Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000

SIGNALIZERS AND BALANCE IN GROUPS OF FINITE MORLEY RANK

JEFFREY BURDGES

Abstract

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. The most successful approach to this conjecture has been Borovik’s program analyzing a minimal counterexample, or simple K^{∗}-group. We show that a simple K^{∗}-group of finite Morley rank and odd type is either aalgebraic of else has Prufer rank at most two. This result signifies a switch from the general methods used to handle a large groups, to the specilized methods which must be used to identify PSL_{2}, PSL_{3}, PSp_{4}, and G_{2}.

The Algebraicity Conjecture for simple groups of finite Morley rank, also known as the Cherlin-Zilber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last 15 years, the main line of attack on this problem has been the Borovik program of transferring methods from finite group theory. This program has led to considerable progress; however, the conjecture itself remains decidedly open. We divide groups of finite Morley rank into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2-subgroups. For even and mixed type the Algebraicity Conjecture has been proven, and connected degenerate type groups are now known to have trivial Sylow 2-subgroups [13]. The present paper is part of the program to analyze a minimal counterexample to the conjecture in odd type, where the Sylow 2-subgroup is divisible-abelian-by-finite. It is the final paper in a sequence proving that such a minimal counterexample, or simple nonalgebraic K^{∗}-group, has Prüfer 2-rank at most two.

High Prüfer Rank Theorem 1. A simple K^{∗}-group of finite Morley rank with Prüfer 2-rank at least three is algebraic.

This will be a consequence of the following so-called trichotomy, which is proved in the present paper. Here the traditional term “trichotomy” refers to the fact that there is also the Prüfer 2-rank ≤ 2 case, which is largely unexplored at present.

Generic Trichotomy Theorem 2. Let G be a simple K^{∗}-group of finite Morley rank and odd type with Prüfer 2-rank ≥ 3. Then either

1. G has a proper 2-generated core, or

2. G is an algebraic group over an algebraically closed field of characteristic not 2.

2000 Mathematics Subject Classification 03C60 (primary), 20G99 (secondary).

Burdges was supported by NSF grant DMS-0100794, and Deutsche Forschungsgemeinschaft grant Te 242/3-1.

The High Prüfer Rank Theorem then follows by applying the next two results.

Strong Embedding Theorem 3 [14]. Let G be a simple K^{∗}-group of finite Morley rank and odd type, with normal 2-rank ≥ 3 and Prüfer 2-rank ≥ 2. Suppose that G has a proper 2-generated core M. Then G is a minimal connected simple group, and M is strongly embedded.

Minimal Simple Theorem 4 [19]. Let G be a minimal connected simple group of finite Morley rank and of odd type. Suppose that G contains a proper definable strongly embedded subgroup M. Then G has Prüfer 2-rank one.

It may seem odd that the first of these results is appearing last. In fact, an earlier version of the trichotomy theorem began this sequence of developments. Namely, Borovik first proved the trichotomy theorem under a tameness assumption in [10], and the present author had explored eliminating tameness in [17]. In [10], Borovik produces the proper 2-generated core with a tame nilpotent signalizer functor theorem [15, Thm. B.30] (see also [17, Thm. 6.2]), an approach mirrored in the present paper. In [17], we show that the “most unipotent part” of a solvable signalizer functor is a nilpotent signalizer functor. This was believed to quickly eliminate tameness from [10]. However, more careful investigations revealed that obtaining a signalizer functor remained problematic.

In §2.1 of the present paper, we resolve this difficulty by constructing signalizer functors of a “sufficiently unipotent” reduced rank. The most serious obstacle is explained in Example 2.1. Our approach forces subsequent analysis to restrict itself to components of the centralizers of involutions which involve sufficiently large fields, a worrying but ultimately harmless restriction. Indeed, all complexities introduced by this approach are dispensed with in §2.1. This seems to be a different approach from that used by finite group theorists, who work with so-called weakly balanced signalizer functors [24, §29]; a similar method might work here as well. We call our approach partial balance.

The first section of this article covers necessary background material, including the definitions of a signalizer functor and the 2-generated core.

The second section contains the delicate definitions of partial balance, and of the associated family Ẽ_{X} of components from the centralizers of involutions. This section also contains a version of Asar’s theorem (Theorem 2.12) which states that Ẽ_{X} ≠ ∅, as well as a criterion for ⟨Ẽ_{X}⟩ = G (Theorem 2.18). Borovik’s earlier unpublished work on the analysis of Lie rank two components [11] has heavily influenced this final result, although partial balance has given these results a more technical flavor.

The third section provides a suitable version of Berkman and Borovik’s Generic Identification Theorem [8]. It is the role of section two to verify the two hypotheses of this argument: reductivity for, and generation by, the centralizers of involutions. Our partial balance approach provides only a weak form of the reductivity hypothesis, which necessitates some alterations in the proof of the Generic Identification Theorem. All such critical changes are confined to §3.1, but there are important modifications throughout §3. The reader unfamiliar with the Generic Identification Theorem should consider exploring §3 before §1 or §2.

This is by no means the end of the story. The Prüfer 2-rank ≥ 3 hypothesis used

signalizers and balance 3

here is weaker than the normal 2-rank ≥ 3 hypothesis originally used by Borovik [10]. As part of the ongoing program in odd type, [12] will show that Borovik’s original trichotomy holds, without the tameness hypothesis. We view [12] as a bridge between the “generic case” which is treated here, and the “quasi-thin” case (the identification of PSp_{4}, G_{2}, and PSL_{3}).

1. Preliminaries

This first section recalls various definitions and facts which are used throughout §2, and less pervasively in §3.

1.1. K-groups

We proceed, in this paper, by analyzing a so-called simple K^{∗}-group of finite Morley rank. A K^{∗}-group is a group whose proper definable simple sections are all algebraic. Similarly, a K-group is a group whose definable simple sections are all algebraic. So the proper subgroups of our K^{∗}-group are clearly K-groups. One major K-group fact used throughout this article is the following generation principle.

Fact 1.1 [10, Thm. 5.14]; see also [16, Thm. 3.25]. Let G be a connected Kgroup of finite Morley rank and odd type. Let V be a four-subgroup acting definably on G. Then

G = ⟨C_{G}^{∘}(v) | v ∈ V^{#}⟩

Another major K-group fact worth recalling at the outset is the following “reductivity” criterion, which requires two definitions.

Definition 1.2. A quasisimple subnormal subgroup of a group G is called a component of G (see [15, p. 118 (2)]). We define E(G) to be the connected part of the product of components of G, or equivalently the product of the components of G^{∘} (see [15, Lemma 7.10iv]). Such components are normal in G^{∘} by [15, Lemma 7.1iii], and indeed E(G) ◁ G.

Definition 1.3. The odd part O(G) of a group G of finite Morley rank is the maximal definable connected normal 2^{⊥}-subgroup of G.

Clearly O(G) is solvable if G is a K-group.

Fact 1.4 [10, Thm. 5.12]. Let H be a connected K-group of finite Morley rank and odd type with O(H) = 1. Then H = F^{∘}(H) ∗ E(H) is isomorphic to a central product of quasisimple algebraic groups over algebraically closed fields of characteristic not 2 and of a definable normal divisible abelian group F^{∘}(H).

This fact motivates signalizer functor theory, whose goal is to show that O(H) = 1, or something similar, when H is the centralizer of an involution. The version we aim at here is Corollary 2.11 below.

4 jeffrey burdges

1.2. Algebraic groups

A key tool in our program is the fact that a group of finite Morley rank acting faithfully as a group of automorphisms of an algebraic group must itself be algebraic.

Definition 1.5. Given an algebraic group G, a maximal torus T of G, and a Borel subgroup B of G which contains T, we define the group Γ of graph automorphisms associated to T and B, to be the group of algebraic automorphisms of G which normalize both T and B.

Fact 1.6 [15, Thm. 8.4]. Let G ⋊ H be a group of finite Morley rank where G and H are definable, G an infinite quasi-simple algebraic group over an algebraically closed field, and C_{H}(G) is trivial. Then, viewing H as a subgroup of Aut(G), we have H ≤ Inn(G)Γ, where Inn(G) is the group of inner automorphisms of G and Γ is the group of graph automorphisms of G, relative to a fixed choice of Borel subgroup B and maximal torus T contained in B.

An algebraic group is said to be reductive if it has no unipotent radical. Such a group is a central product of semisimple algebraic groups and algebraic tori. The centralizer of an involution in a reductive algebraic group over a field of characteristic ≠ 2 is itself reductive.

Fact 1.7 [35, Theorem 8.1]. Let G be a quasisimple algebraic group over an algebraically closed field. Let ϕ be an algebraic automorphism of G whose order is finite and relatively prime to the characteristic of the field. Then C_{G}^{∘}(ϕ) is nontrivial and reductive.

Proof. We know both that C_{H}(ϕ) ≤ C_{H}(ϕ mod Q) as well as [ϕ, C_{H}(ϕ mod Q)] ≤ Q. There is a homomorphism C_{H}(ϕ mod Q) → Q given by x ↦ [ϕ, x]. As Z(G) is finite, t centralizes C_{H}^{∘}(ϕ mod Q), as desired. So C_{G/Z(G)}^{∘}(ϕ) = C_{G}^{∘}(T)/Z(G) We may therefore assume that G is its own universal central extension.

Since ϕ is algebraic and has finite order, G ⋊ ⟨ϕ⟩ is an algebraic group which contains ϕ as an inner automorphism. Since the order of ϕ is finite and relatively prime to the characteristic, ϕ is a semisimple automorphism of G. So the result follows from Theorem 8.1 of [35].

Proof. We may assume that G is its own universal central extension since C_{G/Z(G)}^{∘}(ϕ) = C_{G}^{∘}(ϕ)/Z(G). Since ϕ is algebraic and has finite order, G ⋊ ⟨ϕ⟩ is an algebraic group which contains ϕ as an inner automorphism. Since the order of ϕ is finite and relatively prime to the characteristic, ϕ is a semisimple automorphism of G. So the result follows from Theorem 8.1 of [35].

More specialized facts about algebraic groups will appear in §3.

1.3. Unipotent groups

While there is no intrinsic definition of unipotence in a group of finite Morley rank, there are various analogs of the “unipotent radical”: the Fitting subgroup, the p-unipotent operators U_{p}, for p prime, and their “characteristic zero” analogs U_{0,r} from [17, 16]. We recall their definitions below.

Definition 1.8. The Fitting subgroup F (G) of a group G of finite Morley rank is the subgroup generated by all its nilpotent normal subgroups.

The Fitting subgroup is itself nilpotent and definable [5, 30, 15, Thm. 7.3], and serves as a rough notion of unipotence in some contexts. However, the Fitting subgroup of a solvable group H may not be contained in the Fitting subgroup of a solvable group containing H.

Definition 1.9. A connected definable p-subgroup of bounded exponent in a group H of finite Morley rank is said to be p-unipotent. We write U_{p}(H) for the subgroup generated by all p-unipotent subgroups of H.

Clearly U_{p}(H) need not be solvable when H is a non-solvable algebraic group in characteristic p; however, a p-unipotent K-group is olvable, and hence nilpotent by the following.

Fact 1.10 [22, Cor. 2.16]; [1, Fact 2.36]. Let H be a connected solvable group of finite Morley rank. Then U_{p}(H) ≤ F^{∘}(H) is itself p-unipotent, and hence nilpotent.

