomorphic to an algebraic group over an algebraically closed field, and is a K ∗ -group if every proper definable subgroup is a K-group. A bad group is a connected simple group of finite Morley rank in which every connected soluble subgroup is nilpotent. We say that G involves no bad group if it has no bad group as a section. A bad field is a structure of the form (K, T ) where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, and such that (K, T ) is of finite Morley rank in its full language. We say that G involves no bad field if no bad field is interpretable in G. The
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group G is tame if it involves no bad group or bad field. The main conjecture in the theory of groups of finite Morley rank is that the connected tame simple groups are all algebraic, over algebraically closed fields. The restriction to the tame case puts aside two potential sources of pathology which are not well understood at this time. Bad groups probably do not in fact exist, while the situation with regard to bad fields is less clear. In any case, working inductively, in order to prove that connected simple tame groups of finite Morley rank are algebraic, it suffices to deal with K ∗ -groups exclusively. Thus virtually all of the results in this area are stated and proved in that context. This applies in particular to the results of this paper. If S is a Sylow 2-subgroup of G, then its connected component S ◦ has a simple structure: S ◦ = U ∗ T , a central product with finite intersection, where U is nilpotent of bounded exponent and T is a 2-torus (that is, a divisible abelian 2-group). We say that G is of even type if T = 1, of odd type if U = 1, and of mixed type otherwise. The first major subdivision of the theory is Theorem 3.4: a connected simple tame K ∗ -group is either of even type, or of odd type. A plan of attack for the classification of tame groups exists in both cases, based loosely on the strategy that works in the finite case (more specifically, on a revisionist strategy using induction throughout). The odd case was discussed in much detail in [19]. In the present paper we emphasise on the even case, though mentioning most recent developments in the odd case. As it suffices to attack odd type and even type groups separately, two quite different strategies have been developed to deal with these two cases. Theorem 5.2 below amounts to a division of the odd type case into three rather special subcases. In the case of even type, there is a division in the ‘generic’ and ‘component type’ cases, described in Section 7. None of these two cases was studied so far in any detail, but there is an existing collection of generally useful tools which can be used in further development. We explain in the present paper how we use these tools to approach the division of the even case.
Background We generally refer to the text [21] for standard background material, including the notion of finite Morley rank, connected components, and Sylow theory for the prime 2, the soluble and Fitting radicals, and the generalised Fitting subgroup F ∗ (G). We use many ideas and concepts of finite group theory; the reader may wish to consult Gorenstein [34] for an introduction to the classification theory of finite simple groups. The analogy with algebraic group theory is also very
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important for the present paper. We rely throughout on the general structure theory of algebraic groups. Most of the material needed is quite standard, and is found in Humphreys [39]; early sections of Carter [24] also provide an efficient presentation of a good deal of useful structure theory, some of it in summary form. We mention ω-stable groups; the reader who wishes to learn more about them can consult Poizat [49]. We recommend Chang and Keisler [25] and Hodges [38] for references concerning model theory.
1
Axioms of Morley Rank
The best way to introduce Morley rank is to use an analogy with measure. We know that measure is a countably additive function µ : Σ −→ R≥0 ∪ ∞ defined on a certain set of sets Σ; the latter is supposed to be a σ-algebra, i.e. it is closed under complementation and countable unions. For Morley rank we use the same approach: Morley rank is a function rk : U −→ N defined for all non-emty sets in a set U of sets. We call U an universe, its elements definable sets and assume that U is closed with respect to certain set-theoretic operations. The axioms for a universe U and rank function rk follows. Our axioms are satisfied by dimension of constructible sets (i.e. Boolean combinations of Zariski closed sets) in algebraic geometry.
1.1
Universe
We require the following axioms from a universe: • Closure under Boolean operations. If A and B are definable sets, then the sets A ∩ B, A ∪ B and A \ B are also definable. • Closure under products. If A and B are definable sets, then their Cartesian product A × B and the canonical projections π1 : A × B −→ A, π2 : A × B −→ B are also definable1 . If A = B, then the diagonal ∆ = { (a, a) | a ∈ A } ⊂ A × A is also definable. We assume also that if C is a definable subset in A × B, then the images π1 (C), π2 (C) of C under canonical projections are definable. 1A
function is definable if and only if its graph is.
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• Finite subsets. If A is definable and a ∈ A, then the singleton set { a } is definable. • Factorisation. If E(x, y) is a definable equivalence relation on a definable set A (thus in particular E is a definable subset of A2 ), then the quotient A/E is encoded in the universe as follows: there is a set A¯ ∈ U and a surjective definable function f : A −→ A¯ such that for x, y ∈ A we have: f (x) = f (y) iff E(x, y) holds. It easily follows from these axioms that the union and the intersection of finitely many definable sets are also definable, that the empty set Ø and all the finite and cofinite subsets of a definable set are definable. Also the image and the preimage of a definable function are definable sets. Given a set of sets { A, B, C, . . . }, there is the minimal universe U containing these sets; we shall say that U is generated by the sets A, B, C, . . . . If we have an algebraic structure, say, a group G, we can generate by G the universe U(G) in the following way: take the graph M ⊂ G × G × G of the multiplication map G × G −→ G, (g, h) 7→ gh and the graph I ⊂ G × G of the inversion map G −→ G, g 7→ g −1 , and set U (G) to be equal to the universe generated by the underlying set of G, M and I. We also say that an algebraic structure is definable or (what is the same) interpretable in the universe U , if its underlying set, operations and predicates belong to U . If a structure N is interpretable in U(M), then we say that N is interpretable in M and M interprets N . For example, if N G is a normal definable subgroup of a group G, then the factor group G/N is interpretable in G. Given an algebraic structure M , one can show that every subset of M n which is defined by means of a first order formula (possibly with parameters in M ) belongs to U (M ), i.e. is definable in U(M ) in the sense of our axiomatic definition. For example, if G is a group and a ∈ G, then the centraliser CG (a) and the conjugacy class aG can be defined by the formulae xa = ax and ∃y(y −1 ay = x), correspondingly: CG (a) = aG
=
{ g ∈ G | ga = ag }, { g ∈ G | ∃y(y −1 ay = g) }.
