Contributions to the Classification of Tame K^{∗}-groups of Odd Type

and Other Applications of 2-local Theory

Christine Altseimer

Dissertation

zur Erlangung des Doktorgrades der Mathematischen Fakultät der Albert-Ludwigs-Universität

Freiburg im Breisgau

August 1998

Dekan der mathematischen Fakultät: Prof. Dr. Ludger Rüschendorf

Erster Referent: Prof. Dr. Martin Ziegler

Zweiter Referent: Prof. Dr. Alexandre V. Borovik

Datum der Promotion:

5. November 1998 Wer fertig ist, dem ist nichts recht zu machen,

ein Werdender wird immer dankbar sein. (J. W. von Goethe)

Acknowledgments

My most grateful thanks go to Professor Alexandre Borovik at UMIST, who was my supervisor during the academic year 1996/97, and without whom nothing of this work would have been possible. He proposed the topics of the dissertation and was always there to give his ideas and advice freely. I warmly remember all our mathematical discussions at different places in and around Manchester, which gave me a deeper understanding of mathematical structures and a great enthusiasm to explore them.

I also thank Professor Martin Ziegler, who gave me the freedom to write a dissertation that is only losely connected with his mathematical interests, and who still gave me all the support I needed from him during these last years.

From time to time I was also able to discuss my work with other people, and I would like to thank them all in this place. Namely there were Laci Babai, Eamonn O’Brien, Markus Junker, William Kantor, Charles Leedham-Green, Frank Lübeck, Jürgen Müller and Akos Seress.

For the financial support during the last two years, I thank the Landesgraduiertenförderung Baden-Württemberg and the German Academic Exchange Service (DAAD, HSPII/AUFE).

Zusammenfassung

Involutionen spielen innerhalb der Gruppentheorie eine besonders interessante Rolle. Zum einen sind sie die denkbar einfachsten Studienobjekte und sind uns allen als Spiegelungen vertraut, zum anderen tragen die von ihnen definierten Strukturen einen sehr hohen Informationswert. So wurden beispielsweise viele endliche einfache Gruppen mit Hilfe der Zentralisatoren ihrer Involutionen charakterisiert und, im Falle der sporadischen Gruppen, gefunden; halbeinfache komplexe Lie Algebren lassen sich durch ihr Wurzelsystem klassifizieren, ein Wurzelsystem ist aber im wesentlichen nichts anderes als ein geschlossenes Spiegelsystem.

Diese Arbeit liefert eine Vielzahl von weiteren Beispielen, wie Involutionen die Struktur einer Gruppe bestimmen. Im Mittelpunkt steht dabei die 2-lokale Analyse von Gruppen, d.h. das Studium von Normalisatoren elementar-abelscher 2-Untergruppen.

Zunächst geht es um die Klassifikation der einfachen Gruppen von endlichem Morley-Rang. Gruppen von endlichem Morley-Rang können dabei als Gruppen mit einer “Dimension” [18] verstanden werden, d.h. als Verallgemeinerung algebraischer Gruppen. G. Cherlin [32] vermutete, daß die zwei Klassen von Gruppen im wesentlichen zusammenfallen und unendliche einfache Gruppen von endlichem Morley Rang algebraische Gruppen über algebraisch abgeschlossenen Körpern sind. Ein Schritt zum Beweis der Cherlinschen Vermutung ist die Klassifikation der zahmen Gruppen von endlichem Morley-Rang; das sind Gruppen, bei denen gewisse, modelltheoretische, Konstellationen von vorneherein ausgeschlossen werden. Der grundlegende Ansatz ist, mit einem minimalen Gegenbeispiel zur Cherlinschen Vermutung, einer K^{∗}-Gruppe, zu arbeiten. Zahme K^{∗}-Gruppen von endlichem Morley-Rang zerfallen natürlicherweise in zwei Klassen, die Gruppen von geradem und ungeradem Typ, je nachdem, ob die Sylow 2-Untergruppen denen der algebraischen Gruppen über einem Körper der Charakteristik 2 oder ungleich 2 gleichen.

Das Hauptresultat der Dissertation ist die Charakterisierung einfacher algebraischer Gruppen von Rang 2 über einem algebraisch abgeschlossenen Körper der Charakteristik ungleich 2 in der Klasse der zahmen K^{∗}-Gruppen von endlichem Morley-Rang mit Hilfe von Zentralisatoren von Involutionen in Kapitel 4. So werden wir beweisen, daß eine einfache zahme K^{∗}-Gruppe G von endlichem Morley-Rang isomorph zu PSp_{4}(K) oder G_{2}(K) ist, falls G eine Involution i enthält, so daß C_{G}(i)^{∘}/O(C_{G}(i)) entweder isomorph zu SL_{2}(K) ∗ SL_{2}(K) oder zu PSL_{2}(K) × K^{∗} für einen algebraisch abgeschlossenen Körper K von Charakteristik 0 oder Charakteristik p für eine Primzahl p ≠ 2, 3 ist.

Die Kapitel 2 und 3 dienen der technischen Vorbeitung dieser Sätze. In Kapitel 2 untersuchen wir quasiunipotente Gruppen, die in der Theorie der Gruppen von endlichem Morley-Rang die Rolle unipotenter Untergruppen einer algebraischen Gruppe übernehmen. Wir können zeigen, daß eine zahme auflösbare Gruppe G von endlichem Morley-Rang eine eindeutig bestimmte maximale quasiunipotente Untergruppe besitzt. Der zentrale Satz in Kapitel 3 besagt, daß eine stark eingebettete Untergruppe einer zahmen einfachen K^{∗}-Gruppe G entweder auflösbar ist, oder G von ungeradem Typ und Prüfer 2-Rang 1 ist. Mit Hilfe dieses Satzes kann das Auftreten bestimmter Konfigurationen in zahmen einfachen K^{∗}-Gruppen von ungeradem Typ ausgeschlossen werden.

Der zweite Teil der Arbeit beschäftigt sich mit der Frage, wie man zwei endliche einfache Gruppen unterscheiden kann, wenn man nur die Ordnungen von einer Menge zufälliger Elemente der Gruppe und von deren Produkten kennt. Die bisherigen Algorithmen haben im wesentlichen die Statistik der Elementordnungen benutzt. Der einzige Fall, der sich mit dieser Methode nicht behandeln läßt, ist die Unterscheidung zwischen P Sp_{2n}(q) und Ω_{2n+1}(q), da diese beiden Gruppen nicht nur diesselbe Ordnung haben, sondern im Prinzip auch dieselbe Statistik von Elementordnungen. Im 6. Kapitel stellen wir einen Algorithmus vor, der dieses Problem löst, indem er die Unterschiede in der Struktur von Konjugiertenklassen von Involutionen in diesen beiden Gruppen nützt. In Kapitel 5 werden hierzu die Ordnungen von stark realen Elementen in diesen beiden Gruppen bestimmt.

Contents

Introduction 1

1 Preliminaries 6

1.1 Groups of Finite Morley Rank . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.2 Tame K^{∗}-Groups . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Chevalley Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Finite Groups of Lie type . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Probabilistic Algorithms . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Black box groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

I Groups of Finite Morley Rank 21

2 Quasiunipotent Groups 23 2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Maximal quasiunipotent groups . . . . . . . . . . . . . . . . . . . 28 2.3 Normalizers of quasiunipotent groups . . . . . . . . . . . . . . . . 31

3 2-local Analysis 35 3.1 Strongly embedded subgroups . . . . . . . . . . . . . . . . . . . . 35 3.2 Thompson Rank Formula . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Structure of G . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.2 Computing r_{i} and r_{j} . . . . . . . . . . . . . . . . . . . . . 49

4 Groups of Small Prüfer 2-rank 52

4.1 PSL_{3}(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1.1 Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.2 3- and 2-dimensional case . . . . . . . . . . . . . . . . . . 59 4.1.3 1-dimensional case . . . . . . . . . . . . . . . . . . . . . . 65 4.2 PSp_{4}(K) and G_{2}(K) . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.2 PSp_{4}(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.3 G_{2}(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

II Black Box Groups 95

5 Reflexive Tori 97 5.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Examples of reflexive tori . . . . . . . . . . . . . . . . . . . . . . . 101

6 Distinguishing B_{n} and C_{n} 107 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . 111

Introduction

Involutions of a group are special objects in many ways. On the one hand they are very easy to study and we all know them as reflections. On the other hand, the structure definable by involutions bears a great deal of information. Many finite simple groups have been characterized or, in the case of sporadic groups, even found by means of the centralizers of involutions. Semisimple complex Lie algebras can be classified by their root system, which is nothing else but a closed system of mirrors.

This thesis gives some new examples of how involutions determine the structure of groups. We will be mostly concerned with the 2-local analysis of groups, i.e. the study of normalizers of elementary abelian 2-subgroups. The first part of the thesis gives applications to groups of finite Morley rank, the second part an application to black box groups.

One of the great open problems in model theory is the Cherlin-Zil’ber conjecture. It states that an infinite simple group of finite Morley rank is isomorphic, as an abstract group, to an algebraic group over an algebraically closed field. This would imply that for simple groups the existence of a rank function obeying certain natural conditions would already suffice to obtain all its geometrical structure.

The original model theoretic approach to prove the Cherlin-Zil’ber conjecture was the following. Zil’ber has shown that we can interpret a field in a connected solvable non-nilpotent group of finite Morley rank. As it is of finite Morley rank such a field has to be algebraically closed by Macintyre’s theorem. Hence there is always an algebraically closed field K interpretable in a simple infinite group G of finite Morley rank as long as none of its proper sections is a bad group. Here a bad group is a non-solvable group of finite Morley rank all of whose proper definable and connected subgroups are nilpotent. K will in general bear extra structure which is induced by G. If this structure is definable in K itself, then K is said to be pure. Hrushovski has shown that if a field K is interpretable in a simple group of finite Morley rank, we can interpret the group in K. If K is pure, then the group is already algebraic by a result of Hrushovski again. Zil’ber actually conjectured that all fields of finite Morley rank are pure, which would imply the Cherlin-Zil’ber conjecture for simple groups of finite Morley rank that do not interpret a bad group. However, Hrushovski constructed a field of finite Morley rank that is not pure. It is not known up to now if bad groups or fields in which non-algebraic groups can be interpreted exist.

Since then a new approach was proposed by Borovik, Cherlin and Nesin. Their idea is to apply the methods from the classification of finite simple groups (CFSG) to the context of groups of finite Morley rank. They believe that CFSG actually consists of several more or less independent theories. One theory, “sporadic”, deals with “sporadic objects”, the other “generic theory”, describes structural properties of the majority of finite simple groups, namely, large groups of Lie type. If G is an affine algebraic group and X ⊆ G an affine subvariety, then the the logarithm of the number of GF (q)-points of X is approximately dim X by a Lang-Weil type estimate [31]. If, for example, the center of G has dimension m, then the center of the corresponding finite group over GF (q) has order approximately q^{m}. This enables one to some extent to deduce properties of finite simple groups from properties of simple algebraic groups. On the other hand much of the theory of groups of finite Morley rank is inspired be the part of CFSG that deals with large groups of Lie type.

To actually make this approach work we have to assume that no bad fields are interpretable in the group. A bad field is a ranked structure of the form ⟨K, +, ·, A⟩, where K is an algebraically closed field and A is a proper infinite multiplicative subgroup of K^{∗}. If we call a group of finite Morley rank tame if none of its proper sections is a bad group and if it does not interpret a bad field, then the Cherlin-Zil’ber conjecture splits into three parts.

Conjecture 1 Bad groups do not exist.

Conjecture 2 Bad fields do not exist.

Conjecture 3 A simple tame group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field.

While a proof of the conjectures 1 and 2 is most likely to involve heavy model theory, it is hoped that the proof of Conjecture 3 can be achieved using ideas from finite group theory. The first of these is to analyse a minimal counterexample to Conjecture 3. A group of finite Morley rank is called a K-group, if every infinite simple definable and connected section of the group is an algebraic group over an algebraically closed field. A K^{∗}-group is a group of finite Morley rank in which every proper definable subgroup is a K-group. In this inductive setting the conjecture reduces to the following:

Conjecture 4 An infinite simple tame K^{∗}-group is isomorphic to an algebraic group over an algebraically closed field.

It is known [3] that an infinite simple tame K^{∗}-group G is either of even type, meaning that its Sylow 2-subgroups are nilpotent, definable and of bounded exponent, or of odd type, meaning that the Sylow 2-subgroups are divisible abelian by finite. If G is of even type, Conjecture 4 would imply that G is isomorphic to an algebraic group over an algebraically closed field of characteristic 2. If G is of odd type, then G should be isomorphic to an algebraic group over an algebraically closed field of characteristic not 2. Depending on the type of the group, very different methods are required.

In this thesis we are dealing with the classification of simple tame K^{∗}-groups of odd type. This case splits into three parts: Let G be a simple tame K^{∗}-group of odd type. Either G is small, meaning that normal 2-rank of the group is less than 3, or G contains a proper 2-generated core or G satisfies the so-called B-conjecture and contains a classical involution. The last case could be almost completely described by A. Berkman in [15]. The second case should not appear at all and we will prove a partial result pointing in this direction in Chapter 3. Since, however, the amount of information which we have in the first case is extremly small, this is the most difficult part in the classification of simple tame K^{∗}-groups of finite Morley rank. By Conjecture 4, we know what to expect.

Conjecture 5 An infinite simple tame K^{∗}-group G of odd type and normal 2rank n(G) ≤ 2 is isomorphic to one of the groups PSL_{2}(K), PSL_{3}(K), PSp_{4}(K), G_{2}(K) over some algebraically closed field K of characteristic not 2.

We are mainly concerned with first approaches to prove this conjecture. In analogy with the CFSG, we attempted a characterization of the groups in question with the help of centralizers of involutions.