Thus the p-unipotent radical U_{p} will automatically behave well, inside a solvable group. Its only weakness is that it may be trivial.

Fact 1.11 [15, Thm. 9.29 & §6.4]. Let G be a connected solvable group of finite Morley rank. Then a Sylow p-subgroup P of G is connected, and P = U_{p}(G) ∗ T for a divisible abelian p-group T.

The present paper relies on the theory of “characteristic zero” unipotence introduced in [17]. We now turn our attention to this definition, as well as some facts from [17, 18, 16].

Definition 1.12. We say that a connected abelian group of finite Morley rank is indecomposable if it has a unique maximal proper definable connected subgroup, denoted J(A) (see [17, Lemma 2.4]). We define the reduced rank r̄(A) of a definable indecomposable abelian group A to be the Morley rank of the quotient A/J(A), i.e. r̄(A) = rk(A/J(A)). For a group G of finite Morley rank, and any integer r, we define

U_{0,r}(G) = ⟨ A ≤ G | A is a definable indecomposable group, r̄(A) = r, and A/J(A) is torsion- free ⟩ .

We say that G is a U_{0,r}-group (alternatively (0, r)-unipotent group) if U_{0,r}(G) = G. We also set r̄_{0}(G) = max{r | U_{0,r}(G) ≠ 1}.

We view the reduced rank parameter r as a scale of unipotence, with larger values being more unipotent. By the following fact, analogous to Fact 1.10, the “most unipotent” groups, in this scale, are nilpotent.

Fact 1.13 [16, Thm. 2.21]; [17, Thm. 2.16]. Let H be a connected solvable group of finite Morley rank. Then U_{0,r̄_{0}(H)}(H) ≤ F (H).

6 jeffrey burdges

Fact 1.14 [18, Cor. 4.6]. Let G = HT be a group of finite Morley rank, with H and T definable and nilpotent, and H ◁ G. Suppose that T is a U_{0,r}-group for some r ≥ r̄_{0}(H). Then G is nilpotent.

A good torus is a divisible abelian group of finite Morley rank whose definable connected subgroups are the definable hulls of their torsion. We arrive at a good torus when all our various notions of unipotence are trivial.

Fact 1.15 [16, Thm. 2.19]; [17, Thm. 2.15]. Let H be a connected solvable group of finite Morley rank. Suppose U_{p}(H) = 1 for all p prime, and U_{0,r̄_{0}(H)}(H) = 1. Then H is a good torus.

In a similar vein, the notion of (0, r)-unipotence provides a useful decomposition of a nilpotent group.

Fact 1.16 [18, Cor. 3.6]; [16, Thm. 2.31]. Let G be a connected nilpotent group of finite Morley rank. Then G = D∗B is a central product of definable characteristic subgroups D, B ≤ G where D is divisible and B is connected of bounded exponent. Let T be the torsion part of D. Then we have decompositions of D and B as follows.

D = d(T) ∗ U_{0,1}(G) ∗ U_{0,2}(G) ∗ ⋯
B = U_{2}(G) × U_{3}(G) × U_{5}(G) × ⋯

The next fact tells us when q-unipotence is preserved by taking centralizers, a fact used to produce a signalizer functor in Lemma 2.5 below.

Fact 1.17 [17, Fact 3.4]; [2]. Let G be a connected solvable p^{⊥}-group of finite Morley rank, and let P be a finite p-group of definable automorphisms of G. Then C_{G}(P) is connected.

There is also a “characteristic zero” analog of the foregoing.

Fact 1.18 [17, Lemma 3.6]. Let G be a nilpotent (0, r)-unipotent p^{⊥}-group of finite Morley rank, and let P be finite p-group of definable automorphisms of G. Then C_{G}(P) is (0, r)-unipotent.

In a similar vein, commutator subgroup of connected or (0, r)-unipotent groups tend to retain these properties.

Fact 1.19 [15, Cor. 5.29]. Let H be a definable connected subgroup of a group G of finite Morley rank and let X ⊂ G be any subset of G. Then the group [H, X] is definable and connected.

Fact 1.20 [18, Cor. 3.6]. Let G be a solvable group of finite Morley rank, let S ⊂ G be any subset, and let H be a nilpotent U_{0,r}-group which is normal in G. Then [H, S] ≤ H is a U_{0,r}-group.

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1.4. 2-Local structure

As the goal of our project is to constrain the 2-local structure, we need a few parameters to measure the complexity of a Sylow 2-subgroup. We define the 2-rank m(G) of a group G to be the maximum rank of its elementary abelian 2-subgroups. The Prüfer 2-rank pr(G) is the maximum k such that there is a Prüfer 2-subgroup ℤ(2^{∞})^{k} inside G, and the normal 2-rank n(G) is the maximum 2-rank of a normal elementary abelian 2-subgroup of G. In an odd type group of finite Morley rank, these various ranks are all finite, and we have

m(G) ≥ n(G) ≥ pr(G).

These notions are well-defined because the Sylow 2-subgroups of a group of finite Morley rank are conjugate [9, 15, Thm. 10.11].

We use E_{k}(H) to denote the set of elementary abelian 2-subgroups U ≤ H with m(U) ≥ k. We give E_{2}(H) a graph structure by placing an edge between U, V ∈ E_{2}(H) whenever [U, V] = 1. We say H is 2-connected if the graph E_{2}(H) is connected, and we refer to the components of E_{2}(H) as 2-connected components otherwise.

Fact 1.21 compare [4, 46.2]. Let S be a locally finite 2-group. Then

1. If m(S) > 2 then the graph E_{2}(S) has a unique nonsingleton 2-connected component given by

E_{2}^{0}(S) := {X ∈ E_{2}(S) : m(C_{S}(X)) > 2}, and

E_{2}^{0}(S) contains any X ∈ E_{2}(S) with X ◁ S.

2. If n(S) > 2 then S is 2-connected.

Proof. Since S is locally finite, this reduces to the finite case, found in [4, 46.2].

1.5. Proper 2-generated core

Definition 1.22. Consider a group G of finite Morley rank and a 2-subgroup S of G with m(S) ≥ 3. We define the 2-generated core Γ_{S,2}(G) of G (associated to S) to be the definable hull of the group generated by all normalizers of groups in E_{2}(S):

Γ_{S,2}(G) = d(⟨N_{G}(U) : U ∈ E_{2}(S)⟩)

We also define the weak 2-generated core Γ_{S,2}^{0}(G) of G (associated to S) to be the definable hull of all normalizers of groups in the nonsingleton 2-connected component E_{2}^{0}(S).

Γ_{S,2}^{0}(G) = d(⟨N_{G}(U) : U ∈ E_{2}(S), m(C_{S}(U)) > 2⟩).

We say that G has a proper 2-generated core, or a proper weak 2-generated core, when, for a Sylow 2-subgroup S, Γ_{S,2}(G) < G or Γ_{S,2}^{0}(G) < G, respectively.

Both notions of 2-generated core are well-defined, by the conjugacy of Sylow 2-subgroups. By Fact 1.21-2, the 2-generated core and the weak 2-generated core coincide when n(G) ≥ 3, as is the case for much of the rest of this paper. When they differ, the weak 2-generated core is the more useful notion.

For an elementary abelian 2-group V acting definably on G, we define Γ_{V} (G) to be the group generated by the connected centralizers of involutions in V.

Γ_{V} (H) = ⟨C_{H}^{∘}(v) : v ∈ V^{#}⟩.

Proposition 1.23. Let G be a simple K^{∗}-group of finite Morley rank and odd type, with m(G) ≥ 3, and let S be a Sylow 2-subgroup of G. Suppose that Γ_{E}(G) < G for some E ∈ E_{2}^{0}(S). Then G has a proper weak 2-generated core.

This depends on a lemma.

Lemma 1.24. Let G be a simple K^{∗}-group of finite Morley rank and odd type. Then Γ_{U}(G) = Γ_{V} (G) for any U, V in the same connected component of the graph E_{2}(G).

Proof. It is enough to prove the result for U, V with [U, V] = 1. For any v ∈ V^{#}, simplicity implies that C_{G}^{∘}(v) is a proper subgroup of G, and hence a K-group. Since U normalizes C_{G}^{∘}(v), C_{G}^{∘}(v) = Γ_{U}(C_{G}^{∘}(v)) by Fact 1.1. So Γ_{V} (G) ≤ Γ_{U}(G), and the result follows by symmetry.

Proof of Proposition 1.23. We may assume E ≤ S by conjugacy of Sylow 2subgroups. Since involutions of G have infinite centralizers by [15, Ex. 13 & 15 p. 79], the result will follow from the following claim, and simplicity.

Γ_{S,2}^{0}(G) ≤ N_{G}(Γ_{E}(G)) for any E ∈ E_{2}^{0}(S)

By Lemma 1.24 and Fact 1.21-1, Γ_{E}(G) = Γ_{U}(G) for any U ∈ E_{2}^{0}(S). For any U ∈ E_{2}^{0}(S),

N_{G}(U) ≤ N_{G}(Γ_{U}(G)) = N_{G}(Γ_{E}(G)).

Thus Γ_{S,2}^{0}(G) ≤ N_{G}(Γ_{E}(G)), as desired.

We will encounter a variation of the preceding in the next section (see Lemma 2.15 and Proposition 2.17).

The following black hole principle for proper 2-generated cores reverses the roles of the subgroups Γ_{S,2}^{0}(G) and Γ_{E}(G) in Proposition 1.23.

Lemma 1.25. Let G be an infinite simple K^{∗}-group of finite Morley rank and odd type, and let S be a 2-subgroup of G satisfying m(S) ≥ 3. Then C_{G}^{∘}(x) ≤ Γ_{S,2}^{0}(G) for every x ∈ I(S) with m(C_{S}(x)) > 2.

Proof. There is an E ∈ E_{2}^{0}(S) with x ∈ E and m(E) ≥ 3 by Fact 1.21-1. So there is an E_{1} ∈ E_{2}^{0}(S) with E_{1} ≤ E and E_{1} ∩ ⟨x⟩ = 1. For any y ∈ E_{1}^{#}, we have C_{C_{G}(x)}(y) ≤ C_{G}(y, x) and ⟨y, x⟩ ∈ E_{2}^{0}(S). By simplicity, Fact 1.1 yields

C_{G}^{∘}(x) = Γ_{E_{1}}(C_{G}^{∘}(x)) ≤ Γ_{S,2}^{0}(G)

In particular, given a simple K^{∗}-group G, Lemma 1.25 says

Γ_{E}(G) ≤ Γ_{E,2}(G) for any E ∈ E_{3}(G).

Now Proposition 1.23 and Lemma 1.25 yield the following.

Proposition 1.26. Let G be a simple K^{∗}-group of finite Morley rank and odd type, with m(G) ≥ 3, and let S be a Sylow 2-subgroup of G. If Γ_{E,2}(G) < G for some E ∈ E_{3}(G), then G has a proper weak 2-generated core, i.e. Γ_{S,2}^{0}(G) < G.

1.6. Signalizer functors

Signalizer functors are used in both the finite case and in the finite Morley rank case to produce a dichotomy between a proper 2-generated core, and a reductivity condition for centralizers of involutions.

Definition 1.27. Consider a group G of finite Morley rank, and an elementary abelian 2-subgroup E ∈ E_{3}(G). An E-signalizer functor on G is a family {θ(s)}_{s∈E^{#}} of definable E-invariant 2^{⊥}-subgroups of G satisfying: a. θ(s) ◁ C_{G}(s) for each s ∈ E^{#}. b. θ(s) ∩ C_{G}(t) ≤ θ(t) for any s, t ∈ E^{#}.