Hence CG (a) and aG belong to U (G).
1.2
Rank
Let U be a universe. A function rk : U \ { Ø } −→ N is called a rank if the following axioms are satisfied for all A, B ∈ U .
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• Monotonicity of rank. rk A ≥ n + 1 if and only if there are infinitely many pairwise disjoint, non-empty, definable subsets of A each of rank at least n. • Definability of rank. If f is a definable function from A into B, then, for each integer n, the set { b ∈ B | rk f −1 (b) = n } is definable. • Additivity of rank. If f is a definable function from A onto B and if for all b ∈ B, rk f −1 (b) = n then rk A = rk B + n. • Boundness of finite preimages. For any definable function f from A into B there is an integer m such that for any b in B the preimage f −1 (b) is infinite whenever it contains at least m elements. We say that a universe U is ranked if there is a rank function with the above properties.
1.3
Groups and Fields of Finite Morley Rank
If M is an algebraic structure, we say that M is a ranked structure if U(M) is a ranked universe. In this case, the rank of the definable set M is called the rank of the structure M. A group G is said to be of finite Morley rank if it is ranked. Here are some examples of groups of finite Morley rank. Finite groups. They obviously have Morley rank 0. Algebraic groups. An algebraic group G over an algebraically closed field K is a group of finite Morley rank (and the rank of G itself coincides with the dimension of G over K). Abelian groups of bounded exponent have finite Morley rank. Divisible abelian groups. Torsion-free divisible abelian S∞groups are of finite Morley rank, as well as the quasicyclic groups Z(p∞ ) = n=1 Z/pn Z.
On the other hand, the infinite cyclic group Z, free groups Fn , and simple compact Lie groups, are not groups of finite Morley rank. From the model-theoretic point of view groups of finite Morley rank are a special case of ω-stable groups. A group G has finite Morley rank in the sense of our axiomatic definition if and only if G is ω-stable and the Morley ranks (ordinals defined by means of model theory) of all definable in G sets are finite. Notice, finally, that by Macintyre [43], an infinite field has finite Morley rank if and only if it is algebraically closed.
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Basic Properties of Groups of Finite Morley Rank
When working with ranked groups it is useful to think of them as algebraic groups from which the structure of an algebraic variety has been deleted, while retaining a notion of dimension for the constructible sets. Many group-theoretical constructions in a ranked group yield definable sets: intersections, products, as well as the centralisers and normalisers of definable subgroups are definable. Also definable are the centralisers of elements. This analogy will suffice for understanding of the subsequent text without turning to the book [21], which contains proofs of all the necessary auxiliary results. From now on G stands for a group of finite Morley rank.
2.1
Special Classes of Groups.
We list here some known results on abelian, nilpotent and soluble groups of finite Morley rank. • Abelian groups, Macintyre [42]. An abelian group G of finite Morley rank can be decomposed as G = D ⊕ B where D is a divisible group and B is a subgroup of bounded exponent. • Nilpotent groups, Nesin [47]. Let G be a nilpotent group of finite Morley rank. Then G is a central product D ∗ C where D is definable, connected, characteristic in G and divisible, C is definable and of bounded exponent. • Soluble groups, Zil’ber [52], Nesin [46]. The commutator subgroup G0 of a soluble connected group G of finite Morley rank is nilpotent.
2.2
Minimality Conditions
Groups of finite Morley rank satisfy many group-theoretical minimality properties, and most of them follow from the following general result. Fact 2.1 (Macintyre [42]) A group G of finite Morley rank satisfies the descending chain condition for chains of definable subgroups. In particular, a group G of finite Morley rank contains a unique minimal definable subgroup of finite index, which is called the connected component of G and is denoted by G◦ . A group G is called connected, if G = G◦ .
2.3
The Indecomposability Theorem
The following fact is a consequence of a deep result by Boris Zil’ber, so called Zil’ber’s Indecomposability Theorem [53]. It yields the definability of a wide range of subgroups.
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Fact 2.2 (Zil’ber [53]) If { Hi | i ∈ I } is a family of connected definable subgroups in a group G of finite Morley rank, then the subgroup hHi | i ∈ Ii generated by them is definable and connected.
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Tame Groups: Sylow Theory and Odd/Even Division
3.1
Involutions in Tame Groups
The definition of a tame group was given in the Introduction; the reader can find there a discussion of the following conjecture, central in our work. Tame Conjecture Every infinite simple tame K*-group is a simple algebraic group over an algebraically closed field. From now on we work mostly (though not exclusively) with tame K*-groups. The starting point of our theory is the following analogue of the Feit-Thompson theorem on solvability of groups of odd order [30]. Fact 3.1 [21, Theorem B.1] A connected tame group G without involutions is nilpotent. Therefore, in a infinite tame simple group we always have elements of even order. Moreover, if we introduce a Sylow 2-subgroup of a group G as a maximal 2-subgroup in G, then we have an even stronger result. Fact 3.2 [21, Theorem B.3] Let G be an infinite simple tame group. Then Sylow 2-subgroups in G are infinite. Not surprisingly, the methods of the theory of tame groups concentrate around the properties of involutions and Sylow 2-subgroup.
3.2
Sylow 2-Subgroups
Sylow 2-subgroups in groups of finite Morley rank are not necessary definable. For example, a Sylow 2-subgroup P of a simple algebraic group G over an algebraically closed field of characteristic 6= 2 is not definable, see discussion in [19]. It can be shown that P contains a Pr¨ ufer 2-subgroup T of finite index. (A Pr¨ ufer 2-group is a product of a finite number of copies of the quasicyclic group Z(2∞ ).) Moreover, P is not nilpotent though it is soluble and abelian-by-finite. The exponent of P is infinite and all elementary abelian subgroups of P are finite. On the contrary, in the case of characteristic 2 Sylow 2-subgroups in simple algebraic groups are precisely maximal unipotent subgroups. Therefore they
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are definable, nilpotent, connected, have bounded exponent and contain infinite elementary abelian 2-subgroups. Fact 3.3 [22] All Sylow 2-subgroups in a group G of finite Morley rank are conjugate. If S is one of them, then S has a subgroup S ◦ of finite index with the following property: S ◦ = B ∗ D is a central product of a definable connected nilpotent subgroup B of bounded exponent and of a divisible abelian group D. In particular, S is nilpotent-by-finite. We shall call B the unipotent part of S and D the maximal 2-torus of S. More generally, any divisible abelian 2-subgroup of G is called 2-torus. 2-tori in groups of finite Morley rank are Pr¨ ufer groups.