Chapter 2 and 3 provide the technical machinery which is used to prove the main classification results. Chapter 2 is dedicated to a thorough discussion of the concepts of quasi- and pseudounipotent. These are generalizations of unipotent subgroups of algebraic groups to the context of groups of finite Morley rank. The main result is, that if G is a solvable group of finite Morley rank which does not interpret a bad field then there exists exactly one maximal pseudounipotent (quasiunipotent) subgroup H (U) of G and H and U are characteristic subgroups. Especially maximal pseudounipotent (quasiunipotent) subgroups of a K-group of finite Morley rank which does not interpret a bad field are conjugate.

In Chapter 3 we discuss strongly embedded subgroups of a simple K^{∗}-group of finite Morley rank and odd type that does not interpret a bad field. Strongly embedded subgroups should not exist in this context at all. As a first step we show that strongly embedded subgroups are solvable if the Prüfer 2-rank of G is greater than 1. This implies for example that G does not contain a proper 2-generated core, if G is of normal 2-rank greater than 2 and contains a nonsolvable centralizer of an involution. In Section 3.2 we give an application of the Thompson rank formula, which we already proved in Section 1.1. This formula allows one to compute the Morley rank of a group of finite Morley rank with at least two conjugacy classes of involutions just using the structure of centralizers of involutions.

The main classification results are proved in Chapter 4. We give a partial characterization for PSL_{3}:

Theorem Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that the centralizers of involutions of G are isomorphic to GL_{2}(K) for some algebraically closed field K of characteristic not 2. Then G is isomorphic to PSL_{3}(K), if there exists an involution i and a quasiunipotent subgroup U ≤ C_{G}(i) which is maximal in C_{G}(i) but not maximal in G.

For groups of larger dimension we achieve a better result.

Theorem Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that G contains an involution i, such that either

(i) C_{G}(i)^{∘}/O(C_{G}(i)) ≅ SL_{2}(K) ∗ SL_{2}(K), where the two copies of SL_{2}(K) intersect nontrivially or

(ii) C_{G}(i)^{∘}/O(C_{G}(i)) ≅ PSL_{2}(K) × K^{∗}

for an algebraically closed field K of characteristic neither 2 nor 3. Then G ≅ PSp_{4}(K) or G ≅ G_{2}(K).

The second part of this thesis is concerned with algorithms for black box groups. A black box group G is a finite group whose elements are encoded as 0-1 strings of uniform length N, and the group operations are performed by an oracle (the “black box”) at unit cost. Furthermore a set S of generators is given in this encoding. The main question is how much information about the group can be gained in time polynomial in N. Ideally one would like to be able to compute the composition series of the group or to identify all the simple components. This will in general be an unfeasible task.

If we, for example, consider a finite group G of Lie type of dimension n over a field of q elements, then G can be considered as a black box group since elements can be encoded by a tuple of length N := n^{2} log(q). Assume that q = 2^{m} for some large m. Then – by the Lang-Weil type estimate already mentioned – the share of 2-elements in G is about 1/q^{2} and the share of elements of even order is about 1/q. If we thus generate elements of G randomly using the generating set S, we cannot expect to generate an element of even order in time polynomial in n^{2} log(q). Hence G is a simple group that appears to be solvable by the Feit-Thompson theorem in our setting.

A more feasible task is to try to identify a black box group G, given the promise that it is simple. Much progress has been made recently to solve this problem. Most algorithms which have been developed in this context use the additional assumption, that the orders of elements of G are given.

The algorithms proposed so far are mainly based on the statistics of element orders of G. The crucial case was thus to distinguish PSp_{2n}(q) from Ω_{2n+1}(q) for q odd and n ≥ 3, since not only the orders of these two groups coincide, but also the statistics of orders of elements is for all practical purposes the same in both groups. In Chapter 6 we present an algorithm to solve this problem. The algorithm uses the difference in the 2-local structure of symplectic and orthogonal groups. This is feasible in our context since at least one quarter of the elements of these groups has even order [39] and we can thus generate involutions in polynomial time.

Chapter 5 develops the theoretical background for this problem by investigating orders of strongly real elements in the two groups.

Chapter 1

Preliminaries

1.1 Groups of Finite Morley Rank

Since [18] and [19] give a very good and easily accessible introduction to groups of finite Morley rank, we are only going to introduce the most important notions and to discuss the recent developments. The original source of a result is not given if it can also be found in one of these two works. Citations are made accordingly. A collection of the basic properties of groups of finite Morley rank can also be found in [6].

1.1.1 Basic Properties

There are two equivalent ways to define groups of finite Morley rank. The first one is the original model theoretic definition by means of a topological rank on the Stone-space associated to the Boolean algebra of definable subsets of a model. The second one the axiomatic description as groups with a “dimension” by Borovik. Details of how these two definitions connect can be found in [50] and in [6] as well.

Note that the term “definable” always refers to definability in some ranked universe U (see [18] for definitions and discussion). Model-theoretically speaking a set definable in a group G= ⟨G, ∗, 1, ⋯ ⟩ is a boolean combination of sets of the form A/ ≡ where A ⊂ G^{n} for some n ∈ ℕ and ≡ is definable in G.

One of the most important group theoretic consequences is that a group G of finite Morley rank satisfies the descending chain condition for definable subgroups by [18, 5.2]. This yields that a group G of finite Morley rank contains a unique minimal definable subgroup of finite index, which is called the connected component G^{∘} of G. G is called connected if G = G^{∘}. As a further consequence every subset X of G is contained in a minimal definable subgroup, the definable closure d(X) of X. Note, that in general d(X) ≠ dcl(X). For any subgroup Y of G we call Y^{∘} := Y ∩ d(Y)^{∘} the connected component of Y .

Crucial for the understanding of groups of finite Morley rank is the 2-structure of them. Sylow 2-subgroups of a group G of finite Morley rank are conjugate to each other by [18, 10.11]. If S is one of them, then S^{∘} = B ∗ D is a central product of a definable connected nilpotent subgroup B of bounded exponent and a divisible abelian group D by [18, 6.22]. B and D are characteristic in S. Thus S is nilpotent-by-finite and Z(S) ≠ 1 by [18, exc. 12, p. 14]. We say that G is of even type if D = 1 and that G is of odd type if B = 1.

Involutions can also be used for rank estimates. Let G be any group. Elements of order 2 are called involutions and the set of all involutions is denoted by I(G). Let x ∈ G be a nontrivial element. x is called strongly real if there exists an involution i ∈ G such that x^{i} = x^{−1}. Let Sr(G) be the set of strongly real elements of G. For x ∈ Sr(G) we denote by C_{G}^{∗}(x) the generalized centralizer of x in G, where C_{G}^{∗}(x) := {g ∈ G : x^{g} ∈ {x, x^{−1}}}. We have the following analogue to the Brauer-Fowler theorem.

Fact 1.1 (see [18] and also [6]) Let G be a group of finite Morley rank containing an involution i. Then there exists a strongly real element x = i^{g}i^{h} for some g, h ∈ G such that

rk(G) ≤ 2 rk(C_{G}(i)) + k(x) ≤ 2 rk(C_{G}(i)) + rk(C_{G}(x))

where k(x) := rk(I(C_{G}(x)) if x ∈ I(G) and k(x) := rk(I(C_{G}^{∗}(x)\I(C_{G}(x))) otherwise.

If a group G of finite Morley rank contains at least two conjugacy classes of involutions, the Morley rank of G can be computed locally by the following formula. The underlying fact is, that if two involutions i, j ∈ G are not conjugate, then there exists an involution in d(ij) by [18, 10.2].

Proposition 1.2 (Thompson Rank Formula) Let G be a group of finite Morley rank with at least two distinct conjugacy classes C_{1} = u_{1}^{G} and C_{2} = u_{2}^{G} of involutions. Assume that there exists a definable function f : C_{1}C_{2} ⟶ I(G) such that f(x^{g}) = f(x)^{g} for all x ∈ C_{1}C_{2} and g ∈ G and

f(x) ∈ {g ∈ G : g^{y} = g^{−1} for all y ∈ I(C_{G}^{∗}(x)\C_{G}(x))}.

Let ρ(i) = rk({(c_{1}, c_{2}) ∈ C_{1} × C_{2} : f(c_{1}c_{2}) = i}) for all i ∈ I(G). If X_{1}, ⋯, X_{k} is a definable partition of I(G) into sets such that ρ is constant on each X_{j} then

2 rk(G) = C_{G}(u_{1}) + C_{G}(u_{2}) + max_{1≤j≤k} (ρ(j) + rk(X_{j}))

If G has finitely many conjugacy classes of involutions, we may take the X_{j} to be these classes. In the case where G contains exactly two conjugacy classes of involutions we get the simple equation

rk(G) = max {r_{1} + rk(C_{G}(u_{2})), r_{2} + rk(C_{G}(u_{1}))}

where r_{j} := rk({(κ, λ) : κ ∈ C_{1} ∩ C_{G}(u_{j}), λ ∈ C_{2} ∩ C_{G}(u_{j}) and f(κλ) = u_{j}}) for j = 1, 2.

Proof. Define a function φ : C_{1} × C_{2} ⟶ I(G) such that φ(λ, κ) = f(λκ). Then φ is definable and rk(φ^{−1}(i)) = ρ(i). By [1, 6.1]

rk(C_{1} × C_{2}) = max_{1≤j≤k}{ρ(j) + rk(X_{j})}.

Since rk(C_{j}) = rk(G)−rk(C_{G}(u_{j})) for j = 1, 2 and rk(C_{1}×C_{2}) = rk(C_{1})+rk(C_{2}) the first claim follows. Furthermore ρ is constant on conjugacy classes since f(x^{g}) = f(x)^{g}. Consider now

φ^{−1}(u_{j}) = {(κ, λ) ∈ C_{1} × C_{2} : f(κλ) = u_{j}}

for j = 1, 2. Since λ and κ invert λκ, u_{j} must be inverted as well by choice of f and hence λ, κ ∈ C_{G}(u_{j}). The final equation now follows easily, noting that k = 2, X_{1} = C_{1} and X_{2} = C_{2} in this case. □

Altınel [1] has shown that if G is a group of even type, then d(x) contains exactly on involution for all x ∈ C_{1}C_{2} and we can choose f to be the following function: f(x) = I(d(x)) for all x ∈ C_{1}C_{2}. If G is a group of odd type we should be able to construct such a function f in many cases as well if we have enough information about the conjugacy classes of involutions. This is no great restriction, however, since the Thompson rank formula can only be applied if we have precise information about the centralizers of involutions (compare [55, 5.1]). Section 3.2 gives an example of how the formula can be used.

1.1.2 Tame K^{∗}-Groups

This thesis is mainly concerned with tame K^{∗}-groups of finite Morley rank. A group of finite Morley rank is called tame if none of its proper sections is a bad group and if it does not interpret a bad field. A bad group is a non-solvable group of finite Morley rank all of whose proper definable and connected subgroups are nilpotent and a bad field is a ranked structure of the form ⟨K, +, ·, A⟩, where K is an algebraically closed field and A is a proper infinite multiplicative subgroup of K^{∗}. We call a group of finite Morley rank K-group, if every infinite simple definable and connected section of the group is an algebraic group over an algebraically closed field. A K^{∗}-group is a group of finite Morley rank in which every proper definable subgroup is a K-group. To work with K^{∗}-groups means to work with a minimal counter-example to the Cherlin-Zil’ber Conjecture. The most crucial distinction in the theory of tame K^{∗}-groups is the following:

Fact 1.3 ([3]) Let G be a simple tame K^{∗}-group. Then G is either of odd type or of even type.

We need to introduce some terminology. If E is a finite elementary abelian 2-group, its 2-rank m(E) is the minimal number of generators of E. If H is any subgroup of a group G of finite Morley rank and odd type, its 2-rank m(H) is the maximum of the 2-ranks of elementary abelian subgroups in H. Note that the 2-rank is well-defined, since elementary abelian subgroups of a group of odd type are finite. If H is definable and S is a Sylow 2-subgroup of H, then the normal 2-rank n(H) is the maximum of the 2-ranks of elementary abelian normal subgroups of S. Also, the Prüfer 2- pr(H) of a not necessarily definable subgroup H is the maximum of the Prüfer ranks of Prüfer 2-subgroups of H. (A Prüfer p-group is a product of a finite number of copies of the quasi-cyclic group ℤ(p^{∞}); the number of copies is called its Prüfer rank). Obviously pr(G) ≤ n(G) and if G is a simple algebraic group over an algebraically closed field of characteristic not 2, equality holds.

Let S be a Sylow 2-subgroup in a group G of finite Morley rank. We define the 2-generated core Γ_{S,2}(G) as the definable closure of the group generated by all normalizers N_{G}(U) of all subgroup U ≤ S with m(U) ≥ 2.

A group G of finite Morley rank will be called semi-simple , if G = G′ and G/Z(G) is the direct sum of n non-abelian simple subgroups for some n ∈ ℕ. If n = 1, then G is called quasi-simple . The only proper normal subgroups of a quasi-simple group are the central ones. Quasi-simple algebraic groups are traditionally called simple algebraic groups. By a component of G we mean a subnormal quasi-simple subgroup of G. Components are definable and there are only finitely many of them by [18, 7.10]. The layer of G, L(G), is the product of all components of G. It is definable and normal in G.

The Fitting subgroup F (G) is the group generated by all normal nilpotent subgroups of G. F (G) is nilpotent by [18, 7.3]. The generalized Fitting subgroup F^{∗}(G) is taken to be F^{∘}(G) ∗ L^{∘}(G). Then C_{G}^{∘}(F^{∗}(G)) ≤ F^{∗}(G). G satisfies the B-conjecture, if C_{G}^{∘}(i) = F^{∗}(C_{G}(i)) for any involution i ∈ G. Simple algebraic groups over an algebraically closed field of characteristic not 2 satisfy the Bconjecture.