We observe that the second condition is equivalent to the “balance” condition

θ(s) ∩ C_{G}(t) = θ(t) ∩ C_{G}(s) for any s, t ∈ E^{#}.

In practice, we will only be interested in signalizer functors satisfying the following stronger invariance condition, which is used to produce a proper 2-generated core.

θ(s)^{g} = θ(s^{g}) for all s ∈ E^{#} and all g ∈ G for which s^{g} ∈ E.  (†)

As one would expect, we say θ is a connected or nilpotent signalizer functor if the groups θ(s) are connected or nilpotent, respectively, for all s ∈ E^{#}.

We now show that signalizer functors yield a proper weak 2-generated core.

Theorem 1.28. Let G be a simple K^{∗}-group of finite Morley rank and odd type, and let S be a Sylow 2-subgroup of G. Suppose that, for some E ∈ E_{3}(S), G admits a nontrivial connected nilpotent E-signalizer functor θ satisfying (†) Then G has a proper weak 2-generated core.

The key fact underlying this result is the Nilpotent Signalizer Functor Theorem.

Definition 1.29. We say that an E-signalizer functor on a group G of finite Morley rank is complete if: a. θ(E) = ⟨θ(s) : s ∈ E^{#}⟩ is a solvable p^{⊥}-group, and b. θ(s) = C_{θ(E)}(s) for any s ∈ E^{#}.

Nilpotent Signalizer Functor Theorem 5 [10, 17, 15, Theorem B.30]. Let G be a group of finite Morley rank, and let E ≤ G be a finite elementary abelian 2-group of rank at least 3. Let θ be a connected nilpotent E-signalizer functor. Then θ is complete and θ(E) is nilpotent.

10 jeffrey burdges

We shall work with the proper group θ(E) in the same manner as we did with Γ_{E}(G) in Proposition 1.23.

We also recall a variation on Fact 1.1.

Fact 1.30 [17, Fact 3.7]. Let H be a solvable p^{⊥}-group of finite Morley rank. Let E be a finite elementary abelian p-group acting definably on H. Then

H = ⟨C_{H}(E_{0}) | E_{0} ≤ E, [E : E_{0}] = p⟩

Proof of Theorem 1.28. It suffices to show that Γ_{E,2}(G) < G by Proposition 1.26. The Nilpotent Signalizer Functor Theorem says that θ is complete and θ(E) is nilpotent. Since G is simple, our result will follow from

Γ_{E,2}(G) ≤ N_{G}(θ(E)).

For any U, V ∈ E_{2}(E), we have that θ(U) = θ(V) because

θ(u) ≤ ⟨C_{θ(u)}(v) : v ∈ V^{#}⟩ ≤ θ(V) for any u ∈ U.

by Fact 1.30 and the signalizer functor property. Thus θ(U) = θ(E). For any U ∈ E_{2}(E) and any g ∈ N_{G}(U), our hypothesis (†) yields

θ(E)^{g} = θ(U)^{g} = θ(U^{g}) = θ(U) = θ(E).

Thus Γ_{E,2}(G) ≤ N_{G}(θ(E)) < G, as desired.

2. Balance and components

In the tame setting of [10], Borovik states that O(C_{G}(i)) is a nilpotent signalizer functor. In view of Theorem 1.28, it then follows that either G has a proper 2generated core, if O(C_{G}(i)) ≠ 1, or else C_{G}(i) is “reductive” in the sense of Fact 1.4. It also follows, from Proposition 1.23, that either G has a proper 2-generated core, or else Γ_{Ω_{1}(S^{∘})}(G) = G. These two facts constitute the reductivity and generation conditions of the Generic Trichotomy Theorem [8], so a tame version of the Generic Trichotomy Theorem then follows. In fact, [8] uses these two conditions merely to establish that G is generated by the quasisimple components of the centralizers of toral involutions, and hence by their root SL_{2}-subgroups. The remainder of the argument focuses on these root SL_{2}-subgroups, treating them as an abstract family of root SL_{2}-subgroups for G, and eventually applying the Curtis-Tits Theorem.

In this section, we turn our attention towards “unbalanced groups” where O(C_{G}(i)) is not necessarily a signalizer functor, in order to eliminate the hypothesis of tameness. Instead, we use the “most unipotent” parts of O(C_{G}(i)) as signalizer functors. In Theorem 2.9, these signalizer functors are used to prove a dichotomy between a proper 2-generated core, and our B̃-property (see §2.1 below). Corollary 2.11 then provides a limited form of the reductivity proved in Fact 1.4. However, this weaker form of reductivity does not admit such a quick proof of generation by components. So our version of this result, Theorem 2.18 below, requires a considerably more delicate argument.

2.1. Partial balance

We require an example to explain the failure of balance.

signalizers and balance 11

Example 2.1. Consider a field (k, T, +, ·) of finite Morley rank, with T < k^{∗} torsion-free, and G := SO_{8}(k) (D_{4}). By Table 4.3.1 on p. 145 of [26], there are involutions i, j in G, lying in a common torus, such that

C_{G}(i) ≅ SL_{4}(k) ∗ k^{∗} and C_{G}(j) ≅ SL_{3}(k) ∗ SL_{3}(k)

So O(C_{G}(i)) = O(k^{∗}) = T ≠ 1 and O(C_{G}(j)) = 1. However, every inner involutive automorphism of SL_{n} is a central product with one copy of k^{∗}. So O(C_{G}(·)) is not a signalizer functor. In fact, the reductivity hypothesis of Fact 1.4 fails too, although its conclusions still holds since the centralizer is still reductive.

Our solution to this is to choose a reduced rank r̄^{∗}(·) which is the largest possible problematic reduced rank in k^{∗}, and work above it by using the fact that rk(k^{∗}) > r̄_{0}(k^{∗}).

Definition 2.2. Consider a simple K^{∗}-group G of finite Morley rank and let X be a subgroup of G with m(X) ≥ 3. We write

I^{0}(X) := {i ∈ I(X) : m(C_{X}(i)) ≥ 3}

for the set of involutions from eight-groups in E_{3}(X). We define

r̄^{O}(X) := sup{r̄_{0}(O(C_{G}(i))) : i ∈ I^{0}(X)}

as the supremum of the reduced ranks of the odd parts of the centralizers in G of the involutions in I^{0}(X). We also define r̄^{∗}(X) to be the supremum of r̄_{0}(k^{∗}) as k ranges over the base fields of the quasisimple components of the quotients C_{G}^{∘}(i)/O(C_{G}(i)) associated to involutions i ∈ I^{0}(X).

One can easily check that

r̄^{O}(G) = max_{E∈E_{3}(G)} r̄^{O}(E) and r̄^{∗}(G) = max_{E∈E_{3}(G)} r̄^{∗}(E).

We recall that, for a nonsolvable group L of finite Morley rank, U_{0,r}(L) and U_{p}(L) need not be solvable, as quasisimple algebraic groups are generated by the unipotent radicals of their Borel subgroups. Proposition 2.13 below will shed further light on the definition of r̄^{∗}(·) by providing a converse to this observation.

Definition 2.3. We continue in the notation of Definition 2.2. For a definable subgroup H of G, we define Ũ_{X}(H) to be the subgroup of H generated by U_{p}(H) for p prime as well as by U_{0,r}(H) for r > r̄^{∗}(X). As an abbreviation, we use F̃_{X}(H) to denote F^{∘}(Ũ_{X}(H)), and Ẽ_{X}(H) to denote E(Ũ_{X}(H)). We use Ẽ_{Y}^{X} to denote the set of components of Ẽ_{X}(C_{G}(i)) = E(Ũ_{X}(C_{G}(i))) for i ∈ I^{0}(Y) with Y ≤ X, and we set Ẽ_{X} = Ẽ_{X}^{X}.

Ũ_{X}(H) is the subgroup of H which is generated by its unmistakably unipotent subgroups. These definitions are all sensitive to the choice of X, which is usually a fixed eight-group.

Definition 2.4. We say that a simple K^{∗}-group G with m(G) ≥ 3 satisfies the B̃-property if, for every 2-subgroup X ≤ G with m(X) ≥ 3 and every t ∈ I^{0}(X), the group Ũ_{X}(O(C_{G}^{∘}(t))) is trivial. This is equivalent to (B̃-1) U_{p}(O(C_{G}(t))) = 1 for all t ∈ I^{0}(G) and every prime p. (B̃-2) r̄^{O}(X) ≤ r̄^{∗}(X) for every 2-subgroup X ≤ G with m(X) ≥ 3, and

The B̃-property is an unbalanced alternative to Borovik’s B-conjecture: that O(C_{C}(i)) = 1 for all i ∈ I(G). Although the B̃-property is significantly more delicate than the strong B-property, the next two subsections will establish results about the components in Ẽ_{X} which are similar to Borovik’s.

Our goal in this subsection is to verify that the failure of the B̃-property leads to a proper weak 2-generated core. For this, we need two appropriate signalizer functors.

Lemma 2.5. Let G be a simple K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3, and let E ∈ E_{3}(G). Then {U_{p}(O(C_{G}(t))) | t ∈ E^{#}} is a connected nilpotent E-signalizer functor satisfying

θ(s)^{g} = θ(s^{g}) for all s ∈ E^{#} and all g ∈ G for which s^{g} ∈ E.  (†)

We need the following two facts.

Fact 2.6 [15, Ex. 11 p. 93 & Ex. 13c p. 72]. Let G be a group of finite Morley rank and let H ◁ G be a definable subgroup. If x ∈ G is an element such that x̄ ∈ G/H is a p-element, then xH contains a p-element. Furthermore, if H and G/H are p^{⊥}-groups, then G is a p^{⊥}-group.

Fact 2.7 [2]; [17, Fact 3.2]. Let the group G = H ⋊T be a semidirect product of finite Morley rank. Suppose T is a solvable π-group of bounded exponent and Q◁H is a definable solvable T -invariant π^{⊥}-subgroup. Then

C_{H}(T)Q/Q = C_{H/Q}(T).

Proof of Lemma 2.5. Let θ(t) := U_{p}(O(C_{G}(t))). We observe that θ(s)^{g} = θ(s^{g}) for every involution s ∈ I(G) and every g ∈ G. By Fact 1.10, θ(s) is nilpotent. θ(s) is connected by definition.

Let s, t ∈ E^{#}; in particular [s, t] = 1. Also let K_{s} = O(C_{G}(s)). Since C_{G}(s) is a K-group, Fact 1.4 says C_{G}^{∘}(s)/K_{s} = G_{1} ∗ ⋯ ∗ G_{n} ∗ F is the central product of finitely many quasisimple algebraic groups G_{1}, …, G_{n} and of a definable divisible abelian group F. Since F is abelian, O(C_{F}^{∘} (t)) = 1 by Fact 2.6. We now consider the action of t on the components. For any component G_{k}, either t normalizes G_{k}, or else t swaps G_{k} with another component G_{l} = G_{k}^{t}. In the second case, the centralizer of t is some diagonal subgroup of G_{k}^{t} ∗ G_{k}, i.e. C_{G}^{∘}(t) ≅ {(g, σ(g)) | g ∈ G_{k}} ≅ G_{k} for some automorphism σ of G_{k}. So we may assume that t normalizes each G_{k} with k ≤ m, and C_{G_{m+1}∗···∗G_{n}}^{∘}(t) ≅ G_{\frac{n+m}{2}} ∗ ⋯ ∗ G_{n}. By Facts 1.6 and 1.7, C_{G_{k}}^{∘}(t) is reductive for k ≤ m. So O(C_{G_{k}}^{∘}(t)) is a subgroup of an algebraic torus, and hence divisible abelian. Hence U_{p}(O(C_{G_{1}∗···∗G_{n}}^{∘}(t))) = 1. So U_{p}(O(C_{C_{G}^{∘}(s)/K_{s}}^{∘}(t))) = 1. Since C_{C_{G}^{∘}(s)}(t)K_{s}/K_{s} = C_{C_{G}^{∘}(s)/K_{s}}(t) by Fact 2.7, U_{p}(O(C_{C_{G}^{∘}(t)}(s))) = U_{p}(O(C_{C_{G}^{∘}(s)}(t))) ≤ K_{s}.