3.3
Groups of Odd and Even Type
This is the most important bifurcation point of our theory. Theorem 3.4 [3] Let S be a Sylow 2-subgroup of a simple tame K*-group G and S ◦ = T ∗ B, where T is the maximal 2-torus and B is the unipotent part of S. Then either T = 1 or B = 1. In the two cases of Theorem 3.4, we say that G is of even type if T = 1 and of odd type if B = 1. These two classes of groups will be studied by completely different methods.
4
General Structural Properties
This section is devoted to general background material running parallel to the theory of finite groups, including a considerable amount of reasonably standard material from [21]. We also make use of a good deal of more elementary material from [21] without explicit mention. While most concepts are all closely modeled on the finite case, because of issues of definability and connectedness, some care has to be taken with the ◦ notation. In particular, we write NG (H) for NG (H)◦ , and similarly for other operations on groups.
4.1
Soluble Radical
The soluble radical R(G) is defined as the join of the normal soluble subgroups of G, which in general is only locally soluble. Fact 4.1 (Belegradek [15]; [21, p. 112]) Let G be a group of finite Morley rank. Then the soluble radical of G is definable, and soluble. Fact 4.2 [2, Fact 2.43] Let G be a connected K-group of finite Morley rank. Then G/R(G) is a product of finitely many quasisimple algebraic groups.
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4.2
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Fitting Subgroup
The Fitting subgroup F (G) is defined as the join of the normal nilpotent subgroups of G, which in general is only locally nilpotent. Fact 4.3 (Belegradek [15], Nesin [48]; see also [21, p. 112]) Let G be a group of finite Morley rank. Then the Fitting subgroup of G is definable, and nilpotent.
4.3
Components
A group G will be called quasi-simple, if G = G0 and G/Z(G) is non-abelian simple. Note that the only proper normal subgroups of a quasi-simple group are the central ones. Notice that quasi-simple algebraic groups are traditionally called simple algebraic groups. The following important result shows that quasisimple K-groups are algebraic. Fact 4.4 (Altınel and Cherlin [6]) Let G be a group of finite Morley rank which is a perfect central extension of an algebraic group over an algebraically closed field. Then Z(G) is finite and G is an algebraic group. We note that as an abstract group it is possible for an algebraic group over an algebraically closed field to have perfect central extensions with infinite center. The point is that such groups cannot have finite Morley rank. By a component of G we mean a connected definable subnormal quasisimple subgroup; it is a slight variation from [21] where components were not assumed to be connected. Fact 4.5 (Belegradek [15]) In a group G of finite Morley rank, every quasisimple subnormal subgroup is definable and there are only finitely many of them. Their product L(G) is a definable normal subgroup in G. The subgroup L(G) is called the layer of G. We also denote E(G) = L(G)◦ . Notice that components of G are exactly nontrivial quasisimple normal subgroups of E(G). Every component of G is normal in G◦ .
4.4
Automorphisms
Fact 4.6 [21, Theorem 8.4, p. 124] Let G be a group of finite Morley rank, and H a definable normal subgroup which is isomorphic to a quasisimple algebraic group over an algebraically closed field. Then the subgroup of Aut H induced by the action of G is an extension of the group of inner automorphisms by graph automorphisms. In particular, if G is connected then G = H ∗ CG (H).
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Generalised Fitting Subgroup
We deviate from [21] in our definition of the generalised Fitting subgroup F ∗ (G), which we take to be F ◦ (G) ∗ E(G) rather than F (G) ∗ L(G). This has the side effect that in soluble groups, our F ∗ (G) and F (G) may differ. In particular we cannot claim that in general CG (F ∗ (G)) ≤ F ∗ [21, p. 120]. However one can show that this holds with C ◦ in place of C, and we prefer to work with connected groups as far as possible. In fact we will find it useful to have the following more precise version of this, which however is only known for K-groups. Fact 4.7 [20, Fact 16] Let G be a K-group of finite Morley rank. Then ◦ CG (F ◦ (G)) = Z ◦ (F ◦ (G)) ∗ E(G).
5
Groups of Odd Type
In this section G is a simple tame K*-group of odd type.
5.1
The Core
Let H be a group of finite Morley rank. We denote by O(H) the maximal normal definable connected subgroup of H without involutions; it is called sometimes the core of H. Fact 5.1 If H is a tame K-group, then the subgroup O(H) is nilpotent. If, furthermore, H is of odd type and H = (H/O(H))◦ , then H = F ∗ (H) and F (H) is an abelian group. Moreover, components of F ∗ (H) are simple algebraic groups over algebraically closed fields of characteristic not 2.
5.2
2-Rank and 2-Generated Core
If U is a finite abelian 2-group, m(U ) stands for the minimal number of generators for U . If G is an arbitrary group, its 2-rank m(G) is defined as the maximum of m(U ) for all finite abelian 2-subgroups U ≤ G. It can be shown that for a group G of finite Morley rank and odd type m(G) is finite. Let S be a Sylow 2-subgroup in a group G of finite Morley rank and odd type. The normal 2-rank n(G) = n(S) is set to be equal to the maximum of 2-ranks m(E) of normal abelian subgroups E S. The 2-generated core ΓS,2 (G) is the definable closure of the group generated by all normalisers NG (U ) for all subgroups U ≤ S with m(U ) ≥ 2. (Here the definable closure of a set X ⊂ G is the intersection of all definable subgroups containing X.) Finite simple groups with a proper 2-generated core ΓS,2 (G) < G are known by Aschbacher [10].