Let i ∈ G be an involution. A component A ⊲ L(C_{G}(i)) is called intrinsic if i ∈ Z(A). An involution is called classical if its centralizer contains an intrinsic component isomorphic to SL_{2}(K) for an algebraically closed field K.

Fact 1.4 (Trichotomy Theorem,[19]) Let G be a simple tame K^{∗}-group of odd type. Then one of the following statements is true.

• n(G) ≤ 2.

• G has a proper 2-generated core.

• G satisfies the B-conjecture and contains a classical involution.

The largest part of this thesis will be concerned with first approaches to prove Conjecture 4 for the first case. There is furthermore the outline of a proof in [18] that the second case of the Trichotomy Theorem cannot occur if n(G) ≥ 3. We have proved this result in Corollary 3.15 under the additional assumptions that G contains a non-solvable centralizer of an involution.

Suppose now that z is a classical involution in a simple tame K^{∗}-group G of odd type and that M ≅ SL_{2}(K) is an intrinsic component in C_{G}(z). Let Ω stand for the collection of all conjugates of M in G. Form a graph D with vertex set Ω by joining a vertex J to L if [J, L] = 1.

Fact 1.5 ([15]) Let G be a simple tame K^{∗}-group of finite Morley rank and odd type which satisfies the B-conjecture and contains a classical involution. Assume that the graph D is connected. Then G is isomorphic to one of the following groups: PSL_{n}(K) for n ≥ 5, PSp_{2n}(K) for n ≥ 3, PSO_{n}(K) for n ≥ 9, E_{6}(K), E_{7}(K), E_{8}(K), F_{4}(K), where K is an algebraically closed field of odd or zero characteristic.

The case where D is not connected is more difficult, since it is going to involve small groups like PSL_{3}(K).

1.2 Algebraic Groups

We do presuppose a basic knowledge of affine algebraic groups over an algebraically closed field. The standard reference is [42], where all necessary facts can be found easily.

In our context an algebraic group will always be the group of points over an algebraically closed field K of a linear algebraic group defined over K. If G is a semisimple algebraic group or connected sovable non-nilpotent algebraic group, then the field K can be interpreted in the pure group G by Zil’ber. In other cases (first of all, in the case of commutative groups) we usually have to postulate that the field K is definable in the universe U.

Since we consider algebraic groups with a possibly richer structure inherited from the definable universe U, we have two different concepts of “closed” subsets: “Zarisiki closed” and “definable”. Therefore, if G is an algebraic group over a definable field K, Zariski closed subgroups are definable and the Zariski closure \overline{H} of a subgroup H < G contains the definable closure d(H). If G is an algebraic group without extra structure, definable subsets are definable in G as a pure group and hence already Zariski closed.

We also have two versions of the connected component: the definable connected component and the Zariski closed identity component, the former being contained in the latter. Furthermore the Zariski closure of a definable connected group is Zariski connected. In most cases both do coincide though the proof usually requires some work. If we speak about connected algebraic groups in the following, we always mean Zariski connected groups.

Another important concept which transfers to groups of finite Morley rank is the concept of a Borel group. A subgroup B ≤ G of a group of finite Morley rank is called a Borel subgroup if it is maximal among the connected definable solvable subgroups of G. If G is an algebraic group over a definable field K and B a Borel subgroup in the sense of algebraic group theory, then B^{∘} is a Borel subgroup in the above sense. If on the other hand B is a Borel subgroup in the above sense, the Zariski closure \overline{B} of B is a Borel subgroup in the sense of algebraic group theory by [42, 23.1]. Hence a Borel subgroup in the above sense has finite index in the Borel subgroup in the sense of algebraic group theory containing it.

Let furthermore G be any group. The group generated by all normal solvable subgroups σ(G) is called the solvable radical. If G is a group of finite Morley rank, then σ(G) is definable and solvable by [18, 7.3].

Proposition 1.6 Let G be a K-group of finite Morley rank and B_{1}, B_{2} be two Borel subgroups of G. Then B_{1} and B_{2} are conjugate.

Proof. We may assume, that G is connected. The proposition holds if G is solvable. Let therefore G be nonsolvable and σ the solvable radical of G. Then σ^{∘} ≤ B_{n} for n = 1, 2, as σ^{∘}B_{n} is a connected definable solvable subgroup of G. Set \overline{X} := Xσ/σ for all X ≤ G. Then \overline{G} is the direct product of finitely many simple algebraic groups A_{i} for 1 ≤ i ≤ m by [2]. Since σ is solvable, H ≤ G is solvable if and only if \overline{H} is solvable. Hence \overline{B}_{n} are definable connected solvable subgroups. Let P_{n} ≤ G such that \overline{P}_{n} is a Borel subgroup containing \overline{B}_{n} for n = 1, 2. Then P_{n} is a definable solvable subgroups containing B_{n} and P_{n}^{∘} = B_{n} for n = 1, 2. Especially \overline{B}_{n} = \overline{P_{n}^{∘}} = \overline{P}_{n} for n = 1, 2, as \overline{P_{n}^{∘}} is connected and rk(\overline{P_{n}^{∘}}) = rk(P_{n}^{∘}) − rk(P_{n}^{∘} ∩ σ) = rk(P_{n}) − rk(σ) = rk(\overline{P}_{n}) by Lascar’s equality. Thus \overline{B}_{n} is a Borel subgroups of \overline{G} for n = 1, 2. Consider the projections \overline{B}_{n}^{i} of \overline{B}_{n} onto A_{i} for n = 1, 2 and 1 ≤ i ≤ m. By the above discussion \overline{B}_{n}^{i} is the connected component of a Borel subgroup in the sense of algebraic group theory for all n = 1, 2 and 1 ≤ i ≤ m. As Borel subgroups of connected algebraic groups are conjugate by [42, 21.3], there are g_{i} ∈ A_{i} such that (\overline{B}_{1}^{i})^{g_{i}} = \overline{B}_{2}^{i} for 1 ≤ i ≤ m. Therefore we can conclude that (\overline{B}_{1})^{g} = \overline{B}_{2} for g := g_{1} ⋯ g_{m}. Hence there exists h ∈ G, such that B_{1}^{h} ≤ B_{2}σ. However (B_{2}σ)^{∘} = B_{2} and since B_{1} is connected B_{1}^{h} ≤ B_{2}. The claim follows since B_{1} is maximal. □

Chapter 2 will be essentially devoted to an accurate treatment of algebraic implications of the existence of an extra structure on an algebraic group. The resulting technical complications seem to be unavoidable in work with K^{∗}-groups of finite Morley rank, which explains the need for two different versions of the concept of an unipotent subgroup.

1.3 Chevalley Groups

All classical algebraic groups over a field K are Chevalley groups, i.e. they can be defined as the group of automorphism of a simple Lie algebra L_{K} over K. However, semisimple Lie algebras can be characterized by means of a root system by [41]. For the following compare [25] and [43].

Let V be a Euclidean space of dimension l. We denote by w_{r} the reflection in the hyperplane orthogonal to r. A finite subset Φ of non-zero vectors that spans V is called a root system in V, if it satisfies the following conditions: w_{r}(s) ∈ Φ and 2(r, s)/(r, r) ∈ ℤ for all r, s ∈ Φ and additionally if r, λr ∈ Φ for some λ ∈ ℝ, then λ = ±1. The elements of Φ are called roots. The Weyl group W (Φ) of Φ is the group generated by all reflections w_{r} for r ∈ Φ.

A subset Δ of Φ is called fundamental system, if it is linearly independent and every root in Φ is a linear combination of roots in Δ with coefficients which are either all non-negative or all non-positive. Fundamental systems always exist and their elements are called simple roots. The rank of a root system is the number of simple roots.

If W := W (Φ) is finite, we can characterize W by its Dynkin diagram Γ. The vertex set of Γ is the set of simple roots for some fixed fundamental system Δ. Let m(α, β) := o(w_{α}w_{β}) for α ≠ β. Then m(α, β) ∈ {2, 3, 4, 6} and a pair of vertices α ≠ β is joined by a single edge if m(α, β) = 2, 3, by a double edge if m(α, β) = 4 and a triple edge if m(α, β) = 6. If Γ is connected, we call Φ irreducible. If Φ is irreducible at most two root lengths are possible and we label the graph by directing an arrow from long to short roots.

Fact 1.7 ([41]) If Φ is an irreducible root system of rank n, its Dynkin diagram is one of the following:

[figure: A_{n}, n ≥ 1; …]

[figure: B_{n}, n ≥ 3; …]

[figure: C_{n}, n ≥ 3; …]

[figure: D_{n}, n ≥ 4; …]

[figure: G_{2}]

[figure: F_{4}]

[figure: E_{6}]

[figure: E_{7}]

[figure: E_{8}]

If we associate to every semisimple Lie algebra L a root system in the natural way, then two semisimple Lie algebras over the same field are isomorphic if and only if the corresponding Dynkin diagrams have the same type.

Fact 1.8 ([25]) Let G be a Chevalley group over the Lie algebra L(K), where K is any field.

(i) If L is of type A_{n} for n ≥ 1, then G is isomorphic to P SL_{n+1}(K).

(ii) If L is of type B_{n} for n ≥ 2, then G is isomorphic to P Ω_{2n+1}(K, f), where f is the quadratic form x_{0}^{2} + x_{1}x_{−1} + ⋯ x_{n}x_{−n}.

(i) If L is of type C_{n} for n ≥ 3, then G is isomorphic to P Sp_{2n}(K).

(ii) If L is of type D_{n} for n ≥ 4, then G is isomorphic to P Ω_{2n}(K, f), where f is the quadratic form x_{1}x_{−1} + ⋯ x_{n}x_{−n}.

Furthermore we can associate to every reductive algebraic group a Dynkin diagram, which is irreducible if G is simple. A graph automorphism of G is an automorphism of G which is induced by a nontrivial bijection on the Dynkin diagramm of G.

Fact 1.9 (see e.g. [37]) Let G be a simple algebraic group that allows a nontrivial graph automorphism. Then G is of type A_{n} (n ≥ 2), D_{n} (n ≥ 4) or E_{6}. Furthermore Γ ≤ S_{3} if G is of type D_{4} and Γ ≤ ℤ_{2} in all other cases.

Let Φ be a root system of rank n, Δ a fundamental system. The ℤ-span P of Φ in V is called the root lattice. The root lattice is a free abelian group of rank n with basis Δ. Let Ξ be the group of K-characters of P for some field K. Then χ_{r,λ} ∈ Ξ for all r ∈ Φ, λ ∈ K^{∗}, where

χ_{r,λ}(a) = λ^{2(r,a)/(r,r)} for all r ∈ P .

Let X be the subgroup of Ξ generated by χ_{r,λ} for all r ∈ Φ, λ ∈ K^{∗}. If K is algebraically closed, then X is isomorphic to a maximal torus T of the corresponding Chevalley group G. Furthermore W (Φ) ≅ N_{G}(T)/T, the Weyl group of G. Thus statements about the action of the Weyl group of G on a maximal torus T can be translated into statements about characters of the root lattice and vice versa. One example is the following fact. Let h : X ⟶ T be the isomorphism between X and T and n : N_{G}(T) ⟶ W (Φ) the homomorphism with kernel T, which induces the isomorphy between the two Weyl-groups.

Fact 1.10 ([25]) Let g ∈ N_{G}(T), t ∈ T, then gtg^{−1} = h(χ), where χ(r) = h^{−1}(t)(n(g)^{−1}(r)) for all r ∈ Δ.

Let finally H_{r} ≤ T the subgroup of T generated by h(χ_{r,λ}), λ ∈ K^{∗}. One can associate, in a unique way, two one-dimensional unipotent subgroup X_{r} and X_{−r} of G with H_{r} such that P := ⟨X_{r}, X_{−r}⟩ ≥ H_{r} and P is either isomorphic to SL_{2}(K) or to P SL_{2}(K). Furthermore H_{r} normalizes X_{r} and X_{−r} and there exists an element g ∈ P such that n(g) = w_{r}. We call X_{r} and X_{−r} root subgroups of G.

1.4 Finite Groups of Lie type

The key to the study of finite groups of Lie type is that they are the fixed points of a special endomorphism, a Frobenius map, where G is a connected reductive algebraic group. A thorough discussion of this concept can be found in [27].

Let G be a linear algebraic group over an algebraically closed field of characteristic p. A homomorphism F : G ⟶ G is called a standard Frobenius map, if there exists an injective homomorphism i : G ⟶ GL(n, K) for some n ∈ ℕ, such that i(F (G)) = F_{q}(i(g)) for some q = p^{e} and all g ∈ G where F_{q}(a_{ij}) = (a_{ij}^{q}). A homomorphism F : G ⟶ G is called a Frobenius map if some power of F is a standard Frobenius map.

Notice that a Frobenius map is a surjective homomorphism of algebraic groups and that the set of fixed points G^{F} is a finite subgroup of G. Especially (G, F) is a σ-setup in the sense of Steinberg [52]. The basis of the theory is the following proposition:

Fact 1.11 ([52]) Let G be a connected algebraic group and σ an endomorphism from G onto G with finitely many fixed points. Then the map g ⟶ g^{−1}σ(g) is surjective.

This setup allows us to deduce properties of finite groups of Lie type from analogous properties of algebraic groups. One of the key tools is the following Fact. Let A be a group on which F acts. H^{1}(F, A) denotes A modulo the equivalence relation ≡, where a ≡ b iff a = cbF (c)^{−1} for some c ∈ A. If a, b ∈ A are equivalent we say that they are F -conjugate.