For any t ∈ E^{#}, the group θ(t) is p-unipotent, and so 2^{⊥} and nilpotent. By Fact 1.17, C_{θ(t)}(s) is a connected. So C_{θ(t)}(s) = U_{p}(O(C_{θ(t)}(s))) ≤ U_{p}(O(C_{C_{G}^{∘}(t)}(s))) ≤ U_{p}(K_{s}) = θ(s).

signalizers and balance 13

Lemma 2.8. Let G be a simple K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3. Let E ∈ E_{3}(G) and set r := r̄^{O}(E). If r > r̄^{∗}(E) then U_{0,r}(O(C_{G}(t))) is a connected nilpotent E-signalizer functor again satisfying (†).

Proof. Let θ(t) := U_{0,r}(O(C_{G}(t))). We observe that θ(s)^{g} = θ(s^{g}) for every involution s ∈ I(G) and every g ∈ G. θ(s) is clearly connected and solvable. So θ(t) is nilpotent by Theorem 1.13.

Let s, t ∈ E^{#}; in particular [s, t] = 1. Also let K_{s} = O(C_{G}(s)). Since C_{G}(s) is a K-group, Fact 1.4 says C_{G}^{∘}(s)/K_{s} = G_{1} ∗ ⋯ ∗ G_{n} ∗ F is the central product of finitely many quasisimple algebraic groups G_{1}, …, G_{n} and of a definable divisible abelian group F. Since F is abelian, O(C_{F}^{∘} (t)) = 1. We next show that U_{0,r}(O(C_{G_{1}∗···∗G_{n}}^{∘}(t))) = 1. For any component G_{k}, either t normalizes G_{k}, or else t swaps G_{k} with another component G_{l} = G_{k}^{t}. In the second case, the centralizer of t is some diagonal subgroup of G_{k}^{t} ∗G_{k}, i.e. C_{G}^{∘}(t) ≅ {(g, σ(g)) | g ∈ G_{k}} ≅ G_{k} for some automorphism σ of G_{k}. So we may assume that t normalizes each G_{k} with k ≤ m, and C_{G_{m+1}∗···G_{n}}^{∘}(t) ≅ G_{\frac{n+m}{2}} ∗ ⋯ G_{n}. Consider a connected definable indecomposable abelian subgroup A of O(C_{G_{1}∗···∗G_{m}}^{∘}(t)) with A/J(A) torsion-free. Then there is an k ≤ m with a nontrivial projection map π : A → O(C_{G_{k}/Z(G_{k})}^{∘}(t)). By [17, Lemma 2.9], the image π(A) is also indecomposable abelian, and r̄_{0}(A) = r̄_{0}(π(A)). By Facts 1.6 and 1.7, C_{G_{k}/Z(G_{k})}^{∘}(t) is reductive, and hence O(C_{G_{k}/Z(G_{k})}^{∘}(t)) is an algebraic torus. It follows that

r̄_{0}(A) = r̄_{0}(π(A)) ≤ r̄_{0}(O(C_{G_{k}/Z(G_{k})}^{∘}(t))) ≤ r̄^{∗}(E).

Since r > r̄^{∗}(E), we have U_{0,r}(O(C_{G_{1}∗···∗G_{n}}^{∘}(t))) = 1. Since C_{C_{G}^{∘}(s)}(t)K_{s}/K_{s} = C_{C_{G}^{∘}(s)/K_{s}}(t) by Fact 2.7, U_{0,r}(O(C_{C_{G}^{∘}(t)}(s))) = U_{0,r}(O(C_{C_{G}^{∘}(s)}(t))) ≤ K_{s}.

For any t ∈ E^{#}, the centralizer C_{θ(t)}(s) is a connected (0, r)-unipotent 2^{⊥}-group by Fact 1.18. Thus

C_{θ(t)}(s) ≤ U_{0,r}(O(C_{C_{G}^{∘}(t)}(s))) ≤ U_{0,r}(K_{s}) = θ(s),

and θ is a signalizer functor.

We can now verify the B̃-property, in the absence of a proper 2-generated core.

Theorem 2.9. Let G be a simple K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3. Then either

1. G has a proper weak 2-generated core, or else

2. G satisfies the B̃-property, i.e. Ũ_{X}(O(C_{G}(t)) = 1 for every 2-subgroup X ≤ G with m(X) ≥ 3 and every t ∈ I^{0}(X).

Proof. We first suppose that (B̃-1) fails, i.e. U_{q}(O(C_{G}(i))) ≠ 1 for some involution i ∈ I^{0}(G). There is an E ∈ E_{3}(G) containing i. By Lemma 2.5, θ(t) := U_{p}(O(C_{G}(t))) is a connected nilpotent E-signalizer functor satisfying (†). So G has a proper weak 2-generated core by Theorem 1.28.

We next suppose that (B̃-2) fails, i.e. r̄^{O}(X) > r̄^{∗}(X) for some 2-subgroup X ≤ G with m(X) ≥ 3. There is an E ∈ E_{3}(X) such that r̄^{O}(E) = r̄^{O}(X). Let θ(t) := U_{0,r}(O(C_{G}(t))) where r := r̄^{O}(E) is the largest reduced rank appearing inside O(C(i)) for involutions i ∈ E^{#}. Now θ(t) := U_{0,r}(O(C_{G}(t))) is a connected nilpotent E-signalizer functor satisfying (†) by Lemma 2.8. By the choice of r, θ(i) is nontrivial for some involution i ∈ E^{#}. So G again has a weak proper 2-generated core by Theorem 1.28.

2.2. Existence of components in Ẽ_{X}

In this subsection, we will use the B̃-property to show that G is a group “of component type” in the sense that

Ẽ_{X} ≠ ∅ for every 2-subgroup X ≤ G with m(X) ≥ 3.

We employ the p-unipotent and 0-unipotent signalizer functors found in §2.1, via Theorem 2.9. Our major tool will be the following analog of Fact 1.4 which allows us to exploit the B̃-property.

Lemma 2.10. Let H be a K-group of finite Morley rank and odd type, and let H̃ be the subgroup of H generated by a. U_{0,r}(H) for r > r̄_{0}(O(H)), as well as b. U_{p}(H) for any prime p satisfying U_{p}(O(H)) = 1. Then H̃ = E(H̃) ∗ F^{∘}(H̃) and F^{∘}(H̃) is abelian.

Proof. We first show that [F (O(H)), H̃] = 1. Let A be a definable connected nilpotent subgroup of H with either a. U_{0,r}(A) = A for some r > r̄_{0}(O(H)), or b. U_{p}(A) = A for some prime p for which U_{p}(O(H)) = 1. Then A·F (O(H)) is nilpotent by Fact 1.14. Since r > r̄_{0}(O(H)), we have [A, F (O(H))] = 1 by Fact 1.16. So [F (O(H)), H̃] = 1.

Since F (O(H̃)) is definably characteristic in H̃, F (O(H̃)) ≤ F (O(H)), so F (O(H̃)) ≤ Z(H̃). Hence O(H̃) is nilpotent, and

O(H̃) = F (O(H̃)) ≤ Z(H̃).  (⋆)

Consider K := H̃/O(H̃). By Fact 1.4, K = E(K)∗F (K) and F (K) is abelian. By (⋆), the inverse image of F (K) in H̃ is nilpotent, and thus equals with F^{∘}(H̃). As any quasi-simple component L of E(K) is perfect, such a component L admits only finite central extensions by [3].By (⋆), the inverse image L̂ of L in H̃ is isomorphic to a central product L ∗ O(H̃), which thus contains a component of H̃. Now H̃ = E(H̃) ∗ F^{∘}(H̃), as desired.

Therefore F^{∘}(H̃) satisfies our generation hypotheses. Clearly U_{p}(F^{∘}(H̃)) ≤ U_{p}(O(H)) = 1. By Fact 1.16, F^{∘}(H̃) is a central product of the U_{0,r}(F^{∘}(H̃)) with r > r̄_{0}(O(H)) ≥ r̄_{0}(O(H̃). It follows that F^{∘}(H̃) is abelian since F (K) was abelian.

The B̃-property states that the centralizers of appropriate involutions satisfy the hypotheses of Lemma 2.10.

Corollary 2.11. Let G be a simple K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3 which satisfies the B̃-property. Then, for every 2-subgroup X ≤ G with m(X) ≥ 3 and every i ∈ I^{0}(X), we have Ũ_{X}(C_{G}(i)) = Ẽ_{X}(C_{G}(i)) ∗ F̃_{X}(C_{G}(i)) and F̃_{X}(C_{G}(i)) is abelian.

We can now verify that Ẽ_{X} is nonempty.

signalizers and balance 15

Theorem 2.12. Let G be a simple K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3. Then either

1. G has a proper weak 2-generated core, or else

2. Ẽ_{X} ≠ ∅ for every 2-subgroup X ≤ G with m(X) ≥ 3.

Proof. By Theorem 2.9, we may assume that G satisfies the B̃-property. Consider a 2-subgroup X ≤ G with m(X) ≥ 3. There is an E ∈ E_{3}(X) with r̄^{∗}(E) maximal. So r̄^{∗}(E) = r̄^{∗}(X) and Ẽ_{E} ⊂ Ẽ_{X}.

We first consider the case where C_{G}^{∘}(i) is solvable for all i ∈ E^{#}. In particular, r̄^{∗}(E) = 0. For all i ∈ E^{#}, U_{p}(C_{G}(i)) = U_{p}(O(C_{G}(i)) = 1 and r̄^{O}(E) ≤ r̄^{∗}(E) = 0, since G satisfies the B̃-property. By Fact 1.15, O(C_{G}^{∘}(i)) is a good torus, and hence central in C_{G}^{∘}(i) by [15, Theorem 6.16]. Since C_{G}^{∘}(i)/O(C_{G}^{∘}(i)) is abelian by Fact 1.4. C_{G}^{∘}(i) is nilpotent, and divisible. Now C_{G}^{∘}(i)′ is torsion-free by [15, Theorem 6.9]. As r̄^{O}(E) = 0, C_{G}^{∘}(i) is in fact abelian. By Fact 1.30, C_{G}^{∘}(i) = ⟨C_{C_{G}^{∘}(i)}^{∘}(E_{0}) : E_{0} ≤ E, [E : E_{0}] = 2⟩. As C_{G}^{∘}(i) ≠ 1 by [15, Ex. 13 & 15 p. 79], there is some four-group E_{1} ≤ E with H := C_{G}^{∘}(E_{1}) ≠ 1. Since each C_{G}^{∘}(i) is abelian, H ◁ C_{G}^{∘}(i) for all i ∈ E_{1}^{#}, and thus H ◁ Γ_{E_{1}}(G). Since G is simple, Γ_{E_{1}}(G) ≤ N_{G}(H) < G, and G has a proper weak 2-generated core Γ_{S,2}^{0}(G) < G by Proposition 1.23. So we may assume that C_{G}^{∘}(i) is nonsolvable for some i ∈ E^{#}.