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B-Conjecture
If t is an involution we abbreviate Ct = CG (t). We shall say that G satisfies the B-conjecture, if Ct◦ = F ∗ (Ct ) for any involution t ∈ G. This notion generalises to the context of finite Morley rank the B-conjecture by J. Thompson from finite group theory. It easily follows from the structural properties of algebraic groups that simple algebraic groups over algebraically closed fields of characteristic not 2 satisfy the B-conjecture.
5.4
Classical Involutions
Let t be an involution in G. A component A E(Ct ) is called intrinsic if t ∈ Z(A). Following Aschbacher [12], we call an involution classical if its centraliser contains an intrinsic component isomorphic to SL2 (K) for an algebraically closed field K.
5.5
Trichotomy of the Odd Case
Theorem 5.2 [19] Let G be a simple tame K*-group of odd type. Then one of the following statements is true. (i) n(G) ≤ 2. (ii) G has a proper 2-generated core. (iii) G satisfies the B-conjecture and contains a classical involution. Recall that finite simple groups satisfying clauses (i), (ii), (iii) of the above theorem were completely classified by Gorenstein and Harada [35], Aschbacher [10] and Aschbacher [12], correspondingly.
5.6
Groups of Small Normal Rank
We would like to emphasise the challenge posed by groups of small normal rank (part (i) of Theorem 5.2). Their classification will be a substitute in our theory for the theorem by Gorenstein and Harada [35] on finite groups of sectional 2rank ≤ 4. The expected list of simple tame K*-groups G with n(G) ≤ 2 consists of the groups PSL2 (K), PSL3 (K), PSp4 (K), G2 (K) over algebraically closed fields K of characteristic not 2. The exploration of this area has just started; the following is one of the few available results. Theorem 5.3 (Altseimer [7]) Let C1 and C2 be the nonisomorphic centralisers in PSp4 (K) of two involutions where K is an algebraically closed field of characteristic 6= 2. Let G be a tame K ∗ -group such that G contains 2 conjugacy classes of involutions iG and j G . Assume that CG (i)/O(CG (i)) ' C1 and CG (j)/O(CG (j)) ' C2 . Then G ' PSp4 (K).
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Notice that C1◦ ' SL2 (K) ∗ SL2 (K), C2c irc ' PSL2 (K) × K ∗ and C1 /C1◦ ' C2 /C2◦ ' Z2 ; in particular, C1 and C2 are not connected. The proof of Theorem 5.3 requires some technical refinement of the concept of unipotent subgroup (Altseimer and Berkman [9]). Namely, a definable, connected, nilpotent subgroup U ≤ G is called quasiunipotent, if U does not contain a nontrivial p-torus, i.e an abelian divisible p-subgroup. The main tool for identification of the group G with PSp4 (K) is a theorem by DeBonis and Nesin [29] which classifies, in the finite Morley rank context, BN -pairs of Tits rank 2; the latter result is an analogue of Fong and Seitz [31]. We state it with a modification due to Altseimer. Theorem 5.4 (DeBonis and Nesin [29]) Let G be an infinite, simple tame BN pair of Tits rank 2. Assume further that the regular subgroup of a Borel subgroup B of G is of the form B = U T where U is a maximal quasiunipotent subgroup of G. Then G ' PSL3 (K), PSp4 (K) or G2 (K) for some algebraically closed field K.
5.7
Groups with a Proper 2-Generated Core
We hope that there are no simple tame K*-groups of odd type satisfying case (ii) of the Trichotomy Theorem. Moreover, there is a conjecture (and even a sketch of a proof) that, in the case of n(G) ≥ 3, the existence of a proper 2-generated core in G implies the existence of a strongly embedded subgroup (cf. [21]).
5.8
Groups with a Classical Involution
Suppose that z is a classical involution in a simple tame K*-group group G of odd type and K ' SL2 (F ) is an intrinsic component in CG (z). Let Ω stand for the collection of all conjugates of K in G. Form a graph D with vertex set Ω by joining a vertex J to L if [J, L] = 1. Theorem 5.5 (A. Berkman, work in progress) In this notation, assume that the graph D is connected. If, in addition, the group G satisfies the B-conjecture then G ' PSLn (F ) for n ≥ 5, PSOn (F ) for n ≥ 9, PSp2n (F ) for n ≥ 3, F4 (F ), E6 (F ), E7 (F ), or E8 (F ). If the graph D is disconnected but has an edge, then the expected result is that G ' PSOn (F ) for 5 ≤ n ≤ 8. If D has no edges, then it is expected that G ' PSL3 (F ) or G2 (F ).
6
Groups of Even Type: General Methods
The results of this section make a collection of tools for the classification of tame groups of even type. They are applied mostly to configurations which does not
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normally occur in simple algebraic groups and need to be excluded from consideration, thus allowing to pin down the structure of the group. The finite group theoretic analogues of our tools are known as results on conjugation and fusion, generation, signaliser functors, strong embedding, p-uniqueness, but, for lack of space, we cannot discuss this connection here. Experience to date suggests that, in the tame groups context, our tools are incomparably more powerful than their finite group theoretic counterparts; see, for example, our treatment of p-uniqueness subgroups in Theorem 6.18. A possible explanation of this phenomenon is that structural properties of simple algebraic groups over algebraically closed fields are much more uniform and simple than the corresponding properties of finite simple groups.
6.1
Conjugacy
We restate the Sylow theorem for groups of even type. We tend to work more with the connected components of the Sylow 2-subgroups, which we call Sylow◦ 2-subgroups, than with the Sylow 2-subgroups themselves. A unipotent group is a definable connected nilpotent group of finite exponent. A p-unipotent group is a unipotent p-group. Fact 6.1 [22], [21, Chapter 10] In a group G of even type the Sylow◦ 2-subgroups are the maximal 2-unipotent subgroups; they are definable and nilpotent, and are conjugate in G. A torus in G is a definable divisible abelian subgroup. A p-torus is a divisible abelian p-subgroup (these are almost never definable). The p-rank of a torus is the dimension over Fp of the annihilator of p. The following result provides the best possible analogue, in our context, of conjugacy of maximal tori in algebraic groups. Theorem 6.2 [20, Fact 23] Let G be a K ∗ -group of finite Morley rank of even type. Then any two maximal tori which normalise a nontrivial 2-subgroup are conjugate.