Fact 1.12 ([52]) Let G be a connected linear algebraic group, F a Frobenius map, and M be a nonempty set, on which G and F act. If G permutes the elements of M transitively and F (gm) = F (g)F (m) for all g ∈ G, m ∈ M, then M contains a point m fixed by F. If A := C_{G}(m) is a Zariski-closed subgroup of G then G^{F} has |H^{1}(F, A)| orbits on M^{F}. Furthermore |H^{1}(F, A)| = |H^{1}(F, A/A^{∘})|.

If we apply this proposition to the set M of maximal tori of G, then we get the following result.

Corollary 1.13 Let G be a connected reductive group and F a Frobenius map. G contains a maximal torus T fixed by F. There are |H^{1}(F, W)| classes of maximal tori fixed by F under conjugation by G^{F} where W = N_{G}(T)/T is the Weyl-group of G.

Note that T^{g^{−1}} is fixed by F iff g^{−1}F (g) ∈ N_{G}(T). In this case we say that (T^{g^{−1}})^{F} is a maximal torus of G^{F} twisted by w := g^{−1}F (g)T ∈ W.

To get hold of the intersection of tori we need the following definition. Let G be a connected reductive group with Frobenius map F and let T be an F -stable maximal torus of G. Then T is called non-degenerate if T is the only maximal torus of G containing T^{F}. This is equivalent to T = C_{G}(T^{F})^{∘}. If a maximal torus T is non-degenerate, we have good control over the Weyl-group of T^{F}, especially (N_{G}(T))^{F} = N_{G^{F}} (T^{F}) and N_{G}(T)^{F} /T^{F} ≅ (N_{G}(T)/T)^{F} in this case.

For our purposes it is enough to consider the case, where G is a Chevalley group. Since F fixes every elements of the Weyl-group in this case, H^{1}(F, W) corresponds to the conjugacy classes of W and we get the following order estimates.

Fact 1.14 ([52]) Let G be a Chevalley group and F a standard Frobenius map. Let T be a maximal split torus, w ∈ W = N_{G}(T)/T and T_{w} a maximal torus twisted by w. If q is the number of elements of the base field of G^{F}, then the characteristic polynomial of w evaluated at q gives the order of (T_{w})^{F}. Furthermore |N_{G}(T_{w})^{F} /(T_{w})^{F} | = |C_{W} (w)|

The characteristic polynomials of elements in the Weyl-group of Chevalley groups are given in [25]. We have thus a complete description of the orders of maximal tori in Chevalley groups.

1.5 Probabilistic Algorithms

The last decades of computations have seen several shortcomings of deterministic algorithms and have replaced them in many cases by randomized algorithms, i.e. algorithms which use random numbers. A further step is not only to consider algorithms which depend on random choices of elements of a certain sample but furthermore to consider algorithms where the outcome is only correct with a certain probability. The most famous application of these kinds of algorithms are primality testing algorithms.

Example (Strong Primality Test): Let n be any odd number. We want to know if n is prime.

• Take any base b where 1 ≤ b < n is a natural number and test if b^{n−1} = 1 mod n. If it is not, then n cannot be prime.

• Otherwise factor n − 1 = 2^{k}m, where m is odd.

• Consider the sequence b^{m}, (b^{m})^{2}, ⋯ (b^{m})^{2^{k}}. If the sequence does not contain −1, then n cannot be prime. Otherwise b passes the test.

One can show that if n is not prime at least 1/2 of the bases fails the test [24]. Hence if we chose k bases randomly and all of them pass the test, then n is a prime number with probability greater than 1 − (\frac{1}{2})^{k}.

The strong primality test is a so called one-sided Monte-Carlo algorithm. A Monte Carlo algorithm A is a randomized algorithm, where the result is only correct with a probability 1 − ϵ, where 0 ≤ ϵ < 1/2. Assume that A has only yes and no as possible results. Then A is called one-sided, if the result yes is always correct. A Las Vegas algorithm is a randomized algorithm, which gives a result with probability 1 − ϵ, where 0 ≤ ϵ < 1/2. Here the result has to be always correct.

Let g be a function from ℕ onto ℝ^{+,0}. The Landau symbol O(g) is defined as

O(g) := {f : ℕ ⟶ ℝ^{+,0}| ∃n_{0} ∈ ℕ, c > 0 ∀n ≥ n_{0} : 0 ≤ f(n) ≤ cg(n)}.

Furthermore we say that an algorithm A with entries in E has complexity O(f) , if there exists g ∈ O(f) such that the running time is bounded by g(k(e)) for all e ∈ E, where k : E ⟶ ℕ × ⋯ × ℕ measures the length of the input.

1.6 Black box groups

Black box groups were introduced by Babai and Szemerédi [9] as the ideal setting for some group computational problems and their importance has grown ever since. For the following compare [11] and [13].

A black box group G is a finite group whose elements are encoded as 0-1 strings of uniform length N, and the group operations are performed by an oracle (the “black box”). Moreover, a list of generators of the group is given in this encoding. Given strings representing g, h ∈ G, the black box can compute strings representing gh and g^{−1}, and decide whether g = h at unit cost. We assume that not all strings need to correspond to group elements and that the encoding need not be unique.

Note that the upper bound on |G| is 2^{N}. Furthermore, if we know the generating elements of a subgroup H, H is again a black box group with these generators. If moreover H is normal and recognizable, then G/H is a black box group as well with the same generators as G and all elements of H corresponding to the identity element. A subgroup H of a black box group G is said to be recognizable, if there exists a polynomial time algorithm which can decide if an element g of G is contained in H.

The classical examples of black box groups are permutation groups and matrix groups over finite fields. If G is a permutation group on n letters for some n ∈ ℕ, then any element can be encoded by at most n log(n) bits, and if G ≤ GL_{n}(q) for n ∈ ℕ and q any prime power, then elements can be encoded by strings of length n^{2} log q.

Sometimes it is necessary to strengthen the assumptions. A grey box group is a black box group with an additional oracle. Classically, a grey box group possesses an oracle that gives element orders or does linear algebra in case that the black box group is a matrix group.

The main question is, how much information about G can actually be gained in polynomial time. Can we for example, given the promise that G is a simple group and the name of a finite simple group A, decide if G is isomorphic to A?

There is a large joined effort to solve this and related questions. The most prominent results are by L. Babai, W. Kantor, P. Palfy and A. Seress. If G is a classical group, then the problem has been solved in [46]. Their algorithm even gives a presentation of the black box group G in question. The draw-back is, that their algorithm takes time polynomial in q. However, there are polynomial time algorithms as well.

Fact 1.15 ([47] and [12]) Let G be a grey box group with order oracle isomorphic to a simple group of Lie type whose elements are encoded as strings of length N. Then there exists a Monte-Carlo algorithm which determines in time polynomial in N

- the size of the underlying field, the dimension and the type of the group, if G is known to be classical.

- the isomorphism type of the group, if the size of the underlying field is known.

Both algorithms mentioned in Fact 1.15 rely on the result that finite simple groups of type B_{n} and C_{n} can be distinguished, which we will prove in Section 6.

As most group-theoretic algorithms for larger groups, many black box algorithms assume that they have random elements available, either from the uniform distribution or, as this is hardly feasible, at least from the near uniform distribution. A probability distribution over the finite set G is ϵ-uniform , if the probability of any particular element being selected is (1 ± ϵ)/|G|. The distribution is near uniform, if it is uniform for some 0 < ϵ < 1.

Fact 1.16 ([10]) Let c and C be positive constants, and let ϵ = M^{−c}. If G is a black box group of order at most M, then there is a Monte Carlo algorithm which constructs ϵ-uniformly distributed, independent random elements of G at a cost of O(log^{5} M) group operations. The algorithm fails with probability ≤ M^{−C}. If the algorithm succeeds, the cost for the construction of a single element after the initial preprocessing is O(log M).

In practice, the algorithm from [29] is most widely used, since it is much faster. The theoretical analysis of this algorithm, however, is still missing and seems extremely difficult to achieve.

Part I

Groups of Finite Morley Rank Quasiunipotent Groups

One of the problems one encounters in the classification of simple tame K^{∗}-groups is that one does not know how to generalize the concept of a unipotent subgroup of an algebraic group to groups of finite Morley rank. If all the proper simple sections of the group are algebraic groups over an algebraically closed field of characteristic p one can use p-groups. However, the algebraic groups that appear as simple sections could have characteristic 0 or they could even have different characteristics. Hence A. Berkman has introduced the concept of a quasiunipotent group and proved some basic results about them. To proceed further it became necessary to consider the more subtle notion of a pseudounipotent group as well. Both concepts capture some of the essential meaning of “unipotent”. A. Berkman’s work together with the results of this section have already been published in [7].

2.1 Basic properties

Let G be a group of finite Morley rank. An abelian divisible p-subgroup of G is called p-torus. A definable connected nilpotent subgroup U ≤ G is called quasiunipotent if U does not contain a p-torus for any prime number p. A definable connected nilpotent subgroup H ≤ G is called pseudounipotent (in G) if it satisfies the following condition: If K ≤ H is a definable connected subgroup of H which normalises a definable connected nilpotent subgroup N ≤ G then the product NK is nilpotent.

We will normally fix the group G; the word “pseudounipotent” in the subsequent text means “pseudounipotent in G”.

Lemma 2.1 Let G be a connected algebraic group over an algebraically closed field. Let U be a definable connected unipotent subgroup of G. Then U is quasiunipotent and pseudounipotent.

Proof. Let U be a definable connected unipotent subgroup of G. Then U is nilpotent and since it does not contain a p-torus for any prime p, U is quasiunipotent.

Let V ≤ U be a definable connected subgroup of U and N ≤ G a connected definable nilpotent subgroup of G that is normalized by V. We show that V N is nilpotent. V N is a solvable connected group and the Zariski closure \overline{V N} is thus a Zariski connected solvable algebraic group. Furthermore since the Zariski closure \overline{N} of N is a Zariski connected and nilpotent algebraic group, \overline{N} = W × T is the direct product of its semisimple part T and its unipotent part W by [42, 19.2]. \overline{V N} has, as a Zariski connected solvable algebraic group, a maximal normal unipotent subgroup H by [42, 19.3] which contains all unipotent elements. Thus V W ≤ H and as H is Zariski closed, already \overline{V} W ≤ H. This implies that \overline{V N} = \overline{V} (W × T) = (\overline{V} W) × T is nilpotent. Hence U is pseudounipotent in G. □

Fact 2.2 (Berkman [7]) If G is a connected algebraic group that does not interpret any bad field then quasiunipotent subgroups of G are unipotent.

In Corollary 2.14 we will furthermore prove that if G is a connected semisimple algebraic group that does not interpret a bad field, then pseudounipotent subgroups of G are unipotent.

Lemma 2.3 Let G be a group of finite Morley rank, H ≤ G a definable subgroup of G and φ a definable automorphism of G.

(i) A definable connected subgroup of a quasiunipotent subgroup of G is also quasiunipotent.

(ii) H is quasiunipotent if and only if φ(H) is quasiunipotent.

(iii) If H is quasiunipotent it can be written as H = B × D, where B and D are definable, B is of bounded exponent and D is torsion-free.

(iv) A definable connected subgroup of a pseudounipotent subgroup of G is also pseudounipotent.

(v) H is pseudounipotent if and only if φ(H) is pseudounipotent.

(vi) If G is connected and nilpotent, then G is pseudounipotent.

(vii) If H is pseudounipotent in G, then H is pseudounipotent in any definable subgroup K of G that contains H.

Proof. (iii) follows by [18, 6.12], the rest is clear from the definitions. □

Fact 2.4 (Berkman [7]) Let G be a group of finite Morley rank. Let U, V be definable subgroups of G and N ⊲ U a definable normal subgroup of U.

(i) If U is quasiunipotent, then U/N is also quasiunipotent.

(ii) If U/N and N are quasiunipotent and if U does not interpret a bad field, then U is quasiunipotent.

(iii) Assume that U and V are quasiunipotent and that U normalizes V. If UV does not interpret a bad field, then UV is quasiunipotent.

To study pseudounipotent groups we need criteria for nilpotency.

Fact 2.5 ([38]) Let G be any group. H, K ≤ G subgroups, such that H ⊲ K. If H and K/H′ are nilpotent, then K is nilpotent as well.

Lemma 2.6 Let G be a connected solvable group of finite Morley rank and H ≤ G a definable connected subgroup. Then the following are equivalent

(i) H is pseudounipotent

(ii) HG^{n}/G^{n} is pseudounipotent for all n ≥ 2.

(iii) HG^{n}/G^{n} is pseudounipotent for some n ≥ 2.

Proof. (iii) → (i): Assume that \overline{H} := HG^{n}/G^{n} is pseudounipotent for some n ∈ ℕ, n ≥ 2 and let K ≤ H be a definable connected subgroup which normalises a definable connected nilpotent subgroup N ≤ G. We have to show that NK is nilpotent. Let \overline{K} := KG^{n}/G^{n} and \overline{N} := NG^{n}/G^{n}. Then \overline{K} is a definable connected subgroup of \overline{H} which normalises the definable connected nilpotent group \overline{N}. Since \overline{H} is pseudounipotent \overline{NK} is nilpotent. However G^{n−1} is nilpotent by [18, 9.9]. Thus NKG^{n} is nilpotent by Fact 2.5 and NK is nilpotent as well.