We now fix an i ∈ E^{#} and a component L of C_{G}^{∘}(i)/O(C_{G}^{∘}(i)) so that r̄_{0}(k^{∗}) = r̄^{∗}(E) where k is the base field of L. Since k is algebraically closed, k^{∗} contains torsion, and hence

rk(k_{+}) = rk(k^{∗}) > r̄_{0}(k^{∗}) = r̄^{∗}(E).

Suppose toward a contradiction that Ẽ_{E}(C_{G}(i)) = 1. By Corollary 2.11, we have Ũ_{E}(C_{G}(i)) = Ẽ_{E}(C_{G}(i)) ∗ F̃_{E}(C_{G}(i)) and F̃_{E}(C_{G}(i)) is abelian for every i ∈ E^{#}. So Ũ_{E}(C_{G}(i)) = F̃_{E}(C_{G}(i)) is abelian. If char(k) > 0 then U_{char(k)}(C_{G}^{∘}(i)) ≤ F̃_{E}(C_{G}^{∘}(i)) is abelian. If char(k) = 0 then U_{0,rk(k_{+})}(C_{G}^{∘}(i)) ≤ F̃_{E}(C_{G}(i)) is abelian. Either case contradicts the existence of L.

We also observe that the definition of Ẽ_{X} restricts the fields involved as follows.

Proposition 2.13. Let H be a group of finite Morley rank which is isomorphic to a linear algebraic group over an algebraically closed field k. Then

1. If U_{0,r}(H) ≠ 1 for some r > r̄_{0}(k^{∗}) then char(k) = 0 and rk(k) = r.

2. If U_{p}(H) ≠ 1 then char(k) = p. If H is quasisimple, these conditions imply U_{p}(H) = H and U_{0,r}(H) = H, respectively.

Proof. If char(k) ≠ p, then H has bounded p-rank, so the second point follows. We now consider the first point. Let A be a nontrivial (0, r)-unipotent abelian

group, and let Â be the Zariski closure of A. Then Â = S×U where S is semisimple, and U is the unipotent radical of A. If A has nonunipotent elements, then Ā := AU/U is a nontrivial subgroup of the semisimple group Â/U. As Â/U is linear, Â/U ↪ (k^{∗})^{n} for some n. But U_{0,r}(Ā) = Ā by [17, Lemma 2.11], contradicting r > r̄_{0}(k^{∗}). So A consists of unipotent elements, i.e. A ≤ U (= Â). Hence char(k) = 0. As U is linear, U ↪ (k_{+})^{n} for some n. By [31, Cor. 3.3], there are no definable subgroups of k^{+}. So U_{0,r}(U) = 1 unless r = rk(k).

16 jeffrey burdges

The last remark follows from the fact that quasisimple algebraic groups are generated by the unipotent radicals of their Borel subgroups, or indeed by any conjugacy class.

2.3. Generation by components in Ẽ_{X}

We next show that components in Ẽ_{X} generate G, i.e.

⟨Ẽ_{Ω_{1}(S^{∘})}⟩ = G when pr(S) ≥ 3.

However, these results will be proven in a form usable also when pr(S) < 3.

For any group H of finite Morley rank, any 2-subgroup X acting definably on H, and any V ∈ E_{2}^{0}(X), we define

Γ̃_{X,V} (H) = ⟨Ũ_{X}(C_{H}^{∘}(v)) : v ∈ V^{#}⟩.

Lemma 2.14. Let G be a K^{∗}-group of finite Morley rank and odd type. Let X be a 2-subgroup of G with m(X) ≥ 3. Suppose that G satisfies the B̃-property and that there is a four-group E ∈ E_{2}^{0}(X) which centralizes a Sylow^{∘} 2-subgroup T of G. Let H := ⟨Ẽ_{X}(C_{G}(z)) : z ∈ E^{#}⟩. Then the following hold.

1. For any x, y ∈ E^{#}, we have [F̃_{X}(C_{G}(x)), F̃_{X}(C_{G}(y))] = 1 and the group F̃_{X}(C_{G}(x)) normalizes Ẽ_{X}(C_{G}(y)).

2. Ũ_{X}(O(H)) = 1.

3. Γ̃_{X,E}(G) ≤ N_{G}^{∘}(Ẽ_{X}(H)).

Proof. As a notational convenience, we let F_{x} := F̃_{X}(C_{G}(x)) and let H_{x} := Ẽ_{X}(C_{G}(x)) for x ∈ E^{#}. Since G satisfies the B̃-property, Corollary 2.11 says Ũ_{X}(C_{G}(x)) = H_{x} ∗ F_{x} and F_{x} is abelian. Since y ∈ E^{#} normalizes F_{x} for any x ∈ E^{#}, there is a homomorphism f_{y} : F_{x} → F_{x} given by u ↦ [y, u]. Since E centralizes T and F_{x} ∩ T is the Sylow 2-subgroup of F_{x}, there is no 2-torsion in F_{x} \ ker(f_{y}). So [F_{x}, E] ≤ O(F_{x}) ≤ O(C_{G}^{∘}(x)) by Fact 2.6 (and Fact 1.19). By Fact 1.20, [U_{0,r}(F_{x}), E] is a U_{0,r}-group. Clearly [U_{p}(F_{x}), E] is p-unipotent. Since Ũ_{X}(C_{G}^{∘}(x)) = H_{x} ∗ F_{x}, we have F_{x} = Ũ_{X}(F_{x}). So [F_{x}, E] ≤ Ũ_{X}(O(C_{G}^{∘}(x))) = 1 by the B̃-property. Now F_{x} ≤ Ũ_{X}(C_{G}(y)) for every x, y ∈ E^{#}. Since F_{y} is central in Ũ_{X}(C_{G}(y)), we have [F_{x}, F_{y}] = 1. Also F_{x} normalizes H_{y}, as H_{y} ◁Ũ_{X}(C_{G}(y)). Thus F := ⟨F_{z} : z ∈ E^{#}⟩ is an abelian group of automorphisms of H = ⟨H_{z} : z ∈ E^{#}⟩.

Since H_{x} is characteristic in C_{G}(x) for all x ∈ E^{#}, E normalizes H. Suppose towards a contradiction that Ũ_{X}(O(H)) ≠ 1. So either

(1) K := U_{0,r}(O(H)) ≠ 1 for some r > r̄^{∗}(X), or else

(2) K := U_{p}(O(H)) ≠ 1 for some prime p. In either case, K = Γ_{E}(K) by Fact 1.1. So a K_{x} := C_{K}(x) is nontrivial. In case (1), we may choose r maximal, so K is nilpotent by Theorem 1.13. Hence K_{x} is (0, r)-unipotent by Lemma 1.18. In case (2), K is nilpotent by Fact 1.10. Hence K_{x} is p-unipotent by Fact 1.17. In either case, K_{x} is nilpotent and normalized by H_{x}. Since K_{x} ≤ Ũ_{X}(C_{G}(x)), and H_{x} is semisimple, we have K_{x} ≤ F_{x}, in contradiction to the B̃-property. Thus Ũ_{X}(O(H)) = 1, as desired.

For the last part, we may assume H < G is a K-group. So H = Ẽ_{X}(H) ∗ F̃_{X}(H) by Corollary 2.11. Since Ẽ_{X}(H) is characteristic in H, F normalizes Ẽ_{X}(H). So Γ̃_{X,E}(G) = F · H ≤ N_{G}^{∘}(Ẽ_{X}(H)).

We now provide the promised variant of Lemma 1.24 above.

signalizers and balance 17

Lemma 2.15. Let G be a K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3, and which satisfies the B̃-property. Let T be a Sylow^{∘} 2-subgroup of G, and let X ≤ C_{G}(T) be a 2-subgroup which centralizes T and has m(X) ≥ 3. Then Γ̃_{X,U}(G) = Γ̃_{X,V} (G) for any two four-groups U, V ∈ E_{2}^{0}(X).

We need the following algebraic fact.

Fact 2.16. Let G be a quasisimple algebraic group over an algebraically closed field of characteristic not 2 which is not of type (P)SL_{2}, and let V be a four-group of algebraic automorphisms of G which centralizes a maximal 2-torus of G. Then

G = ⟨E(C_{G}(x)) : x ∈ V^{#}⟩.

Proof. Let T be a maximal 2-torus of G centralized by V. For i ∈ V^{#}, Fact 1.7 says C_{G}^{∘}(i) is reductive. Hence C_{G}^{∘}(i) = F_{i} ∗ H_{i} where H_{i} := E(C_{G}^{∘}(i)) and F_{i} := F^{∘}(C_{G}^{∘}(i)) is an algebraic torus. So F_{i} is the Zariski closure of F_{i} ∩ T, and hence is centralized by V. Now F := ⟨F_{x} | x ∈ V^{#}⟩ is an algebraic torus normalizing H = ⟨H_{x} | x ∈ V^{#}⟩.

We now show that H is reductive. Suppose that H has a nontrivial unipotent radical U. By Fact 1.1, U_{j} := C_{U}(j) ≠ 1 for some j ∈ V^{#}. But U_{j} ◁ C_{G}^{∘}(j), contradicting the reductivity of C_{G}^{∘}(j). So H must be reductive.

Now Γ_{V} (G) = F H ≤ N_{G}^{∘}(E(H)). We observe that H_{j} ≠ 1 for j ∈ V^{#}, since G ≅ (P)SL_{2}. So H is not solvable, and E(H) ≠ 1. Since Γ_{V} (G) = G by Fact 1.1, it follows that E(H) = G because G is quasisimple, and hence H = G.

Proof of Lemma 2.15. We may assume that [U, V] = 1 since U and V lie in the same 2-connected component. It is enough to show that Ũ_{X}(C_{G}(u)) ≤ Γ̃_{X,V} (Ũ_{X}(C_{G}(u)) for any u ∈ U^{#}. Since G satisfies the B̃-property, we have Ũ_{X}(C_{G}(u)) = Ẽ_{X}(C_{G}(u))∗F̃_{X}(C_{G}(u)) by Corollary 2.11. By Fact 1.16, the abelian group F_{u} := F̃_{X}(C_{G}(u)) may be written as a product of various U_{p}(F_{u}), with p prime, and U_{0,r}(F_{u}), with r > r̄^{∗}(X). By Fact 1.18, C_{U_{0,r}(F_{u})}(v) is (0, r)-unipotent. By Fact 1.17, C_{U_{p}(F_{u})}(v) is p-unipotent. So C_{F_{u}}(v) ≤ Ũ_{X}(C_{G}(u)). Fact 1.1 yields

F̃_{X}(C_{G}(u)) ≤ Γ̃_{X,V} (Ũ_{X}(C_{G}(u)).

Now consider a component L ◁ Ẽ_{X}(C_{G}(u)). It suffices to show that L = Γ̃_{X,V} (L). In the case that L ≅ (P)SL_{2}, Fact 2.16 yields

L = ⟨E(C_{L}(x)) : x ∈ V^{#}⟩.