6.2
The Thompson Rank Formula
In finite group theory the Thompson Order Formula gives a useful computation of the order of a group having at least two conjugacy classes in terms of data that can be computed locally [34]. In the study of groups of finite Morley rank of even type, an analogous computation gives the rank, rather than the order, and seems even more useful than in the finite case. We will refer to this as the Thompson Rank Formula. In particular experience to date suggests that situations calling in the finite case for use of the Thompson Transfer Lemma can be handled in our case using the Thompson Rank Formula. Since there is no analogue of transfer in our context, this is extremely fortunate.
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There is a also a version of the Thompson Rank Formula for groups of odd type Altseimer [8]. Here we will restrict ourselves to the even type case. We will make use of the following general principle. If X ⊂ G then the definable closure d(X) of X is the intersection of all definable subgroups containing X; it is a definable subgroup, by Fact 2.1. Lemma 6.3 [4] Let G be a group of even type of finite Morley rank. (1) If i and j are nonconjugate involutions then there is a unique involution in d(hiji). (2) The function f (i, j) which associates to each pair (i, j) of nonconjugate involutions the unique involution of d(hiji) is definable. Theorem 6.4 [4] Let G be a group of finite Morley rank g and of even type and I the set of all involutions in G. Let C1 , C2 be two distinct conjugacy classes in G. Let θ : C1 × C2 −→ I be the map defined by the rule: θ(x, y) is a unique involution in d(hxyi). For i ∈ I let ρ(i) = rk θ−1 (i). Let cl = rk CG (x) for x ∈ Cl , l = 1, 2. If X1 , . . . , Xk is a definable partition of I into sets such that ρ is constant on each Xj , with value rj , then 2g = c1 + c2 + max (rj + rk Xj ). 1≤j≤k
In particular if G has finitely many conjugacy classes of involutions then we may take the Xj to be these classes, and writing cj for rk CG (x), where x ∈ Cj , we get: g = c1 + c2 + max (rj − cj ). 1≤j≤k
This inequality is very useful because it restricts severely the size of the group G in which these computations are being made. In some cases these restrictions yield rapid contradictions, and in other cases they will serve to “pin down” the structure of G. The Thompson Rank Formula plays the prominent role in the proofs of the Weakly Embedded Subgroup and Strongly Closed Abelian Subgroup Theorems (Theorems 6.7 and 6.13).
6.3
Subgroups of Groups of Even Type
The following two results are the summary of properties of K-groups of even type as used in the sequel. Fact 6.5 Let G be a group of finite Morley rank. Then O(G) commutes with every 2-unipotent subgroup U . Denote by O2 (H) the maximal normal connected definable 2-subgroups of a group H of finite Morley rank.
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Fact 6.6 Let H be a connected tame K-group of even type. Then F ◦ (H) = O(H) × O2 (H) and H/F ◦ (H) = T ∗ L1 ∗ · · · ∗ Lk is a central product of a torus T without involutions and a finite number of simple algebraic groups L1 , . . . , Lk over algebraically closed fields of characteristic 2.
6.4
Weak Embedding
A proper definable subgroup M of G is weakly embedded in G if M has an infinite Sylow 2-subgroup, while M ∩ M g has a finite Sylow 2-subgroup for any g ∈ / M . It can be shown that G contains a weakly embedded subgroup ◦ if G contains a proper definable subgroup H such that NG (U ) ≤ H for any unipotent 2-subgroup U ≤ H [4, Fact 26]. The following result is very useful and easily applied in practice. Theorem 6.7 [4, Theorem 1.3] If G is a connected simple tame K ∗ -group of even type which has a weakly embedded subgroup, then G ' SL2 (K) for some algebraically closed field K of characteristic 2. The proof of Theorem 6.7 is a (highly non-trivial!) reduction to the existence of a strongly embedded subgroup in G, cf. [2], [21, p. 185]. In their turn, tame K*-groups of even type with a strongly embedded subgroup were classified by Altınel [2]; his work generalises a result by Bender [16] from finite group theory. It is based on methods of DeBonis and Nesin [28]. At several points the proof of Theorem 6.7 involves very concrete computations. In general the theory of groups of finite Morley rank is less technical than finite group theory. But, it seems, certain key calculation are unavoidable in either case. In particular, the following configuration has to be studied at the level of the group multiplication table. The first insight concerning its use came from Mazurov [45]; technical details were modelled on Landrock and Solomon [41], though our proof uses some different ideas. Theorem 6.8 Let X = S λ T be a group of finite Morley rank, where S is a 2-unipotent group. Assume that S has a definable subgroup A such that A λ T ' K+ λ K ∗ for some algebraically closed field K of characteristic 2, with the multiplicative group acting naturally on the additive group. Let α be a definable involutory automorphism of X such that CX (α) = A λ T . Then S is isomorphic to one of the following groups. (i) If S is abelian then either S is homocyclic and A = { s ∈ S | s2 = 1 }, or S = E ⊕ E α , where E ' K+ . In the latter case, A = { xxα | x ∈ E }.
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(ii) If S is nonabelian then S is an algebraic group over K whose underlying set is K × K × K and the group multiplication is as follows: for a1 , b1 , c1 , a2 , b2 , c2 ∈ K, p p √ (a1 , b1 , c1 )(a2 , b2 , c2 ) = (a1 + a2 , b1 + b2 , c1 + c2 + a1 a2 + b1 b2 + b1 a2 ) where is either 0 or 1. In this case α acts by
(a, b, c)α = (a, a + b, a + b + c +
√ ab).