(i) → (ii): Let H be pseudounipotent and n ∈ ℕ, n ≥ 2. Let furthermore \overline{K} be a definable connected subgroup of \overline{H} = HG^{n}/G^{n} which normalises the definable connected nilpotent group \overline{N}. We have to show that \overline{N}\overline{K} is nilpotent. Let K, N be the subgroups of G containing G^{n} such that K/G^{n} = \overline{K} and N/G^{n} = \overline{N}. Set K_{1} := (K ∩ H)^{∘}. Then K = K ∩ HG_{n} = (K ∩ H)G_{n} ≥ K_{1}G^{n}. As K is connected and K and K_{1}G^{n} have the same rank, already K = K_{1}G^{n}. Furthermore K and N are connected groups since G^{n} is connected. They are nilpotent by Fact 2.5 since G^{n−1} is nilpotent by [18, 9.9]. As K normalises N, K_{1} ≤ H normalises N as well and NK_{1} is nilpotent by assumption. Hence NK_{1}G^{n}/G^{n} = NK/G^{n} = \overline{NK} is nilpotent as well. □

Corollary 2.7 Let G be a solvable connected group of finite Morley rank and H ≤ G a pseudounipotent subgroup. Then HG^{n} is pseudounipotent for all n ≥ 2.

Proof. Let n ∈ ℕ, n ≥ 2. As H is pseudounipotent, HG^{n}/G^{n} is pseudounipotent by Lemma 2.6. Thus HG^{n} is pseudounipotent as well by Lemma 2.6 again, as G^{n} is connected by [18, 5.29]. □

Lemma 2.8 Let G be any group, H ≤ G a subgroup. Then the following are equivalent:

(i) H is nilpotent.

(ii) HZ_{m}(G)/Z_{m}(G) is nilpotent for all m ∈ ℕ.

(iii) HZ_{m}(G)/Z_{m}(G) is nilpotent for some m ∈ ℕ.

Thus HZ_{m}(G) is nilpotent for all m ∈ ℕ if H is nilpotent.

We give a short proof for this easy fact in lack of a suitable reference.

Proof. First note that (i) implies (ii) and that (ii) implies (iii). Hence it is only left to show that (iii) implies (i). We consider first the case where H = G. Let m ∈ ℕ and \overline{X} := XZ_{m}(H)/Z_{m}(H) for all X ≤ G. Then

(∗)  H/Z_{m+n}(H) ≅ \overline{H}/Z_{n}(\overline{H})

and

(∗∗)  Z_{m+n+1}(H)/Z_{m+n}(H) ≅ Z_{n+1}(\overline{H})/Z_{n}(\overline{H})

for all n ∈ ℕ. We prove (∗) and (∗∗) by simultaneous induction on n. (∗) and (∗∗) are true for n = 0 since Z_{0}(H) = 1. For the induction step notice that

H/Z_{m+n+1}(H) ≅ (H/Z_{m+n}(H))/(Z_{m+n+1}(H)/Z_{m+n}(H))

and hence by induction hypothesis

H/Z_{m+n+1}(H) ≅ (\overline{H}/Z_{n}(\overline{H}))/(Z_{n+1}(\overline{H})/Z_{n}(\overline{H})) ≅ \overline{H}/Z_{n+1}(\overline{H})

proving (∗). Especially

Z(H/Z_{m+n+1}(H)) ≅ Z(\overline{H}/Z_{n+1}(\overline{H})) = Z_{n+2}(\overline{H})/Z_{n+1}(\overline{H})

and (∗∗) follows. Assume now that \overline{H} is nilpotent. Then there exists n ∈ ℕ such that \overline{H} = Z_{n}(\overline{H}). However this implies that H = Z_{m+n}(H) by (∗) and H is nilpotent as well.

Let finally H < G and assume that HZ_{m}(G)/Z_{m}(G) is nilpotent for some m ∈ ω. Since again by induction

Z_{n+1}(G) ∩ H = {x ∈ H : [G, x] ≤ Z_{n}(G)}
≤ {x ∈ H : [H, x] ≤ Z_{n}(G) ∩ H}
≤ {x ∈ H : [H, x] ≤ H}
= Z_{n+1}(H)

for all n ∈ ℕ,

(H/Z_{m}(G) ∩ H)/(Z_{m}(H)/Z_{m}(G) ∩ H) ≅ H/Z_{m}(H)

is nilpotent and the claim follows. □

Lemma 2.9 Let G be a group of finite Morley rank and H a connected definable subgroup. Then the following are equivalent

(i) H is pseudounipotent

(ii) HZ_{n}(G)/Z_{n}(G) is pseudounipotent for all n ≥ 2.

(iii) HZ_{n}(G)/Z_{n}(G) is pseudounipotent for some n ≥ 2.

Proof. Similar to the proof of Lemma 2.6 using Lemma 2.8 instead of Fact 2.5. In proving (i) → (ii) we have to use the connected component of Z_{n}(G), since Z_{n}(G) need not be connected. The result then follows by rank considerations. □

Corollary 2.10 Let G be a group of finite Morley rank and H ≤ G a pseudounipotent subgroup. Then HZ_{n}(G)^{∘} is pseudounipotent for all n ∈ ℕ. Especially Z_{n}(G)^{∘} is pseudounipotent for all n ∈ ℕ.

Proof. Let n ∈ ℕ, n ≥ 2. As H is pseudounipotent, HZ_{n}(G)/Z_{n}(G) is pseudounipotent by Lemma 2.9. Thus HZ_{n}(G)^{∘} is pseudounipotent as well by Lemma 2.9 again. □

2.2 Maximal quasiunipotent groups

If B is a connected solvable algebraic group, the set of unipotent elements forms a maximal unipotent group and is characteristic in B. We are going to show that the analogue property holds for groups of finite Morley rank. We prove that a solvable group G of finite Morley rank contains a unique maximal quasiunipotent (pseudounipotent) subgroup Q(G) respectively P (G). Furthermore Q(G) and P (Q) are characteristic subgroups of G and Q(G) ≤ P (Q).

We call a quasiunipotent (pseudounipotent) subgroup U of a group G of finite Morley rank maximal quasiunipotent (maximal pseudounipotent),if U = V for all quasiunipotent (pseudounipotent) subgroups U ≤ V of G. Note that maximal quasiunipotent and maximal pseudounipotent subgroups exist by ascending chain condition on connected subgroups.

Proposition 2.11 Let G be a group of finite Morley rank which does not interpret a bad field and U ≤ G a quasiunipotent subgroup. Then U is pseudounipotent.

Proof. Let U ≤ G be a quasiunipotent subgroup of G and let V ≤ U be a definable connected subgroup that normalises the definable connected nilpotent subgroup N ≤ G. V N is a connected solvable group. We show that V N is nilpotent. Set F := F (V N). V N/F^{∘} is a divisible abelian group by [18, 9.21] and since N ≤ F^{∘}, V N = V F^{∘}. As V is quasiunipotent it can be written as V = B × D where B and D are definable, B is of bounded exponent and D is torsion-free by Lemma 2.3. Thus V N/F^{∘} = V F^{∘}/F^{∘} ≅ BD/(F^{∘} ∩ BD) is a divisible group and B ≤ F^{∘}. This implies that V N = DF^{∘}.

Consider \overline{V N} := V N/Z_{m}(V N), where Z_{m}(V N) is the hyper-center of V N. Then Z(\overline{V N}) = 1 and \overline{V N} = \overline{D}\overline{F}^{∘}, where \overline{D} is quasiunipotent by Fact 2.4. Furthermore F (\overline{V N}) is the direct product of a connected definable subgroup C of bounded exponent and a definable connected torsion-free subgroup T by [18, ex.3, p.148]. Thus \overline{F}^{∘} ≤ F (\overline{V N}) is quasiunipotent as well. Since G does not interpret a bad field, \overline{V N} is quasiunipotent by Fact 2.4 and hence nilpotent. Thus V N is nilpotent by Lemma 2.8. □

Let G be a group of finite Morley rank. We call

Q(G) = ⟨U| U ⊲ G, U is quasiunipotent ⟩

the quasiunipotent radical of G and

P (G) = ⟨H| H ⊲ G, H is pseudounipotent ⟩

the pseudounipotent radical of G .

Lemma 2.12 Let G be a group of finite Morley rank.

(i) Q(G) and P (Q) are nilpotent groups.

(ii) Q(G) = U_{1} ⋯ U_{n} and P (G) = H_{1} ⋯ H_{m} are the products of finitely many normal quasiunipotent respectively pseudounipotent subgroups.

If G does not interpret a bad field then

(ii) Q(G) is quasiunipotent,

(iii) Q(G) ⊲ P (G).

Proof. As Q(G), P (G) ≤ F (G), they are nilpotent by [18, 7.3]. They are products of finitely many normal quasiunipotent respectively pseudounipotent subgroups by Zil’bers indecomposability theorem [18, 5.28]. If finally G does not interpret a bad field, Q(G) is quasiunipotent by Fact 2.4 and Q(G) ⊲ P (G) by Proposition 2.11. □

It is not known if P (G) is pseudounipotent for every group of finite Morley rank.

Proposition 2.13 Let G be a solvable group of finite Morley rank that does not interpret a bad field. Then F (G)^{∘} is the unique maximal pseudounipotent subgroup of G. Especially P (G) = F (G)^{∘} is pseudounipotent.

Proof. We can take G to be connected, since all pseudounipotent groups are connected. The proof proceeds by induction on the solvable length of G. The claim is trivial if G is nilpotent. Let G be 2-step solvable and non-nilpotent. Consider \overline{G} := G/Z_{m}(G) where Z_{m}(G) is the hyper-center of G. Then \overline{G} is a 2-step solvable connected centerless group of finite Morley rank. Furthermore F (\overline{G})^{∘} is a pseudounipotent subgroup of G by [18, ex.3, p.148] and Proposition 2.11. Since \overline{G}′ ≤ F (\overline{G})^{∘} by [18, 9.9], every subgroup of \overline{G} which contains F (\overline{G})^{∘} is normal in G. Especially every maximal pseudounipotent subgroup of G containing F (\overline{G})^{∘} is equal to F (\overline{G})^{∘} and F (\overline{G})^{∘} is a maximal pseudounipotent group.

Let K ≤ \overline{G} be a pseudounipotent group. Since K normalises F (\overline{G})^{∘}, KF (\overline{G})^{∘} is nilpotent and normal in \overline{G}. Hence KF (\overline{G})^{∘} ≤ F (\overline{G})^{∘} and K ≤ F (\overline{G})^{∘}. Thus F (\overline{G})^{∘} is the unique maximal pseudounipotent subgroup of \overline{G}. Since by Lemma 2.8 \overline{F (G)} = F (\overline{G}), Lemma 2.9 implies that F (G)^{∘} is the unique maximal pseudounipotent subgroup of G.

Let G be n-step solvable with n ≥ 3. By induction F (G/G^{n−1})^{∘} is the unique maximal pseudounipotent subgroup of G/G^{n−1}. Let H ≤ G be such that H/G^{n−1} = F (G/G^{n−1})^{∘}. Then H = F (G)^{∘} by Fact 2.5 since G^{n−2} ≤ G′ is nilpotent by [18, 9.9]. F (G)^{∘} is thus the unique maximal pseudounipotent subgroup of G by Lemma 2.6. □

Corollary 2.14 Let G be a connected algebraic group that does not interpret a bad field. The maximal pseudounipotent subgroups of G are Z(G)^{∘}U^{∘}, where U is the unipotent radical of a Borel subgroup. Thus, if G is semisimple, pseudounipotent and definable connected unipotent subgroups coincide.

Proof. Let H be a maximal pseudounipotent subgroup of G and B a Borel subgroup of G in the sense of algebraic group theory containing H. Then B = U⋊T by [42, 19.3], where U is the unipotent radical of B and T is a maximal torus. Thus H ≤ P (B) by Proposition 2.13. As on the other hand U^{∘} is pseudounipotent by Lemma 2.1 and Proposition 2.11, U^{∘} ≤ P (B) as well. Since the Zariski closure \overline{P (B)} of P (B) is a Zariski connected nilpotent closed subgroup of B, it is the direct product of its unipotent part U = \overline{U}^{∘} and a semisimple part T_{1}. As T_{1} ≤ C_{B}(U) = Z(B) and Z(G) = Z(B) by [42, 22.2], H ≤ P (B) ≤ UZ(G). However, Z(G)^{∘}U^{∘} is pseudounipotent by Corollary 2.10, i.e. H = Z(G)^{∘}U^{∘} by maximality of H. □

Lemma 2.15 Let G be a nilpotent group of finite Morley rank that does not interpret a bad field. Then Q(G) is the unique maximal quasiunipotent subgroup of G.

Proof. We can take G to be connected since all quasiunipotent groups are connected. First assume that G is divisible. We prove the claim by induction on the solvable length of G. If G is abelian, then every quasiunipotent subgroup of G is normal in G and the claim follows by Lemma 2.12.

Assume that G is not abelian. G′ is connected and torsionfree by [18, exc 5., p.102]. Hence G′ is quasiunipotent. Furthermore Q(G/G′) is the unique maximal quasiunipotent subgroup of G/G′ by induction hypothesis. Let G′ ≤ H ≤ G such that H/G′ = Q(G/G′). Then H is a maximal quasiunipotent group by Fact 2.4, i.e. H = Q(G).

Let finally G be a not necessarily divisible connected nilpotent group of finite Morley rank. Then G = D∗C where D is a definable connected characteristic and divisible subgroup of G and C is a definable connected characteristic subgroup of G of bounded exponent by [18, 6.8]. C is quasiunipotent. Hence Q(G) = Q(D)∗C is the unique maximal quasiunipotent subgroup of G. □

Proposition 2.16 Let G be a solvable group of finite Morley rank that does not interpret a bad field. Then Q(G) is the unique maximal quasiunipotent subgroup of G.

Proof. Let U be a maximal quasiunipotent subgroup of G. Then U is pseudounipotent by Proposition 2.11 and hence U ≤ P (G) by Proposition 2.13. However, as P (G) is nilpotent by Proposition 2.13, it contains the unique maximal quasiunipotent subgroup Q(P (G)) by Lemma 2.15. Thus U = Q(P (G)) by maximality of U. Especially Q(P (G)) = Q(G) is the unique maximal quasiunipotent subgroup of G. □

Corollary 2.17 Let G be a K-group of finite Morley rank that does not interpret a bad field. If U_{1}, U_{2} are two maximal quasiunipotent subgroups of G, then U_{1} and U_{2} are conjugate.