By Proposition 2.13, we have E(C_{L}(x)) = Ẽ_{X}(C_{L}(x)), and hence L = Γ̃_{X,V} (L).

In the case that L ≅ (P)SL_{2}, (P)SL_{2} has no graph automorphisms, so every x ∈ V^{#} acts by some inner automorphism by Fact 1.6. We observe that T ≤ C_{G}(u), and thus T is a Sylow^{∘} 2-subgroup of C_{G}(u). So L ∩ T is a Sylow^{∘} 2-subgroup of L. Since (P)SL_{2} contains no four-group centralizing a torus, there is now some x ∈ V^{#} which centralizes L, and the claim follows.

We now prove a version of Proposition 1.23.

Proposition 2.17. Let G be an infinite simple K^{∗}-group of finite Morley rank 18 jeffrey burdges

and odd type with m(G) ≥ 3. Let T be a Sylow^{∘} 2-subgroup of G, and let X ≤ C_{G}(T) be a 2-subgroup which centralizes T with m(X) ≥ 3. Suppose that Γ̃_{X,E}(G) < G for some E ∈ E_{2}^{0}(X). Then G has a proper weak 2-generated core.

Proof. By Proposition 1.23, it is enough to show that Γ_{E}(G) < G. Let A ∈ E_{3}(X) be an eight-subgroup of X containing E. By Lemma 1.25, Γ_{E}(G) ≤ Γ_{A,2}(G). So the result will follow from the following claim and simplicity.

Γ_{A,2}(G) ≤ N_{G}(Γ̃_{X,E}(G))

By Lemma 2.15 and Fact 1.21-1, Γ̃_{X,E}(G) = Γ̃_{X,U}(G) for any U ≤ A. For any four-group U ≤ A,

N_{G}(U) ≤ N_{G}(Γ̃_{X,U}(G)) = N_{G}(Γ̃_{X,E}(G)).

Thus Γ_{A,2}(G) ≤ N_{G}(Γ̃_{X,E}(G)), as desired.

We now prove that our components generate G.

Theorem 2.18. Let G be a K^{∗}-group of finite Morley rank and odd type with m(G) ≥ 3. Suppose that there is a four-group E ∈ E_{2}^{0}(G) which centralizes a Sylow^{∘} 2-subgroup T of G, and that there is an eight-group X ∈ E_{3}(C_{G}(T)) containing E. Then either

1. G has a proper weak 2-generated core, or

2. ⟨Ẽ_{E}^{X}⟩ = ⟨Ẽ_{X}(C_{G}(x)) : x ∈ E^{#}⟩ = G.

We need the following fact about involutive automorphisms of algebraic groups, which follows immediately from Table 4.3.1 on p. 145 of [26] and Fact 1.6.

Fact 2.19. Let G be a quasisimple algebraic group over an algebraically closed field of characteristic not 2 and let α be a definable involutive automorphism of G. If G ≅ (P)SL_{2}, then E(C_{G}(α)) ≠ 1.

Proof of Theorem 2.18. We may assume that G satisfies the B̃-property by Theorem 2.9.

We first show that Ẽ_{X}(C_{G}^{∘}(z)) ≠ 1 for some z ∈ E^{#} (note m(E) ≥ 2). We may assume that Ẽ_{X} ≠ 1 by Theorem 2.12, so there is a component L ≤ Ẽ_{X}(C_{G}(x)) ≠ 1 for some x ∈ X^{#}. We observe that E normalizes L since X is an eight-group containing E. So any z ∈ E^{#} acts on L via an algebraic automorphism, by Fact 1.6, and hence C_{L}^{∘}(z) is reductive by Fact 1.7. By Proposition 2.13, Ũ_{X}(U) = U for any unipotent group in L. As quasi-simple groups are generated by their unipotent subgroups, Ũ_{X}(C_{G}(z)) ≥ Ẽ_{X}(C_{L}(z)) = E(C_{L}^{∘}(z)). If L ≅ (P)SL_{2}, then E(C_{L}^{∘}(z)) ≠ 1 by Fact 2.19, and hence Ẽ_{X}(C_{G}(z)) ≠ 1 by Corollary 2.11. So we may assume L ≅ (P)SL_{2}. Since (P)SL_{2} has no graph automorphisms, any z ∈ E^{#} acts via inner automorphism, by Fact 1.6. Since (P)SL_{2} contains no four-group centralizing a torus, there is now some z ∈ E^{#} which centralizes L, and Ũ_{X}(C_{G}(z)) is nonabelian. By Corollary 2.11, Ũ_{X}(C_{G}(z)) = Ẽ_{X}(C_{G}(z)) ∗ F̃_{X}(C_{G}(z)) and F̃_{X}(C_{G}(z)) is abelian. So Ẽ_{X}(C_{G}^{∘}(z) ≠ 1 for some z ∈ E^{#}.

Let H := ⟨Ẽ_{X}(C_{G}(x)) : x ∈ E^{#}⟩. Since Ũ_{X}(O(H)) = 1 by Lemma 2.14, H = Ẽ_{X}(H) ∗ F̃_{X}(H) and F̃_{X}(H) is abelian by Corollary 2.11. Since Ẽ_{X}(C_{G}^{∘}(x)) ≠ 1, we have Ẽ_{X}(H) ≠ 1.

Since E centralizes T, Lemma 2.14 says Γ̃_{X,E}(G) ≤ N_{G}^{∘}(Ẽ_{X}(H)). Hence Γ̃_{X,E}(G) ≤ N_{G}^{∘}(Ẽ_{X}(H)) < G, and the theorem follows from Proposition 2.17.

3. The Generic Trichotomy Theorem

We now turn our attention toward proving the following, our main result.

Generic Trichotomy Theorem 6. Let G be a simple K^{∗}-group of finite Morley rank and odd type with pr(G) ≥ 3. Then either

1. G has a proper 2-generated core, i.e. Γ_{S,2}(G) < G, or else

2. G is an algebraic group over an algebraically closed field of characteristic not 2.

Our strategy is to replicate the proof by Berkman and Borovik of the Generic Identification Theorem [8], being careful to use only “safe” components, under the assumption that (1) does not occur. So we adopt the following standing hypotheses and notation.

Hypothesis 3.1. We consider a simple K^{∗}-group G of finite Morley rank and odd type with pr(G) ≥ 3, and fix a Sylow 2-subgroup S of G. We also suppose that the 2-generated core of G is not proper, i.e. Γ_{S,2}(G) = G.

As n(2) ≥ pr(G) ≥ 3, we have Γ_{S,2}^{0}(G) = Γ_{S,2}(G) = G by Fact 1.21-2. So, by Theorem 2.9,

G satisfies the B̃-property.  (3.1)

3.1. Root SL_{2}subgroups

The first stage in our analysis is to select, and establish the properties of a family of abstract “root SL_{2}-subgroups” of G. The root SL_{2}-subgroups of an algebraic group associated to a maximal torus T may be defined as those Zariski closed subgroups of G which are normalized by T and are isomorphic to (P)SL_{2}, or alternatively in terms of groups generated by opposite root groups. We employ several facts about root SL_{2}-subgroups of algebraic groups.

Fact 3.2 cf. [8, Fact 2.1]. Let G be a quasisimple algebraic group over an algebraically closed field. Let T be a maximal torus in G and let K, L be Zariski closed subgroups of G that are isomorphic to SL_{2} or PSL_{2} and are normalized by T. Then

1. Either K and L commute or ⟨K, L⟩ is a quasisimple algebraic group of type A_{2}, C_{2}, or G_{2}.

2. The subgroups K and L are root SL_{2}-subgroups of ⟨K, L⟩.

3. If ⟨K, L⟩ is of type G_{2}, then G = ⟨K, L⟩.

More generally, a semisimple subgroup of a simple algebraic group G which is normalized by a maximal torus T is called subsystem subgroup of G, associated to T. Berkman and Borovik refer to the full classification of semisimple subsystem subgroups [33, 2.5] (see also §3.1 of [32]) for the proof of this fact. The elementary argument here is based on the following fact.

Fact 3.3 [32, Prop. 3.1]; [33, 2.5]. Let G be a simple algebraic group, let T be a maximal torus of G, and let X be a closed connected subgroup of G which contains T. Then X = DZU where D is a subsystem subgroup normalized by T, Z is a torus, and U is the unipotent radical of X.

Proof of Fact 3.2. By Fact 3.3, ⟨K, L⟩ = DZU where D is a subsystem subgroup, Z is a torus, and U is the unipotent radical of ⟨K, L⟩. There is an automorphism ϕ of the root system for G which sends any root α ∈ I to its negative −α, and ϕ translates to an automorphism ϕ of the group G such that ϕ normalizes T and ϕ(X_{α}) = X_{−α} [20]. Since K and L each contain one positive and one negative root from I, we find that K, L, and ⟨K, L⟩ are all normalized by ϕ. If U is nontrivial, it contains a root group X_{α}. Since U is characteristic in ⟨K, L⟩, the nilpotent group U must also contain X_{−α}, and hence contains a copy of (P)SL_{2} [20]. So U = 1, and ⟨K, L⟩ = D. Since D is semisimple, D ≅ A_{1} ∗ A_{1}, A_{2}, C_{2}, or G_{2}, as desired.

Fact 3.4 see [21, p. 19]. Let G be a semisimple algebraic group over an algebraically closed field, and fix a maximal algebraic torus T of G. Then the following hold.

1. G is generated by its root SL_{2}-subgroups associated with T.

2. The intersection T ∩ K of T and a root SL_{2}-subgroup K associated to T is a maximal algebraic torus of K.

Proof. The fact that G is generated by those root SL_{2}-subgroups which are normalized by T can be found on p. 19 of [21]. For the second part, we observe that the maximal algebraic tori of N_{G}(K), one of which is T and one of which extends a maximal algebraic torus of K, are conjugate in N_{G}(K).

We also need to know that a root SL_{2}-subgroup is “cut out” by the centralizer of a 2-torus in the associated maximal torus, We remark that this is an essential point if one hopes to apply Fact 3.2, but it remains somewhat obscure in [8].

Fact 3.5. Let G be a quasisimple algebraic group over an algebraically closed field of characteristic not 2. Let T be a maximal algebraic torus of G, and let L be a root SL_{2}-subgroup of G normalized by T. Then L = E(C_{G}(C_{S^{∘}}(L))) where S^{∘} is the Sylow^{∘} 2-subgroup of T.

Proof. We may assume that G is not isomorphic to (P)SL_{2} because otherwise G = L. Let S be a Sylow 2-subgroup of G such that S^{∘} ≤ T. Any connected definable group of automorphisms of G must be inner by Fact 1.6. Since L is normalized by S^{∘}, we have pr(C_{S^{∘}}(L)) = pr(G) − 1. By Fact 1.7 (see also [21, Thm. 3.5.4]), C_{G}(C_{S^{∘}}(L)) is reductive. So pr(K) ≤ 1 where K := E(C_{G}(C_{S^{∘}}(L))). Since L ≤ K and L and K are both algebraic subgroups, we have L = K.