In particular, if S is nonabelian then S has exponent 4. We use in the proof of Theorem 6.8 another result of a hair-splitting kind, the paper by Davis and Nesin [27] on Suzuki 2-groups, as well as some technique from the paper. Their result is an analogue of Higman [37], one of the most important and unusual theorems on finite 2-groups. Here a Suzuki 2-group is a pair (S, T ) where S is a 2-unipotent group and T is an abelian group that acts on S by group automorphisms and which is transitive on the involutions of S. Davis and Nesin proved, in particular, (and we use this fact) that if, in the finite Morley rank context, T acts freely on the set of involutions in S then S is abelian. An interesting feature of our proof of Theorem 6.7 is that the detailed information about subgroups in G obtained with the help of Theorem 6.8 is fed into the Thompson Rank Formula (Theorem 6.4). This yields striking restrictions on the Morley rank and, consequently, structure of G.
6.5
Killing the Cores of 2-Local Subgroups
The following is a fairly direct consequence of Theorem 6.7. It concerns the normalisers of nontrivial definable 2-subgroups, which are called 2-local subgroups. Theorem 6.9 [4, Theorem 10.20] If G is a simple tame K ∗ -group of even type and H is a 2-local subgroup then O(H) = 1 and F ◦ (H) = O2 (H).
6.6
The Alperin-Goldschmidt Conjugacy Theorem
Some analogues of the Alperin-Goldschmidt conjugacy theorem for finite groups [1, 32] are given by Corredor [26]. It involves the notion of a tame◦ intersection, which is a variation on standard terminology with no relationship to tameness for ω-stable groups: Sylow◦ 2-subgroups P, Q of G are said to form a tame◦ ◦ intersection in G if NP◦ ([P ∩ Q]◦ ) and NQ ([P ∩ Q]◦ ) are Sylow◦ 2-subgroups of NG (P ∩ Q). Fact 6.10 (Corredor [26]) If G is a tame K ∗ -group of even type, and if A, B are conjugate connected subgroups of a Sylow◦ 2-subgroup P of G, then there is a finite chain of connected subgroups Di of P (for 1 ≤ i ≤ n) and elements ◦ xi ∈ NG (Di ) (for 1 ≤ i ≤ n), and xi ∈ NG (Di ) unless Di = P ; so that:
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(1) A ≤ D1 and Ax1 ···xn−1 = B; (2) Ax1 ···xi ≤ Di+1 for i ≤ n. (3) For i < n: • Di = (P ∩ Pi )◦ for some Sylow◦ p-subgroup of G such that P and Pi form a tame◦ intersection in G; ◦ (Di )/Di ' SL2 (K) for some algebraically closed • If Di 6= P then NG ◦ (Di ) ≤ Di . field K of characteristic 2, and CG
6.7
Pushing Up
An analogue of a theorem of Baumann [14] is a powerful tool for study of 2local subgroups. In particular, it combines nicely with the Alperin-Goldschmidt Theorem. This involves the notion of a continuously characteristic subgroup of a definable subgroup of G. If K ≤ H and K, H are both definable in G, then we say that K is continuously characteristic in H, relative to G, if K is normalised ◦ by NG (H). The proof is modelled on Stellmacher [51]. Though the proof has not been written yet in detail, but there is a strong evidence that Stellmacher’s arguments can be transferred to the tame groups context with minor changes. Theorem 6.11 (P. Watson, work in progress) Let G be a group of finite Morley rank and H a tame definable subgroup of even type with H/O2 (H) ' SL2 (K) for some algebraically closed field K of characteristic 2. Let S be a Sylow◦ 2-subgroup of H. Then either • there is a definable connected subgroup U0 ≤ U , U0 6= 1, which is normal in H and is continuously characteristic in S, relative to G, or • S is an abelian group of exponent 2. Corollary 6.12 Let G be a tame group of even type, G◦ a subgroup containing the normaliser of a Sylow◦ 2-subgroup S as well as the connected component of the normaliser of any continuously characteristic subgroup of S. Then any pair of connected subgroups of S which are conjugate in G are conjugate in G◦ . Proof. By Alperin-Goldschmidt, the conjugacy can be achieved in a series ◦ of steps involving elements of NG (S) or NG (U ) where H = NG (U ) is of the ◦ ◦ type described in Fact 6.10. Thus NG (U ) ≤ NG (U0 ) for some continuously characteristic subgroup U0 of S. By hypothesis all these elements belong to G◦ . 2 This may be expressed more succinctly in terms of the characteristic generated core, defined as follows. If T is a Sylow◦ 2-subgroup of G, then the
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characteristic generated core C(G; T ) relative to T is the subgroup of G generated by connected components of normalisers of continuously characteristic subgroups of T . Now we may rephrase Corollary 6.12: a group which contains C(G; T ) and NG (T ) will control fusion of connected subgroups of T . This simple observation will play an important role in the classification of groups with a proper characteristic core, Theorem 6.14 below.
6.8
Strongly Closed Abelian Subgroups
If A is a nontrivial connected subgroup and A ≤ B ≤ G then A is said to be strongly closed in B if for any connected subgroup C ≤ A and g ∈ G, C g ≤ B implies C g ≤ A. The following is an analogue of Goldschmidt [33]. Theorem 6.13 [5] Let G be a connected simple tame K ∗ -group of even type and S a Sylow 2-subgroup of G. If S ◦ contains a strongly closed abelian definable subgroup A then G ' SL2 (K) with K an algebraically closed field of characteristic 2. This important result finds many uses; in particular, is used in the proof of the Global C(G; T )-Theorem (Theorem 6.14) and in the proof of the existence of standard components in tame groups of even component type (Theorem 7.3).
6.9
Global C(G; T )-Theorem
The line of development that started with the Weakly Embedded Theorem culminates in the following analogue of a very difficult result by Aschbacher. Theorem 6.14 (P. Watson, work in progress) Let G be a tame K ∗ -group of even type, and assume that C(G; T ) 6= G for some Sylow◦ 2-subgroup T of G. Then G has a weakly embedded subgroup. The proof of the theorem is a combination of pushing up with construction of a strongly closed abelian subgroup in a Sylow◦ 2-subgroup of G (which is easy when M = C(G; T ) is a 2-local subgroup, because then Z ◦ (O2 (M )) is strongly closed in T by Corollary 6.12).