Proof. Since U_{1}, U_{2} are nilpotent, they are contained in Borel subgroups B_{1} and B_{2} of G respectively. Furthermore U_{n} = Q(B_{n}) are characteristic subgroups of B_{n} for n = 1, 2 by Proposition 2.16. Since B_{1} and B_{2} are conjugate by Proposition 1.6, Q(B_{1}) = U_{1} and Q(B_{2}) = U_{2} are conjugate as well. □

2.3 Normalizers of quasiunipotent groups

The normalizer of a maximal unipotent subgroup U of a connected algebraic group G is U ⋊ T for an algebraic torus T by [42, 19.3, 23.1, 30.4]. We are going to show that a similar result is valid for normalizers of maximal quasiunipotent subgroups in groups of finite Morley rank under the special assumptions that we will need in Chapter 4.

Proposition 2.18 Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Let T be a definable connected solvable subgroup. If Q is a nontrivial quasiunipotent subgroup of G that is normalised by T then there exists a maximal quasiunipotent subgroup Q ≥ V of G such that V is normalized by T.

Proof. Let 1 ≠ Q be a quasiunipotent subgroup of G that is normalized by T. Then QT is a definable connected solvable subgroup of N_{G}(Q). Let B be a Borel subgroup of N_{G}(Q) that contains QT. Then Q(B) is the unique maximal quasiunipotent subgroup of B by Proposition 2.16 and Q ≤ Q(B)⊲B. Especially Q(B) is normalized by T. Furthermore, as N_{G}(Q) is a K-group, the Borel subgroups are conjugate by Proposition 1.6 and Q(B) is a maximal quasiunipotent subgroup of N_{G}(Q). Let U ≥ Q(B) be a maximal quasiunipotent subgroup of G. As U is nilpotent and connected either U = Q or N_{U}(Q)/Q is infinite by [18, 6.3]. As N_{U}(Q) = Q(B), either Q is a maximal quasiunipotent subgroup of G, or there exists a quasiunipotent subgroup Q(B) of N_{G}(Q) which is strictly greater than Q and normalized by T. We can now repeat our argument with Q(B) instead of Q. After finitely many steps we arrive at a maximal quasiunipotent group V of G which is normalized by T by ascending chain condition on connected subgroups. □

Lemma 2.19 Let G be an algebraic group that does not interpret a bad field. Let H ≤ G be a definable subgroup and U ≤ H a maximal quasiunipotent subgroup of H. Then N_{H}(U)^{∘} is solvable.

Notice that if H is Zariski closed, Lemma 2.19 is an easy corollary of Lemma 2.2 and [43, 23.1 and 30.4].

Proof. Set N := N_{H}(U)^{∘}. Assume that N is not solvable. Let σ be the solvable radical of N and V ≤ N a subgroup of N such that V/σ is a maximal quasiunipotent subgroup of N/σ. As N/σ is the direct sum of simple algebraic groups over algebraically closed fields by [2], V/σ is infinite. Furthermore V^{∘} is a connected solvable group with maximal quasiunipotent subgroup U by Proposition 2.16 as U ≤ σ. Let \overline{V} be the Zariski closure of V^{∘}. \overline{V} is a Zariski connected solvable group. Let W be the unipotent radical of \overline{V}. Then \overline{V} = W ⋊ T for some algebraic torus T ≤ \overline{V} by [42, 19.3]. Thus

T ≅ \overline{V} /W ≥ V^{∘}W/W ≅ V^{∘}/(V^{∘} ∩ W)

As (V^{∘} ∩ W)^{∘} is quasiunipotent by Lemma 2.1 and U ≤ V^{∘} ∩ W, U = (V^{∘} ∩ W)^{∘} by maximality of U. Thus (V^{∘} ∩ W)/U is finite and

T ≅ V^{∘}/(V^{∘} ∩ W) ≅ (V^{∘}/U)/((V^{∘} ∩ W)/U).

As T is abelian, [V^{∘}/U, V^{∘}/U] ≤ (V^{∘}∩W)/U. Since furthermore the commutator of a connected group is connected by [18, 5.29], V^{∘}/U is a connected abelian group. Especially V^{∘}/U is divisible by [18, 6.7], as T is divisible. Thus

V/σ = V^{∘}σ/σ ≅ V^{∘}/(σ ∩ V^{∘}) ≅ (V^{∘}/U)/((σ ∩ V^{∘})/U)

is a divisible quasiunipotent group and hence torsion-free by Lemma 2.3. Set X := (V^{∘} ∩ σ)(V^{∘} ∩ W). Then X ⊲ V^{∘} and

V^{∘}/X ≅ (V^{∘}/(V^{∘} ∩ σ))/(X/(V^{∘} ∩ σ))

is torsion-free by [18, ex. 11, p. 93]. However,

V^{∘}/X ≅ (V^{∘}/(V^{∘} ∩ W))/(X/(V^{∘} ∩ W))

is isomorphic to a connected definable section of the algebraic torus T. As G does not interpret a bad field, V^{∘}/X = 1 and V^{∘} = X. Especially

V^{∘}σ/σ ≅ V^{∘}/(V^{∘} ∩ σ) ≅ (V^{∘} ∩ W)/(V^{∘} ∩ W ∩ σ)

is finite as U ≤ V^{∘} ∩ W ∩ σ. Contradiction. □

Let G be any group. D ≤ G is called four-subgroup, if D is an elementary abelian subgroup of order 4. We let D^{∗} := D\{1} denote the set of involutions in D.

Proposition 2.20 Let G be a K^{∗}-group of finite Morley rank and odd type that does not interpret a bad field and V a maximal quasiunipotent group of G. Assume that V is normalized by a four-subgroup D such that C_{G}(l)^{∘} for all l ∈ D^{∗} is an algebraic group over an algebraically closed field and D ≤ C_{G}(D)^{∘}. Then N_{G}(V)^{∘} = V ⋊ T, where T ≤ C_{G}(D)^{∘} is an algebraic torus of C_{G}(l)^{∘} for all l ∈ D^{∗}.

Proof. By Fact 2.4, Q ≤ V for any quasiunipotent subgroup Q ≤ N_{G}(V)^{∘} and V is the unique maximal quasiunipotent subgroup of N_{G}(V)^{∘}. By [19, 5.14], N_{G}(V)^{∘} = ⟨C_{N_{G}(V)}(l)^{∘}| l ∈ D^{∗}⟩ where N_{l} := C_{N_{G}(V)}(l)^{∘} is a connected definable subgroup of an algebraic group for all l ∈ D^{∗}. Let U_{l} be a maximal quasiunipotent subgroup of N_{l} for all l ∈ D^{∗}. As U_{l} ≤ V, U_{l} = V ∩ C_{G}(l)^{∘}. Thus U_{l} is a characteristic subgroup of N_{l} for all l ∈ D^{∗} and V = ⟨U_{l}| l ∈ D^{∗}⟩ by [19, 4.6].

Let l ∈ D^{∗}. Then N_{l} = N_{N_{l}}(U_{l}) is solvable by Lemma 2.19. Furthermore D ≤ N_{l}, as D normalizes U and D ≤ C_{G}(D)^{∘}. Let \overline{N}_{l} be the Zariski closure of N_{l}. As \overline{N}_{l} is a Zariski connected solvable group, N_{l} = W ⋊ T by [42, 19.3], where W is the unipotent radical of \overline{N}_{l} and T a maximal torus. Since D consists of semisimple elements, we can chose T to contain D by [42, 19.4]. W^{∘} is a quasiunipotent subgroup of C_{G}(l)^{∘} by Lemma 2.1 and hence contained in U_{l}. Thus \overline{N}_{l}^{∘} ∩ N_{l} = U_{l} ⋊ (T ∩ N_{l}) since T is connected by [18, ex. 3, p. 78]. As G does not interpret a bad field, T ∩ N_{l} ≤ T is an algebraic torus itself and actually T ≤ N_{l}. Finally T ≤ C_{G}(D)^{∘} and thus T ≤ N_{k} for all k ∈ D^{∗}. This implies that N_{G}(V)^{∘} = ⟨U_{l}| l ∈ D^{∗}⟩ ⋊ T = V ⋊ T. □

Corollary 2.21 Let G be a K^{∗}-group of finite Morley rank and odd type that does not interpret a bad field. Let B be a definable connected solvable subgroup of G. Assume that there exists a four-subgroup D ≤ B such that either

(i) C_{B}(l)^{∘} is an algebraic group over an algebraically closed field for all l ∈ D^{∗} or (ii) B is a Borel subgroup of G and C_{G}(l)^{∘} is an algebraic group over an alge-

braically closed field for all l ∈ D^{∗},

then there exists an algebraic torus T such that B = Q(B) ⋊ T.

Proof. B has connected Sylow 2-subgroups by [18, 9.29], especially D ≤ C_{G}(D) for any four-subgroup D ≤ B. Furthermore Q(B) is the unique maximal quasiunipotent subgroup of B by Proposition 2.16. If (i) holds then B = N_{B}(Q(B))^{∘} = Q(B)⋊T for an algebraic torus T by Proposition 2.20. If (ii) holds then Q(B) is a maximal quasiunipotent subgroup of G as B is a Borel subgroup of G by Proposition 2.16. Thus N_{G}(Q(B))^{∘} = Q(B) ⋊ T for an algebraic torus T by Proposition 2.20. As Q(B) ≤ B this implies that B = Q(B) ⋊ (B ∩ T) and the claim follows since G does not interpret a bad field. □

2-local Analysis

Let G be any group. A subgroup H is said to be a local subgroup if H = N_{G}(A) for a nontrivial definable abelian subgroup A ≤ G. If A is a p-group for some prime number p, H is said to be a p-local subgroup. In particular, the centralizer of an involution t ∈ G is a 2-local subgroup because C_{G}(t) = N_{G}(⟨t⟩). As in the classification of finite simple groups, a great deal of information about groups of finite Morley rank can be deduced from information about centralizers of involutions and, especially in the odd type case, normalizers of the connected component of Sylow 2-subgroups.

3.1 Strongly embedded subgroups

One of the standard technics in the CFSG is to prove that a subgroup of a finite simple group G is either strongly embedded — and hence known by the classification result of H. Bender [14] — or already equals G. A proper definable subgroup M of a group G is said to be strongly embedded if it contains involutions and for every g ∈ G \ M, M ∩ M^{g} does not contain involutions. Altınel has shown in [2], that any simple tame K^{∗}-group of even type that has a strongly embedded subgroup is isomorphic to SL_{2}(K) for an algebraically closed field K of characteristic 2. If Conjecture 3 is true, then this is the only place, where strongly embedded subgroups appear in the theory of simple tame K^{∗}-groups. The main aim of this section is to prove the following partial result.

Theorem 3.1 Let G be a simple K^{∗}-group of odd type that does not interpret any bad field and such that pr(G) ≥ 2. Let M < G be a strongly embedded subgroup. Then M^{∘} is solvable.

Corollary 3.2 Let G be a simple tame K^{∗}-group with a strongly embedded subgroup M. Then M is solvable or G is of odd type and Prüfer 2-rank 1.

Proof. G is either of odd or of even type by Fact 1.3. If G is of odd type and pr(G) ≥ 2, then M is solvable by Theorem 3.1. If G is of odd type, then M is solvable by [2]. □

Groups of finite Morley rank with a strongly embedded subgroup have only one conjugacy class of involutions by [18, 10.19]. Furthermore a proper subgroup M of a group G of finite Morley rank is strongly embedded if and only if M contains involutions, C_{G}(t) ≤ M for any t ∈ I(M) and N_{G}(S) ≤ M for any Sylow 2-subgroup S of M by [2]. Their structural properties are discussed in [2] and slightly further explored in [6]. We can describe strongly embedded subgroups differently, if G is a connected K^{∗}-group of finite Morley rank and odd type such that pr(G) ≥ 2.

Proposition 3.3 Let G be a connected K^{∗}-group of finite Morley rank and odd type such that pr(G) ≥ 2. Assume that M < G is a strongly embedded subgroup of G. Then

(i) M^{∘} = ⟨C_{G}(t)^{∘}| t ∈ D^{∗}⟩ for any four-subgroup D of M and

(ii) M is a maximal proper definable subgroup of G.

Thus, if G is a K-group, then it cannot contain a strongly embedded subgroup.

Proof. Any Sylow 2-subgroup S of M is already a Sylow 2-subgroups of G by [2]. Thus there exists a four-subgroup D ≤ M, as pr(G) ≥ 2. Then M^{∘} = ⟨C_{M}(t)^{∘}| t ∈ D^{∗}⟩ by [19, 5.14]. On the other hand C_{G}(t)^{∘} ≤ M^{∘} for all t ∈ D^{∗} as M is strongly embedded, which gives us that M^{∘} = ⟨C_{G}(t)^{∘}| t ∈ D^{∗}⟩.

We show that M is a maximal proper definable subgroup of G. Let N be a definable subgroup of G such that M ≤ N < G. Then N is strongly embedded as well by [2]. Hence N^{∘} = ⟨C_{G}(t)^{∘} : t ∈ D^{∗}⟩ = M^{∘} by [18, 5.14] again. Especially N ≤ N_{G}(M^{∘}). Let g ∈ N_{G}(M^{∘}). Then M^{∘} ≤ M ∩ M^{g} contains involutions as I(M) = I(M^{∘}) by [2]. Since M is strongly embedded, this implies that N_{G}(M^{∘}) ≤ M and hence N = N_{G}(M^{∘}) = M.