We now proceed with the analysis of groups satisfying Hypothesis 3.1.

signalizers and balance 21

Lemma 3.6. For any i ∈ Ω_{1}(S^{∘})^{#} and any definable connected quasisimple algebraic L ≤ E(C_{G}^{∘}(i)) which is normalized by S^{∘}, we have

1. S^{∘} ∩ L is a Sylow^{∘} 2-subgroup of L,

2. T_{L} := C_{L}(S^{∘} ∩ L) is a maximal algebraic torus of L,

3. S^{∘} = C_{S^{∘}}^{∘}(L)(S^{∘} ∩ L), and pr(G) = pr(S^{∘}) = pr(C_{S^{∘}}^{∘}(L)) + pr(S^{∘} ∩ L).

For this, we need the following fact about algebraic groups.

Fact 3.7. Let G be a quasisimple algebraic group over an algebraically closed field of characteristic not 2, and let T be a Sylow^{∘} 2-subgroup of G. Then C_{G}(T) is a maximal algebraic torus of G.

Proof. In an algebraic group over an algebraically closed field, the maximal algebraic torus is the Zariski closure of T, and is thus centralized by anything centralizing T. But a maximal algebraic torus is self-centralizing by [27, 24.1]. So the result follows.

Proof of Lemma 3.6. Since S^{∘} is a Sylow^{∘} 2-subgroup of N_{G}^{∘}(L), the group S^{∘} ∩ L is a Sylow^{∘} 2-subgroup of L. By Fact 3.7, T_{L} is a maximal algebraic torus of L. By Fact 1.6, the connected definable group d(S^{∘}) acts by inner automorphisms on L, so the third condition follows.

Lemma 3.8. For any component M ∈ Ẽ_{S^{∘}}, M is normalized by S^{∘}.

Proof. Clearly M ◁ C_{G}^{∘}(i) for some i ∈ Ω_{1}(S^{∘}) (see [15, Lemma 7.1iii]). Since S^{∘} ≤ C_{G}^{∘}(i), M is normalized by S^{∘}.

Definition 3.9. Let Σ be the set of all root SL_{2}-subgroups of components K ∈ Ẽ_{S^{∘}} which are associated to T_{K}, i.e. Σ is the set of all Zariski closed subgroups of the components K ∈ Ẽ_{S^{∘}} which are normalized by T_{K} and are isomorphic to (P)SL_{2}.

Since the root SL_{2}-subgroups of K generate K by Fact 3.4-1, Theorem 2.18 yields the following.

⟨Σ⟩ = ⟨Ẽ_{S^{∘}}⟩ = G  ((⋆))

We view the subgroups in Σ as abstract root SL_{2}-subgroups for G.

Lemma 3.10. For any L ∈ Σ, we have

1. L is normalized by S^{∘},

2. L = E(C_{G}(C_{S^{∘}}(L))), and

3. L is a Zariski closed subgroup of any definable quasisimple K < G which contains L and which is normalized by S^{∘}.

Proof. Let R_{L} := C_{S^{∘}}(L) and let M ∈ Ẽ_{S^{∘}} be a component containing L as a root SL_{2}-subgroup associated with T_{M}. By Lemma 3.6-3, S^{∘} = C_{S^{∘}}(M)(M ∩ S^{∘}). Since L is normalized by M ∩ S^{∘} ≤ T_{M}, L is normalized by S^{∘}. By Fact 3.5, L = E(C_{M}(M ∩ R_{L})). Fix i ∈ Ω_{1}(S^{∘})^{#} with M ◁ C_{G}^{∘}(i). Clearly E(C_{G}(R_{L})) ≤ M because any other component of C_{G}^{∘}(i) meets R_{L} in an infinite 2-torus. As i ∈ R_{L}, L = E(C_{G}(R_{L})).

22 jeffrey burdges

Now for any definable quasisimple K < G which contains L and which is normalized by S^{∘}, the group R_{L} acts on K by inner automorphisms by Fact 1.6, so L = E(C_{K}(R_{L})) is Zariski closed.

Lemma 3.11 cf. [8, Lemma 3.1]. For any distinct K, L ∈ Σ,

1. C_{S^{∘}}(K) ∩ C_{S^{∘}}(L) ≠ 1 and M := ⟨K, L⟩ is a K-group.

2. Either K and L commute or M is an algebraic group of type A_{2}, B_{2} = C_{2}, or G_{2}.

3. S^{∘} ∩ M = (S^{∘} ∩ K) ∗ (S^{∘} ∩ L) is a Sylow^{∘} 2-subgroup of M.

4. K and L are root SL_{2}-subgroups of M normalized by T_{M}.

5. [T_{K}, T_{L}] = 1.

Proof. Let R_{L} := C_{S^{∘}}^{∘}(L). Since S^{∘} normalizes K and L by Lemma 3.10-1, and K, L ≅ SL_{2}, we know that pr(R_{K}), pr(R_{L}) = pr(G)−1 and S^{∘} = R_{K}R_{L} by Lemma 3.6-3. So

pr(G) = pr(R_{K}) + pr(R_{L}) − pr(R_{K} ∩ R_{L})
= 2 pr(G) − 2 − pr(R_{K} ∩ R_{L})

and

pr(R_{K} ∩ R_{L}) = pr(G) − 2 ≥ 1.

Thus C_{S^{∘}}(K) ∩ C_{S^{∘}}(L) ≠ 1 and M has a nontrivial center. Since G is simple, M < G is a K-group.

Let i ∈ I(C_{S^{∘}}(K) ∩ C_{S^{∘}}(L)). By Corollary 2.11 (and the B̃-property), K, L ≤ Ẽ_{S^{∘}}(C_{G}(i)). If the groups belong to different components of C_{G}^{∘}(i), then they commute. If they both belong to the same component H ∈ Ẽ_{S^{∘}}, then H is a quasisimple algebraic group normalized by S^{∘} by Lemma 3.8. By Lemma 3.10-3, K and L are Zariski closed in H. By Fact 3.2-1, M = ⟨K, L⟩ is an algebraic group of type A_{2}, B_{2} = C_{2}, or G_{2}. In either case, S^{∘} normalizes M, so S^{∘} ∩M is a Sylow^{∘} 2-subgroup of M and T_{M} is a maximal “algebraic” torus of M by Lemma 3.6.

For (4) and (5), we may assume that [L, K] ≠ 1 and M is a quasisimple algebraic group. By Fact 3.2-2, K and L are root SL_{2}-subgroups of M. By Fact 3.4-2, T_{M} = T_{K} ∗ T_{L}, so [T_{K}, T_{L}] = 1.

We give Σ a graph structure by placing an edge between L, K ∈ Σ when [L, K] ≠ 1. Since G is simple and ⟨Σ⟩ = G (⋆), the graph Σ is connected. By Lemma 3.11-2, any adjacent L, K ∈ Σ algebraic groups over the same algebraically closed field. So all the elements of Σ are algebraic groups over a common algebraically closed field 𝔽. Since G has odd type, char(𝔽) ≠ 2. In particular, rk(K) = rk(L) for all K, L ∈ Σ.

From this point on, our argument reduces to that given by Berkman and Borovik in [8], following the presentation of [7]. Indeed, we may now lighten our standing hypotheses to the following.

Hypothesis 3.12. We consider a simple group G of finite Morley rank and odd type with pr(G) ≥ 3, and fix a Sylow 2-subgroup S of G. Also choose some family Ẽ of algebraic components from the centralizers of involutions in S^{∘}, Let Σ be the set of root SL_{2}-subgroups, from components in Ẽ, which are associated to S^{∘}, in the sense of Definition 3.9. Suppose that Lemmas 3.6, 3.10, and 3.11 are satisfied, and also that

⟨Σ⟩ = G.  ((⋆))

We give the analysis in full below.

3.2. Weyl group

We now turn our attention to the Weyl group of G, continuing under Hypothesis 3.12.

Lemma 3.13. The natural torus T := ⟨T_{L} : L ∈ Σ⟩ is divisible abelian. So T_{L} = T ∩ L.

Proof. By Lemma 3.11-5, the algebraic tori T_{K} for K ∈ Σ all commute, so the result follows.

Definition 3.14. For any L ∈ Σ, W (L) := N_{L}(T_{L})/T_{L} is the Weyl group of L and has order 2. We may identify W (L) ≅ N_{L}(T)/C_{L}(T) with its image in W := N_{G}(T)/C_{G}(T), by Lemma 3.13. Now let r_{L} denote the single involution inside W (L), and define W_{0} := ⟨r_{L} ∈ W : L ∈ Σ⟩.

Lemma 3.15 cf. [8, Lemma 3.5]. For any L, K ∈ Σ, [K, L] = 1 if and only if [r_{K}, r_{L}] = 1.

Proof. It suffices to check this in ⟨K, L⟩. So the result follows from Fact 3.2-2 and Fact 3.4.

We will analyze W_{0} by examining its action on S^{∘} and T.

Lemma 3.16 cf. [8, Lemmas 3.6 & 3.7]. S^{∘} is the Sylow 2-subgroup of T and C_{G}(S^{∘}) = C_{G}(T). In particular, W_{0} acts faithfully on S^{∘}.

Proof. By Lemma 3.13, T is divisible abelian, so its Sylow 2-subgroup is connected by [15, Theorem 9.29].

Let D := S^{∘} ∩ T. Suppose toward a contradiction that D < S^{∘}. For all K ∈ Σ, [S^{∘}, r_{K}] = S^{∘} ∩ K, so r_{K} acts trivially on S^{∘}/D. Let b ∈ S^{∘} satisfy |b/D| ≥ 4 and let a ∈ S^{∘} satisfy a^{|W_{0}|} = b. Then c = Π_{w∈W_{0}}a^{w} satisfies b/D = c/D and |c| ≥ 4. Since S^{∘} = C_{S^{∘}}(K) ∗ (S^{∘} ∩ K) and |C_{S^{∘}}(K) ∩ (S^{∘} ∩ K)| ≤ |Z(K)| = 2, we have [C_{S^{∘}}(r_{K}) : C_{S^{∘}}(K)] ≤ 2. Since c ∈ C_{S^{∘}}(r_{K}), c^{2} ∈ C_{S^{∘}}(K) for all K ∈ Σ, and c^{2} ≠ 1. So c^{2} ∈ C_{G}(⟨Σ⟩) = Z(G), contradicting the simplicity of G. Thus S^{∘} is the Sylow 2-subgroup of T, and C_{G}(S^{∘}) ≥ C_{G}(T).

For the reverse direction, consider x ∈ C_{G}(S^{∘}). Then, for every L ∈ Σ, x centralizes C_{S^{∘}}(L). So x normalizes L = E(C_{G}(C_{S^{∘}}(L))) by Lemma 3.10-2. Since x centralizes the maximal 2-torus S^{∘} ∩ L, x must act on L as an element of T_{L} by Fact 1.6. Thus x ∈ C_{G}(T) and C_{G}(S^{∘}) ≤ C_{G}(T).

We use the action of W_{0} on S^{∘} to obtain a complex representation.

Lemma 3.17 cf. [8, §3.3]. W_{0} has a faithful irreducible representation R over ℂ of dimension pr(G) ≥ 3 in which the r_{L} act as reflections for L ∈ Σ.

For this, we employ a Tate module over the 2-adics.

Fact 3.18 [6, 8, §3.3]. Let T be a p-torus of Prüfer p-rank n in a group of finite Morley rank. Then End(T) can be faithfully represented as the ring of n × n matrices over the p-adic integers ℤ_{p} ≅ End(ℤ(p^{∞})).