6.10
Generation
Extremely nice generation properties of K-groups make one of the reasons why proofs in our theory are much shorter than in the finite group theory. See, for example, how Theorem 6.16 is used in the proof of the p-Uniqueness Theorem (Theorem 6.18).
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Theorem 6.15 Let G be a connected K-group of finite Morley rank of even type, and p an odd prime. If G contains no nontrivial p-unipotent subgroup, and A is an elementary abelian p-group of rank 2 acting definably on G, then ◦ G is generated by the subgroups CG (x) for x ∈ A \ { 1 }. Proof. If G is simple this is found in Seitz [50], and for G soluble it is [20, Fact 33]. The general case is done by induction on the Morley rank of G. 2 From Theorem 6.15 we deduce the following result, which will be used to obtain a weakly embedded subgroup in certain cases. Theorem 6.16 Let G be a simple tame K ∗ -group of even type, p an odd prime, and H a proper definable connected subgroup of G containing a nontrivial unipotent 2-subgroup. If A is an elementary abelian p-group of rank 2 acting definably ◦ on H, then H is generated by the subgroups CG (x) for x ∈ A \ { 1 }.
6.11
p-Uniqueness Subgroups
Let G be a group of finite Morley rank, H a proper definable subgroup of G containing a nontrivial unipotent 2-subgroup, and p an odd prime. We say that H is a p-uniqueness subgroup of G with respect to a p-torus R of p-rank two, if ◦ H contains CG (x) for every p-element x of R. The following lemma allows us to refer to a p-uniqueness subgroup without explicit mention of the torus involved. It easily follows from conjugacy of maximal p-tori in K-groups. Lemma 6.17 If p is odd, and H is a p-uniqueness subgroup in a tame K ∗ group of even type, with respect to one p-torus of p-rank 2 in H, then the same applies with respect to any such torus. The proof of the following theorem is so simple, especially in comparison with its finite group theoretical prototype, that we give it here. Theorem 6.18 If H is a p-uniqueness subgroup of the tame K ∗ -group G of even type then, for some Sylow◦ 2-subgroup T of G, C(G; T ) ≤ H. As a corollary of Theorem 6.14, notice that if G has a p-uniqueness subgroup then G has a weakly embedded subgroup and thus known. Theorem 6.18 is crucial in the proof of the existence of standard components in groups of even component type (Theorem 7.3).
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Proof of Theorem 6.18. Let R be a p-torus of p-rank 2 in H. As H is a K-group of even type, R normalises some maximal unipotent 2-subgroup T of H, which by assumption is nontrivial. We claim that T is a Sylow◦ 2-subgroup of G. If not, then T is not maximal 2-unipotent in NG (T ), which is a K-group of even type containing R. Again, R will normalise a maximal unipotent 2subgroup U of NG (T ) containing T . Then U is generated by CU◦ (x) as x varies over elements of R of order p; by assumption these subgroups are contained in H; and thus U = T , a contradiction. Thus R normalises a Sylow◦ 2-subgroup contained in H, and hence normalises its continuously characteristic subgroups. By Theorem 6.11, the connected components of the normalisers of these characteristic subgroups lie inside the group generated by centralisers of elements of R of order p, and hence inside H, as claimed. 2
7
Standard Components
Recall that by a component of G we mean a connected subnormal quasisimple subgroup, or equivalently a quasisimple normal subgroup of E(G).
7.1
Components and L-Balance
The next result is referred to as the L-balance property: Theorem 7.1 [20, Facts 18 and 19] Let G be a K-group of finite Morley rank of even type, and U a 2-subgroup of G. Then E(CG (U )) ≤ E(G). If G or U is connected then E(CG (U )) E(G). The proof of this theorem can be reduced to the case of a simple algebraic group G of characteristic 2, where it becomes an expression of the fundamental fact that the centralisers of 2-subgroups of algebraic groups over algebraically closed fields of characteristic 2 have no components. This last result is a wellknown consequence of a result of Borel and Tits on the relation between unipotent subgroups and parabolic subgroups [18], as explained for example in Gorenstein, Lyons and Solomon [36, §3]. Since we wish to prove that our simple tame K*-group G is in fact a Kgroup, it is desirable to establish first that 2-local structure of G resembles that of K-groups. For simple K-groups G the assertion of Theorem 7.1 becomes E(CG (U )) = 1 for all non-trivial 2-subgroups U < G. Taken together with the triviality of the cores of 2-local subgroups (Theorem 6.9), this becomes F ∗ (N ◦ (U )) = O2 (N ◦ (U )) for every 2-subgroup 1 6= U < G.
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In the latter case, a group G will be called a group of characteristic 2 type (as in finite group theory) or generic type. If G fails to be a group of generic type, then, for some 2-subgroup U , E(CG (U )) 6= 1. If t is an involution in U and Ct = CG (t) then, by L-balance, E(CG (U )) E(Ct ) hence E(Ct ) 6= 1 and the centraliser Ct of an involution t has components. In that case the group G is said to be of component type. Thus we come to the main dichotomy of the even case: A simple tame K*-group G of even type is either of generic type, or of component type. Notice that, in view of the Borel-Tits Theorem [18], groups of generic type include all existing examples of simple tame groups of even type.