As finally G = ⟨C_{G}(t)^{∘}| t ∈ D^{∗}⟩ for any four-subgroup D ≤ G, if G is a K-group by [19, 5.14], G cannot contain a strongly embedded subgroup in this case. □

The reverse direction is true as well.

Proposition 3.4 Let G be a connected K^{∗}-group of finite Morley rank and odd type such that pr(G) ≥ 2. Let N := ⟨C_{G}(l)^{∘}| l ∈ D^{∗}⟩ for a four-subgroup D of G. Then either N is normal in G or N_{G}(N) is a strongly embedded subgroup of G.

Proof. Assume that M := N_{G}(N) < G. Then M is a K-group which contains D and M^{∘} = N by [19, 5.14]. We are going to show that C_{G}(t) ≤ M for any t ∈ I(M) and that N_{G}(S) ≤ M for any Sylow 2-subgroup S of M.

Let t ∈ I(M) be any involution. Then t is contained in Sylow 2-subgroup S of M. As Z(S) ≠ 1 by [18, 6.22], there exists s ∈ S such that D_{1} := ⟨t, s⟩ is a four-subgroup of S with D_{1} ∩ Z(S) ≠ 1. Thus M^{∘} = ⟨C_{M}(k)^{∘}| k ∈ D_{1}^{∗}⟩ by [19, 5.14]. Let L := ⟨C_{G}(k)| k ∈ D_{1}^{∗}⟩. Then C_{G}(t) ≤ L. We show that L ≤ M. As Z(S) ∩ D_{1} ≠ 1, S ≤ L. Let g ∈ M such that D^{g} ≤ S. Then

L^{∘} = ⟨C_{L}(l)^{∘}| l ∈ (D^{g})^{∗}⟩ ≤ N^{g} = N.

by [19, 5.14] again. Thus M^{∘} ≤ L^{∘} ≤ N = M^{∘}, L^{∘} = N and L ≤ N_{G}(N) = M. Hence C_{G}(t) ≤ M for any involution t ∈ M.

Let furthermore S be any Sylow 2-subgroup of M. As C_{G}(t) ≤ M for all t ∈ M, pr(M) = 2. As N_{G}(S) ≤ N_{G}(S^{∘}), it is sufficient to show that N_{G}(S^{∘}) ≤ M. Let g ∈ N_{G}(S^{∘}). Then either g ∈ C_{G}(s) ≤ M for some involution s ∈ S^{∘} or there exists an elementary abelian subgroup E ≤ S^{∘} of order at least 4 such that g ∈ N_{G}(E). Since M^{∘} = ⟨C_{G}(l)^{∘} : l ∈ E\{1}⟩ by the first part of the proof, this implies that g ∈ N_{G}(M^{∘}) = M. □

We are now able to exhibit two possible examples of strongly embedded subgroups.

Remark 3.5 Let H be a K-group of odd type that contains an elementary abelian 2-subgroup E of order 8. Then

H^{∘} = ⟨C_{H}(D)^{∘}| D ≤ E, [E : D] = 2⟩.

Proof. Let D_{1} ≤ E be a four-subgroup. Then H^{∘} = ⟨C_{H}(i)^{∘}| i ∈ D_{1}^{∗}⟩ by [19, 5.14.]. However, E ≤ C_{H}(i) for any i ∈ D_{1}^{∗} and thus

C_{H}(i)^{∘} = ⟨C_{H}(D)^{∘}| i ∈ D ≤ E, [E : D] = 2⟩

by [19, 5.14.] again. □

Proposition 3.6 Let G be a simple K^{∗}-group of odd type such that pr(G) ≥ 2 and assume that G contains an elementary abelian subgroup E of order 8. If G contains a proper 2-generated core Γ, then G contains a strongly embedded subgroup M := N_{G}(Γ^{∘}).

Proof. We may assume that Γ = Γ_{S,2}, where S is a Sylow 2-subgroup of G that contains E. As pr(G) ≥ 2, S ≤ N_{G}(S^{∘}) ≤ Γ by definition and Γ^{∘} = ⟨C_{Γ}(D)^{∘}| D ≤ E, [E : D] = 2⟩ by Remark 3.5. Since N_{G}(D) ≤ Γ for all foursubgroups D ≤ E,

Γ^{∘} = ⟨C_{G}(D)^{∘}| D ≤ E, [E : D] = 2⟩ = ⟨C_{G}(i)^{∘}|i ∈ D_{1}^{∗}⟩

for any four-subgroup D_{1} ≤ E. The claim now follows by Proposition 3.4. □

To give the second example, we need some more definitions. Let G be a group of finite Morley rank. We say that G is a 2^{⊥}-group if G does not contain involutions and we denote by O(G) the core of G, the maximal normal definable connected 2^{⊥}-subgroup of G. The core exists since the product of two normal definable 2^{⊥}-subgroups is a 2^{⊥}-subgroup itself by [18, ex. 11, p. 93]. Remember, that if G is a K-group that does not interpret a bad field, then O(G) is nilpotent by [19, 5.8].

It is well known that in a simple algebraic group G over an algebraically closed field of characteristic not 2 we have O(C_{G}(i)) = 1 for any involution i ∈ G. This should also hold in a simple tame K^{∗}-group G of odd type. In Corollary 3.16 we prove this conjecture in a special case.

Let G be a group of finite Morley rank and θ a function from I(G) into the set of definable subgroups of G. We say that θ is a signalizer functor for G if θ(s) ≤ O(C_{G}(s)) is a connected definable normal subgroup of C_{G}(s) for all s ∈ I(G) and if furthermore for any commuting involutions s, t ∈ G

θ(t) ∩ C_{G}(s) = θ(s) ∩ C_{G}(t).

In particular, θ is a signalizer functor for any simple K^{∗}-group G of odd type if θ(t) = O(C_{G}(t)) for any involution t ∈ G by [18, B.29]. A signalizer functor θ is called complete, if for any elementary abelian subgroup E ≤ G of order ≥ 8 the subgroup θ(E) = ⟨θ(t), t ∈ E\{1}⟩ is a connected 2^{⊥}-subgroup and C_{θ(E)}(s) = θ(s) for any s ∈ E^{∗}. We call θ nilpotent if all the subgroups θ(t) for t ∈ I(H) are nilpotent.

Fact 3.7 ([17]) Any nilpotent signalizer functor θ on a group G of finite Morley rank is complete.

Proposition 3.8 Let G be a simple K^{∗}-group of odd type that does not interpret a bad field. Assume that pr(G) ≥ 2 and that G contains an elementary abelian subgroup E of order 8. Let θ be the signalizer functor defined by θ(s) = O(C_{G}(s)) for all s ∈ I(G). Then either O(C_{G}(s)) = 1 for all s ∈ E^{∗} or N_{G}(N_{G}(θ(E))) is a strongly embedded subgroup of G.

Proof. As G does not interpret a bad field, θ is a nilpotent signalizer functor and thus complete by Fact 3.7. Hence θ(E) is nilpotent. Let D be any foursubgroup of E. As D acts on θ(E) by a definable group automorphism

θ(E) = ⟨C_{θ(E)}(t), t ∈ D^{∗}⟩

by [19, 4.6]. Since θ is complete this implies that

θ(E) = ⟨O(C_{G}(t)), t ∈ D^{∗}⟩

In particular N_{G}(D) ≤ N_{G}(θ(E)). Thus

N_{G}(θ(E))^{∘} = ⟨C_{G}(D)^{∘}| D ≤ E, [E : D] = 2⟩ = ⟨C_{G}(i)^{∘}|i ∈ D_{1}^{∗}⟩

for any four-subgroup D_{1} ≤ E by Remark 3.5. The claim now follows by Proposition 3.4. □

To prove Theorem 3.1, we need the following crucial fact about centralizers of involutions, which is also one of the main tools to classify groups of small Prüfer 2-rank.

Fact 3.9 ([19]) Let G be a simple K^{∗}-group of finite Morley rank and odd type that does not interpret a bad field. Let i ∈ I(G). Then C_{G}(i)^{∘}/O(C_{G}(i)) is a central product of an abelian divisible group T and a semisimple group H all of whose components are simple algebraic groups over algebraically closed fields of characteristic different from 2.

Lemma 3.10 Let H be a group of finite Morley rank. For any X ≤ H and h ∈ H let \overline{X} := XO(H)/O(H) and \overline{t} := tO(H).

(i) If t ∈ I(H) then \overline{t} ∈ I(\overline{H}) and if \overline{s} ∈ I(\overline{H}) then there exists an involution in sO(H).

(ii) O(\overline{H}) = 1.

(iii) Let t, s ∈ I(H). If [\overline{t}, \overline{s}] = 1, then there exists u ∈ I(sO(H)) such that [t, u] = 1.

(iv) Let t ∈ I(H). Then I(tO(H)) = t^{O(H)}. Especially t, s ∈ I(H) are conjugate in H if and only if \overline{t}, \overline{s} are conjugate in \overline{H}.

Proof. Let t ∈ I(H). Since O(H) is a 2^{⊥}-subgroup of H, t ∉ O(H) and \overline{t} is an involution in \overline{H}. Since furthermore sO(H) contains a 2-element by [18, ex 11, p.93] if \overline{s} is an involution, (i) follows.

To show (ii), let B ≥ O(H) be such that \overline{B} = O(\overline{H}). Then B is a definable, connected normal subgroup of H since O(H) and O(\overline{H}) are definable and connected. Furthermore B cannot contain an involution by (i) and B = O(H) by maximality of O(H).

Let t, s ∈ I(H). If [\overline{t}, \overline{s}] = 1, then

\overline{s} ∈ C\overline{H}(\overline{t}) = \overline{C_{H}(t)} ≅ C_{H}(t)/(C_{H}(t) ∩ O(H))

by [3, 2.52], i.e. we can find an involution u ∈ C_{H}(t) such that \overline{s} = \overline{u} which gives us (iii).

Let finally to ∈ I(tO(H)). Then to and t cannot commute with an involution in the definable closure of their product since d(tot) = d(o^{t}) ≤ O(H). Thus they must be conjugate in O(H) by [18, 10.2]. □

Lemma 3.11 Let H be a group of finite Morley rank and odd type. Let S be a Sylow 2-subgroup of H and N ⊲ H a definable solvable subgroup. Then S^{∘}N/N is the connected component of a Sylow 2-subgroup of H/N.

Proof. Let P be a Sylow 2-subgroup of H/N that contains S^{∘}N/N. Let furthermore F be a subgroup of H which contains N such that F/N = d(P^{∘}). Then F is a definable solvable group and S^{∘} ≤ F. Thus S^{∘} is the connected component of a Sylow 2-subgroup of F and S^{∘}N/N is the connected component of a Sylow 2-subgroup of F/N = d(P^{∘}) by [5]. Hence S^{∘}N/N = P^{∘} and the claim follows. □

Lemma 3.12 Let G be a simple K^{∗}-group of finite Morley rank and odd type that does not interpret a bad field. Let i ∈ G be an involution and C := C_{G}(i)^{∘}/O(C_{G}(i)) = H ∗ T as in Fact 3.9. If Q is a Sylow 2-subgroup of H and S a Sylow 2-subgroup of C that contains Q, then S^{∘} = Q^{∘}R where R is the Sylow 2-subgroup of T. Especially pr(C) = pr(H)+pr(T), were either T is trivial or pr(T) ≥ 1.

Proof. Since C is the central product of H and T, H ∩ T ≤ Z(H) ≤ Z(C) and H ∩ T is finite, as H is semisimple. Let Q be a Sylow 2-subgroup of H and S a Sylow 2-subgroup of C that contains Q. Since T is a central subgroup of C, the Sylow 2-subgroup R of T is contained in S and Q ∗ R ≤ S. T is as a divisible group connected by [18, ex. 3, p. 78]. Thus R is connected as well by [18, 9.29] and Q^{∘}R ≤ S^{∘}.

As T is abelian S^{∘}T/T is the connected component of a Sylow 2-subgroup of C/T by Lemma 3.11 where S^{∘}T/T ≅ S^{∘}/(S^{∘} ∩ T) = S^{∘}/R. On the other hand Q^{∘}(H ∩ T)/(H ∩ T) ≅ Q^{∘}/(Q^{∘} ∩ T) is the connected component of a Sylow 2-subgroup of H/(H ∩ T) by Lemma 3.11 again. As C/T = HT/T ≅ H/(H ∩ T) actually S^{∘}/R ≅ Q^{∘}/(Q^{∘} ∩ T) = Q^{∘}/(Q^{∘} ∩ R) ≅ Q^{∘}R/R and S^{∘} = Q^{∘}R. As pr(C) = pr(S), pr(H) = pr(Q) and pr(T) = pr(R) this implies that pr(C) = pr(H) + pr(T).

Assume now that pr(T) = 0. As R is connected, R = 1. Especially T ≤ O(C). However O(C) = 1 by Lemma 3.10 and T is trivial. □

Proposition 3.13 Let G be a simple K^{∗}-group of odd type that does not interpret a bad field. Then N_{G}(P^{∘}) is solvable-by-finite for any Sylow 2-subgroup P of G.

Proof. Since P^{∘} is a 2-torus |N_{G}(P^{∘})/C_{G}(P^{∘})| < ∞ by [18, 6.16]. Hence it is enough to show that C_{G}(P^{∘})^{∘} is solvable.