Proof of Lemma 3.17. For every L ∈ Σ and every 2-torus X ≤ S^{∘} disjoint from L, X must act on L as elements of S^{∘} ∩ L by Fact 1.6, so there is some 2-torus Y ≤ C_{S^{∘}}(L) with X ≤ Y (S^{∘} ∩ L). So S^{∘} = C_{S^{∘}}(r_{L}) ∗ (S^{∘} ∩ L). By Lemmas 3.10-1 and 3.6, r_{L} inverts S^{∘} ∩ L = [S^{∘}, r_{L}]. Thus r_{L} acts as a “reflection” on S^{∘}.

By Lemma 3.16, W_{0} acts faithfully on S^{∘}. By Fact 3.18, W_{0} has a faithful representation over the ring of 2-adic integers ℤ_{2} which has dimension pr(S^{∘}) ≥ 3, By tensoring with ℂ, W_{0} has a faithful representation R over ℂ which has dimension pr(S^{∘}) ≥ 3. The r_{L}’s continue to act as reflections in this representation.

Now suppose towards a contradiction that W_{0} acts reducibly on R. Since the representation R is completely reducible, R = R_{1} ⊕R_{2} where R_{1} and R_{2} are proper W_{0}-invariant subspaces.

Suppose that W_{0} acts trivially on R_{i}. Then there is a 2-torus R̂_{i} centralized by all r_{L}, and G ≤ C_{G}(R̂_{i}), a contradiction. So we may assume that W_{0} acts non-trivially on both R_{1} and R_{2}.

For L ∈ Σ, the −1-eigenspace [R, r_{L}] of r_{L} belongs to one of the two subspaces, either R_{1} or R_{2}. So r_{L} acts as a reflection on that subspace and centralizes the other. Let Σ_{i} := {L ∈ Σ : [R, r_{L}] ≤ R_{i}} for i = 1, 2. For L ∈ Σ_{1} and K ∈ Σ_{2}, we have [r_{L}, r_{K}] = 1, and thus [L, K] = 1 by Lemma 3.15, in contradiction with the fact that Σ is connected.

To further constrain W_{0}, we next obtain representations of W_{0} over almost all finite fields.

Lemma 3.19 cf. [8, §3.4]. For primes q > max(|W_{0}| , 3) with q ≠ char(L) for any L ∈ Σ, W_{0} has a faithful irreducible representation over ℤ/qℤ, where, for any L ∈ Σ, the involution r_{L} acts by reflection.

Proof. Consider the elementary abelian q-group E_{q} generated by all elements of order q in T. W_{0} clearly acts on E_{q}. Let N = N_{G}(T). Since C_{G}(T) ≤ C_{N}(E_{q}), we may show that W_{0} acts faithfully by showing that C_{N}(E_{q}) ≤ C_{G}(T).

For any x ∈ C_{N}(E_{q}) ≤ N, x acts on Σ by conjugation. For any L ∈ Σ, if L^{x} ≠ L then L and L^{x} either commute or generate a quasisimple group as root SL_{2}-subgroups by Lemma 3.11-2. In either case, |L ∩ L^{x}| ≤ 2, in contradiction to the fact that L ∩ E_{q} = L^{x} ∩ E_{q}. So x normalizes L, and the element x acts on T ∩ L as an element of N_{L}(T ∩ L) by Fact 1.6. Since the Weyl group of SL_{2} inverts the torus, any element of N_{L}(T ∩ L) \ C_{L}(T ∩ L) inverts some element of E_{q}. So x centralizes T ∩ L for all L ∈ Σ. Now x centralizes T = ⟨T ∩ L|L ∈ Σ⟩, and W_{0} acts faithfully on E_{q}.

We also observe that W_{0} acts by reflections on E_{q} because, for every L ∈ Σ, [E_{q}, r_{L}] has order q and is inverted by r_{L}, i.e. |E_{q} ∩ L| = q.

signalizers and balance 25

Now suppose toward a contradiction that W_{0} acts reducibly on E_{q}. Since q > |W_{0}|, the representation is completely reducible, and E_{q} = R_{1} ⊕ R_{2} where R_{1} and R_{2} are proper W_{0}-invariant subspaces of E_{q}.

Suppose that W_{0} acts trivially on R_{i}. For any L ∈ Σ, R_{i} acts by inner automorphisms on L by Fact 1.6. We recall that nontrivial Weyl group elements in W (L) invert the torus T_{L}. Since R_{i} centralizes T_{L}, we know that R_{i} acts via conjugation by elements of T_{L}. Since W_{0} centralizes R_{i}, we find that R_{i} centralizes L. So R_{i} ≤ Z(⟨Σ⟩) = Z(G), a contradiction. So we may assume that W_{0} acts nontrivially on both R_{1} and R_{2}.

For L ∈ Σ, the eigenspace [E_{q}, r_{L}] of r_{L} belongs to one of the two subspaces, either R_{1} or R_{2}. So r_{L} acts as a reflection on that subspace and centralizes the other. Let Σ_{i} := {L ∈ Σ : [E_{q}, r_{L}] ≤ R_{i}} for i = 1, 2. Then Σ = Σ_{1} ∪ Σ_{2}. For L ∈ Σ_{1} and K ∈ Σ_{2}, we have [r_{L}, r_{K}] = 1, and thus [L, K] = 1 by Lemma 3.15, in contradiction with the fact that Σ is connected.

The two preceding lemmas provide sufficient information to identify the Weyl group W_{0}.

Lemma 3.20 cf. [8, Lemma 3.11]. There exists an irreducible root system I of type A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, or F_{4} on which W_{0} acts as a crystallographic reflection group.

This lemma follows from the following major fact, which depends on a detailed analysis of the irreducible complex reflection groups [34, 23].

Fact 3.21 [7, Thm. 2.3]. Let W be a finite group, I ⊆ W a subset, and n an integer, satisfying the following conditions.

(i) The set I generates W, consists of involutions, and is closed under conjugation in W;

(ii) The graph Δ_{I} with vertices I and edges (i, j) for noncommuting pairs i, j ∈ I

is connected; (iii) For all sufficiently large prime numbers ℓ, W has a faithful representation V_{ℓ}

over the finite field 𝔽_{ℓ} in which the elements of I operate as complex reflections, with no common fixed vectors.

Then one of the following occurs.

(a) W is a dihedral group acting in dimension n = 2, or cyclic of order two.

(b) W is isomorphic to an irreducible crystallographic Coxeter group, that is, A_{n}, B_{n}, C_{n}, D_{n} (n ≥ 3), E_{n} (n = 6, 7,or 8), or F_{n} (n = 4),

(c) W is a semidirect product of a quaternion group of order 8 with the symmetric group Sym_{3}, acting naturally, represented in dimension 2.

If, in addition, over some field, W has an irreducible representation of dimension at least 3, in which the elements of I act as reflections, then case (b) applies.

Proof of Lemma 3.20. We observe that {r_{L} : L ∈ Σ} is a normal subset of W_{0} which generates W_{0}. The noncommuting graph on this set is connected by Lemma 3.15. So Lemmas 3.19 and 3.17, complete the verification of the hypotheses of Fact 3.21.

We also show that all reflections in W_{0} come from our root SL_{2}-subgroups.

Lemma 3.22 cf. [8, Lemma 3.12]. Every r ∈ W_{0} which is a reflection in the representation R over ℂ has the form r_{K} for some K ∈ Σ.

Recall that the reflections of a Coxeter group correspond to roots in the associated root system (see [29, Lemma 5.7]), and hence there are at most two conjugacy classes of reflections.

Fact 3.23 [28, 10.4 Lemma C]. A finite irreducible reflection group of type A_{n}, D_{n}, E_{6}, E_{7}, or E_{8} has only one conjugacy class of reflections. A finite irreducible reflection group of type B_{n}, C_{n}, F_{4}, and G_{2} has two conjugacy classes of reflections, corresponding to the short and long roots.

Since the roots of only one length are closed under the action of the Coxeter group, they form the root system for a proper subgroup.

Fact 3.24. The subgroup of B_{n}, C_{n}, F_{4}, or G_{2} generated by the reflections associated to roots of only one length is a proper subgroup.

Proof of Lemma 3.22. By Fact 3.23, there are at most two conjugacy classes of reflections in I, corresponding to the short and long roots. So we may assume that I has more than one root length, i.e. W_{0} ≅ B_{n}, C_{n}, or F_{4}, and that the set S := {r_{L} : L ∈ Σ} consists of only one of these conjugacy classes. By Fact 3.24, ⟨S⟩ < W_{0}, a contradiction.

3.3. Identification

We continue the analysis of the preceding subsections, loosely following [8, §3.6]. We will invoke the Curtis-Tits theorem which may be expressed as follows: a simply connected quasisimple algebraic group is the free amalgam of the system of subgroups and inclusion maps corresponding to all root SL_{2} subgroups and subgroups generated by pairs of such subgroups, taken relative to a fixed maximal torus [25]. The Generic Identification Theorem of Berkman and Borovik proceeds by passing from the full system of groups and subgroups to the collection of subsystems corresponding to pairs of roots, which are now known. A flexible form of this result is based on a result of Timmesfeld [36].

Fact 3.25 [7, Prop. 2.3]. Let Φ be an irreducible root system (of spherical type) and rank at least 3, and let Π be a system of fundamental roots for Φ. Let X a group generated by subgroups X_{r} for r ∈ Π, Set X_{rs} = ⟨X_{r}, X_{s}⟩. Suppose that X_{rs} is a group of Lie type Φ_{rs} over an infinite field, with X_{r} and X_{s} corresponding root SL_{2}- subgroups with respect to some maximal torus of X_{rs}. Then X/Z(X) is isomorphic to a group of Lie type via a map carrying the subgroups X_{r} to root SL_{2}-subgroups.

We now conclude the proof of the Generic Trichotomy Theorem, working, as usual, under Hypothesis 3.12. By Lemma 3.20, I is the desired irreducible root system of spherical type and rank at least 3. For every vertex i ∈ I, there is an r_{i} ∈ W_{0} which is a reflection in the representation R over ℂ. There is a also reflection

signalizers and balance 27

r_{L} for every L ∈ Σ, By Lemma 3.22, there is an L_{i} ∈ Σ such that r_{i} = r_{L_{i}}, and

⟨L_{i}|i ∈ I⟩ = ⟨Σ⟩ = G.

For i, j ∈ I, the group M := ⟨L_{i}, L_{j}⟩ is of Lie type by Lemma 3.11-2 when [L_{i}, L_{j}] ≠ 1. If [L_{i}, L_{j}] = 1 then M = L_{i} ∗ L_{j} which has Lie type because L_{i} and L_{j} are algebraic over the same field. By Lemma 3.11-4, L_{i} and L_{j} are root SL_{2}-subgroups corresponding to a maximal torus T_{M} of M. Now G is a Chevalley group by Fact 3.25, as desired.

Acknowledgments

I thank my adviser Gregory Cherlin for direction during my thesis work, of which this article is an outgrowth, and for guidance during its later developments. I am grateful to Tuna Altınel for his careful reading of this material, and numerous corrections and suggestions. I am also grateful to my collaborators on [14] and [19], without whom this article would merely provide the Generic Trichotomy Theorem, as well as Alexandre Borovik and Ayşe Berkman for advice, and allowing me to use their arguments from [8].

Financial support for this work comes from NSF grant DMS-0100794 and DFG grant Te 242/3-1. I also thank IGD at Université Lyon I for their hospitality.

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Jeffrey Burdges School of Mathematics, The

University of Manchester PO Box 88, Sackville St., Manchester

M60 1QD, England

Jeffrey.Burdges@manchester.ac.uk