7.2
The Aschbacher Component Theorem
A definable subgroup L of G is a prestandard component for G if L lies in the centraliser of some involution, and is a component of any centraliser of an involution in which it lies. L is a standard component for G if it is a prestandard component for G, and does not commute with any of its conjugates in G. For groups of even component type we prove an analogue of the Aschbacher Component Theorem [11] which asserts the existence of a standard component. This still leaves us with the need to prove that there exists no simple tame K*group of even component type with a standard component in the centraliser of an involution. This is a formidable problem because there are obvious examples of standard components in non-simple groups. For example, if G is the group PSLn (K), n ≥ 3, extended by a group of order 2 generated by the inversetranspose automorphism t, then E(Ct ) = PSOn (K) is a standard component in Ct . Interestingly, the Aschbacher Component Theorem became almost redundant in our study of groups of odd type (Theorem 5.2, see [19] for details). This happened because, in the the theory of finite groups, the odd case was the natural setting for the component analysis. As it has already happened on many occasions, theorems which are ‘natural’ in the finite case become simple lemmas in the finite Morley rank context. Let G be a connected simple tame K ∗ -group of even type, L(G) the set of all components of the centraliser of any involution in G, and we assume that L(G) is nonempty. We begin the proof of the Aschbacher Component Theorem by showing that there is a prestandard component in L(G). Lemma 7.2 Chose a component A ∈ L(G) so that (a) Morley rank of A is maximal, and (b) A has, of all choices in (a), the maximal Morley rank of a Sylow 2subgroup of CG (A) (which is definable, as CG (A) is of even type). Then A is prestandard.
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Theorem 7.3 [20] A prestandard component is standard. In particular, if the centraliser of some involution of a connected simple tame K ∗ -group G of even type has a component, then in some centraliser of an involution of G there is a standard component. This is an analogue of a result of Aschbacher [11] for finite groups. It is intended to be applied in the classification of tame simple groups of even type, in much the same way that the finite version is applied in finite group theory. To be really useful, this result should be supplemented by the further result that the centraliser of a standard component has finite Sylow 2-subgroups, which is an analogue of a result of Aschbacher and Seitz [13]; however this has not been yet undertaken. In the proof of Theorem 7.3 we start from a prestandard component A which is not standard and construct a group D = Ax1 ∗ · · · ∗ Axk generated by components conjugate to A and commuting with each other. In this setting we work with the centralisers of elements of odd prime order in D. After some rather elaborate analysis we either construct a p-uniqueness subgroup in G (see Theorem 6.18), or prove that a Sylow◦ 2-subgroup of G contains a strongly closed abelian subgroup, which is impossible in view of Theorem 6.13.
8
Directions of Further Development
We reached the point where we divided tame simple K*-groups of even type into two cases, groups of generic type (they cover all the existing examples of simple tame groups of even type), and groups of component type (we expect that they do not exist). Their classification (or exclusion from consideration) are substantial problems analogous to one encountered in the finite case. But we believe that they should be somewhat more manageable since the inductive hypothesis involves connected algebraic groups over algebraically closed fields rather than finite simple groups. In the case of groups of component type we expect that the following subdivision will be useful. We have, by virtue of Lemma 7.2 and Theorem 7.3, that either (i) all components in L(G) are isomorphic to groups PSL2 over algebraically closed fields of characteristic 2, or (ii) G has a standard component A in the centraliser of some involution of rank (in the sense of the theory of algebraic groups) at least 2. Interestingly, case (i) has been already studied, under some extra assumptions, in the proof of Theorem 6.13. We believe that it will be relatively easy to prove that, under the assumptions of (i), Sylow◦ 2-subgroups of G are abelian, which will contradict Theorem 6.13 and prove that no group satisfies (i). This would be an analogue of works by Mason [44] and Griess, Solomon and Seitz [40].
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In case (ii) we can find in A an elementary abelian p-group V of order p2 for an odd prime p and work with the centralisers of p-elements in G. There is a hope that the systematic use of the Generation Property (Theorem 6.16) and other general results described in Section 6 will allow to construct in G a p-uniqueness subgroup, thus proving that G does not exist. One still missing tool is the analogue of the Aschbacher-Seitz [13]: If A is a standard component in a simple tame K*-group G of even type then a Sylow 2-subgroup in CG (A) is finite. The experience of finite group theory suggests that our collection of tools should be more or less adequate to carry out the analysis. This will leave us with the generic case. Again, we believe that we developed a powerful enough machinery for the analysis of the structure of 2-local subgroups and the centralisers of p-elements in a tame simple K*-group G of even type. Our aim is in proving that either G has a BN -pair and thus known by [21, p. 282] and [29], or G has a system of Phan-Curtis-Tits generators [36]. No work has been undertaken so far in this direction.
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[26] L.-J. Corredor, A generalization of Alperin-Goldschmidt’s theorem to some ω-stable groups, J. Symbolic Logic, to appear. [27] M. Davis and A. Nesin. Suzuki 2-groups of finite Morley rank (or over quadratically closed fields), submitted to Comm. Algebra. [28] M. J. DeBonis and A. Nesin. On CN groups of finite Morley rank. J. London Math. Soc., (2) 50 (1994) 532–546. [29] M. J. DeBonis and A. Nesin. On BN -pairs of Tits rank 2 and of finite Morley rank, submitted. [30] W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. [31] P. Fong and G. M. Seitz, Groups with a (B, N )-pair of rank 2. I,II, Invent. Math. 21 (1973), 1–57; 24 (1974), 191–239. [32] D. M. Goldschmidt, A conjugation family for finite groups, J. Algebra 16 (1970), 138–142. [33] D. M. Goldschmidt, 2-fusion in finite groups, Ann. Math. 99 (1974), 70–117. [34] D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, 1982. [35] D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most 4 elements, Mem. Amer. Math. Soc. 147 (1974). [36] D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, Part I, Chapter A: Groups of Lie type, Mathematical Surveys and Monographs, AMS, Providence, to appear ca. 1998. [37] G. Higman, Suzuki 2-groups, Illinois J. Math. 7 (1963), 79–96. [38] W. Hodges, Model Theory, Encyclopaedia of Mathematics and its Applications, vol. 42, Cambridge University Press, 1993. [39] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, Berlin-New York, 2nd edition, 1981. [40] R. Griess, D. Solomon and G. Seitz, Bender groups as standard components, Trans. Amer. Math. Soc. 238 (1978), 179–211. [41] P. Landrock and R. Solomon, A characterization of the Sylow 2-subgroups of P SU (3, 2n ) and P SL(3, 2n ), Technical Report 13, Aarhus Universitet, Matematisk Institut, January 1975. [42] A. Macintyre, On ω1 -categorical theories of abelian groups, Fund. Math. 70 (1971), 253–270.
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