Let i ∈ P^{∘} be any involution. As P is infinite by [19], i exists. We write \overline{X} = XO(C_{G}(i))/O(C_{G}(i)) for any subgroup X ≤ C_{G}(i). If C := \overline{C_{G}(i)^{∘}} is solvable, the claim follows, since C_{G}(P^{∘}) ≤ C_{G}(i). Assume now that C is not solvable. Then C = H ∗ T by Fact 3.9, where T is an abelian divisible group and H is a nontrivial semisimple group all of whose components H_{n} for 1 ≤ n ≤ k are simple algebraic groups over algebraically closed fields of characteristic different from 2. \overline{P}^{∘} is the connected component of a Sylow 2-subgroup of C by Lemma 3.11. Thus \overline{P}^{∘} = Q^{∘}R, where Q is a Sylow 2-subgroup of H and R the Sylow 2subgroup of T by Lemma 3.12. Let Q_{n} for 1 ≤ n ≤ k be the projections of Q^{∘} onto H_{n}. As Q^{∘} consists of commuting 2-elements, Q_{n} is an abelian 2-group and hence contained in a maximal torus T_{n} of H_{n} for all 1 ≤ n ≤ k by [42, 15.4]. As Q_{n} are the connected components of Sylow 2-subgroups of the simple algebraic groups H_{n} for all 1 ≤ n ≤ k, C_{H}(Q^{∘}) = T, where T := T_{1} ⋯ T_{k}. Thus C_{H}(\overline{P}^{∘}) = T as well and \overline{C_{C_{G}(i)^{∘}}(P^{∘})} ≤ C_{C}(\overline{P}^{∘}) is abelian. Hence C_{C_{G}(i)^{∘}}(P^{∘})^{∘} is a connected solvable group. However, C_{G}(P^{∘})^{∘} ≤ C_{G}(i)^{∘} and C_{G}(P^{∘})^{∘} ≤ C_{C_{G}(i)^{∘}}(P^{∘})^{∘} is solvable. □

We need one more result to prove Theorem 3.1.

Lemma 3.14 Let N be a group of finite Morley rank and H ⊲ N a definable normal subgroup such that H^{∘} is a 2^{⊥}-group. Then N^{∘} = C_{N}(i)^{∘}H^{∘} for all involutions i ∈ H.

Notice that the Lemma reduces to the Frattini argument [18, 10.12], if ⟨i⟩ is a Sylow 2-subgroup of H.

Proof. Let i ∈ I(H). Then H^{∘} = C_{H^{∘}}(i)H^{−} where H^{−} is the set of elements of H^{∘} inverted by i by [18, ex. 14, p. 73]. Furthermore

(∗) rk(H^{∘}) = rk(C_{H^{∘}}(i)) + rk(H^{−}).

We claim that H^{−} = ii^{H^{∘}} = iI(iH^{∘}). Let h ∈ H^{−}. As H^{∘} is as a 2^{⊥}-group 2-divisible by [18, ex. 11, p. 72], there exists an element d ∈ H^{∘} such that h = d^{2}. On the other hand h^{i} = h^{−1} and thus already d = d^{−1} by [18, ex. 12, p. 72]. Thus d ∈ H^{−} and t = d^{2} = [i, d] ∈ ii^{H^{∘}}. Furthermore for all d ∈ H^{∘}, ii^{d} ∈ iI(iH^{∘}) as i^{d} = i[i, d] ∈ iH^{∘}. The claim follows since iI(iH^{∘}) is a subset of H^{∘} inverted by i.

i^{N} on the other hand is the disjoint union of finitely many sets I(j_{k}H^{∘}) for 1 ≤ k ≤ d where d ≤ |H/H^{∘}| and j_{k} ∈ i^{N}. Thus we may assume that rk(i^{N}) = rk(I(iH^{∘})) = rk(H^{−}) and

(∗∗) rk(N) = rk(C_{N}(i)) + rk(i^{N}) = rk(C_{N}(i)) + rk(H^{−}).

(*) and (**) imply that

rk(N) = rk(C_{N}(i)) + rk(H^{∘}) − rk(C_{N}(i) ∩ H^{∘}) = rk(C_{N}(i)H^{∘}).

Hence N^{∘} = C_{N}(i)^{∘}H^{∘} for any involution i ∈ N. □

Proof of Theorem 3.1. Let σ be the solvable radical of M^{∘}. Since M does not interpret a bad field O(M^{∘}) is nilpotent and hence, as a definable connected subgroup, contained in σ^{∘}. Notice that, since O(M^{∘}) is characteristic in M^{∘}, O(M^{∘}) ≤ O(M) ≤ M^{∘} and thus O(M^{∘}) = O(M). As furthermore M is a strongly embedded K-group

O(C_{G}(t)) = O(C_{M}(t)) ≤ O(M) ≤ σ^{∘}

for all involutions t ∈ M by [19]. σ^{∘} is a characteristic subgroup of M^{∘} and hence normal in M. Since M contains one conjugacy class of involutions as a strongly embedded subgroup by [18, 10.19], this implies that either I(M) ⊆ σ^{∘} or σ^{∘} does not contain involutions.

(a) If σ^{∘} contains involutions, then M^{∘} is solvable.

Proof. Assume that I(M) ⊆ σ^{∘}. Let S_{1} be a Sylow 2-subgroup of σ^{∘} and S ≥ S_{1} a Sylow 2-subgroup of M. Then S is a Sylow 2-subgroup of G by [2]. Furthermore S_{1} is connected by [18, 9.29] and thus a divisible subgroup of the abelian group S^{∘}. Hence S_{1} has a complement B in S^{∘} by the theorem of Baer [34, 15.1]. As, however, all involutions are contained in σ^{∘}, B has to be trivial, S_{1}^{∘} = S^{∘} and S^{∘} is a Sylow 2-subgroup of σ^{∘}. By the Frattini argument [18, 10.12] M = N_{M}(S^{∘})σ^{∘}. Since N_{M}(S^{∘})/C_{M}(S^{∘}) is finite by [18, 6.16], M^{∘} = C_{M}(S^{∘})^{∘}σ^{∘}. However C_{G}(S^{∘})^{∘} is solvable by Corollary 3.13. Thus M^{∘} is already solvable and M^{∘} = σ^{∘}. □

(b) If σ contains involutions, then either I(M) is finite or M^{∘} is solvable.

Proof. We may assume by (a) that σ^{∘} is a 2^{⊥}-group and that I(M) ⊆ σ as σ is normal in M. As M^{∘} = C_{G}(i)^{∘}σ^{∘} for any involution i ∈ M by Lemma 3.14,

\overline{M}^{∘} := M^{∘}/σ^{∘} ≅ C_{G}(i)^{∘}/(C_{G}(i)^{∘} ∩ σ^{∘}) = C_{G}(i)^{∘}/O(C_{G}(i))

for any involution i ∈ M. Since I(M) = I(M^{∘}) by [2], \{iσ^{∘}|i ∈ I(M)\} ⊆ Z(\overline{M}^{∘}). If I(M) is infinite, then M^{∘} = σ, since M^{∘} would otherwise contain infinitely many commuting involutions by Lemma 3.10. Hence M^{∘} is solvable in this case. □

(c) If I(M) is finite, then either pr(G) = 1 or M^{∘} is solvable.

Proof. If I(M) is finite, then M^{∘} = C_{G}(i)^{∘} for any involution i ∈ M as rk(M) = rk(C_{G}(i)) + rk(I(M)). Hence M^{∘}/O(M) is a central product of an abelian divisible group T and a semisimple group H all of whose components are algebraic groups over algebraically closed fields of characteristic not 2 by Fact 3.9. We assume that M^{∘} is not solvable. Then O(M) = σ^{∘} by (a) and T = 1, as T is connected. Let L_{1}, ⋯, L_{k} be the components of M^{∘}/O(M) = H. By Lemma 3.10 all involutions of H are central in H. Especially |I(L_{l})| = 2^{pr(L_{l})−1}, as Z(L_{l}) is the intersection of all maximal tori in L_{l} for 1 ≤ l ≤ k by [42, ex. 2, p. 162] and |I(H)| = 2^{pr(H)−1}, as Z(H) is the intersection of all maximal tori in H by [42, ex. 2, p. 162] again. Let g ∈ M. As g acts as automorphism on M^{∘}/O(M), for all 1 ≤ l ≤ k there exists 1 ≤ s ≤ k such that L_{l}^{g} = L_{s} by [18, 7.1] and thus I(L_{l})^{g} = I(L_{s}). If i_{1} ∈ I(L_{1}), then I(M^{∘}/O(M)) = i_{1}^{M} = {I(L_{1}), ⋯, I(L_{k})} since M, and hence M/O(M) by Lemma 3.10, contains one conjugacy class of involutions. Set p := pr(L_{1}). Then pr(H) = kp and |{I(L_{1}), ⋯, I(L_{k})}| ≤ k2^{p−1}. This can only happen for k = 1, since |I(H)| = 2^{kp−1} > k2^{p−1} ≥ |i_{1}^{M}| for k > 1. Hence H is already quasi-simple. However, the only quasi-simple algebraic group in which all involutions are central is SL_{2}(K) for an algebraically closed field of characteristic not 2. (Compare e.g. [37]). As pr(H) = pr(M^{∘}/O(M)) = pr(M^{∘}) by Lemma 3.11, the claim follows. □

Suppose that σ does not contain involutions. Then M^{∘} is not solvable since pr(M) = pr(G) ≥ 2 and 1 ≠ \overline{M} := M/σ contains one conjugacy class of involutions as in the proof of Lemma 3.10. As furthermore M is a K-group \overline{M^{∘}} is the direct sum of simple algebraic groups over algebraically closed fields of characteristic not 2 by [2]. Let L_{1}, ⋯, L_{k} be the components of \overline{M}^{∘}. As L_{l} contains involutions for any 1 ≤ l ≤ k, \overline{M} acts transitively on the components and all components are isomorphic by [18, 7.10].

We claim that L_{1} contains one conjugacy class of involutions. For assume that i, j in I(L_{1}) are not conjugate in L_{1}. Then there exists m ∈ \overline{M} such that i^{m} = j. As \overline{M^{∘}} is the direct product of conjugates of L_{1}, this implies that L_{1}^{m} = L_{1}. Thus C_{L_{1}}(i)^{m} = C_{L_{1}}(j). However, non-conjugate involutions of simple algebraic groups have non-isomorphic centralizers of involutions by [37, 4.3]. Contradiction.

The only simple algebraic groups with one conjugacy class of involutions are PSL_{2}(K), PSL_{3}(K) or G_{2}(K) for an algebraically closed field K of characteristic not 2. Hence we may assume that \overline{M}^{∘} is isomorphic to a direct sum of copies of these groups. On the other hand M^{∘} = C_{G}(i)^{∘}H for a connected 2^{⊥}-group H which can be chosen independently from i ∈ I(M) by [2, 3.10] since M is strongly embedded. Hence \overline{M}^{∘} is the product of the centralizer of the involution \overline{ı} and a Borel subgroup B of \overline{M}^{∘} in the sense of algebraic group theory which contains the connected nilpotent group Hσ/σ. As H is independent from i ∈ I(M), we can choose i such that \overline{ı} ∈ B. Let B = U ⋊ T, where U is the unipotent radical of B and T a maximal torus which contains \overline{ı} by [42, 19.3]. Let U_{l} be the projections of U into L_{l} and T_{l} be the projections of T into L_{l} for 1 ≤ l ≤ k. Then \overline{ı} = i_{1} ⋯ i_{k} where i_{l} ∈ I(T_{l}) and thus T_{l} ≤ C_{L_{l}}(i_{l}) for 1 ≤ l ≤ k. Hence dim L_{l} ≤ dim C_{L_{l}}(i_{l}) + dim U_{l} for 1 ≤ l ≤ k. We may assume that i_{1} ≠ 1.

If L_{1} ≅ PSL_{2}(K), then dim L_{1} = 3, dim C_{L_{1}}(i_{1}) = 1 and dim U_{1} = 1.

If L_{1} ≅ PSL_{3}(K), then dim L_{1} = 8, dim C_{L_{1}}(i_{1}) = 4 and dim U_{1} = 3.

If L_{1} ≅ G_{2}(K), then dim L_{1} = 14, dim C_{L_{1}}(i_{1}) = 6 and dim U_{1} = 6.

Thus none of these cases can occur. Contradiction. □

Corollary 3.15 Let G be a simple K^{∗}-group of odd type that does not interpret a bad field. Assume that pr(G) ≥ 2 and that G contains an elementary abelian subgroup of order 8. If there exists an involution i ∈ G such that C_{G}(i) is not solvable, then G does not contain a proper 2-generated core.

Proof. Assume that G does contain a proper 2-generated core. Then G contains a strongly embedded subgroup M by Proposition 3.6. If there exists an involution i ∈ G such that C_{G}(i) is not solvable, we may assume that i ∈ M, as G contains only one conjugacy class of involutions. Then C_{G}(i) ≤ M since M is strongly embedded and M is not solvable. Contradiction to Theorem 3.1. □

Corollary 3.16 Let G be a simple K^{∗}-group of odd type that does not interpret a bad field. Assume that pr(G) ≥ 2 and that G contains an elementary abelian subgroup of order 8. If there exists an involution i ∈ G such that C_{G}(i) is not solvable, then O(C_{G}(t)) = 1 for all t ∈ E^{∗}.

Proof. By Proposition 3.8 either O(C_{G}(t)) = 1 for all t ∈ E^{∗} or G contains a strongly embedded subgroup M. The second case cannot occur, however, as in the proof of Corollary 3.15, which implies the claim. □

Corollary 3.17 Let G be a simple K^{∗}-group of odd type that does not interpret a bad field. Assume that pr(G) ≥ 2. If there exists an involution i ∈ G such that C_{G}(i) is not solvable, then G = ⟨C_{G}(k)^{∘}| k ∈ D_{1}^{∗}⟩ for any four-subgroup D_{1} ≤ G.

Proof. D_{1} ≤ G be a four-subgroup of G and set M := ⟨C_{G}(k)^{∘}| k ∈ D_{1}^{∗}⟩. By Proposition 3.4 either G = N_{G}(M) or M is a strongly embedded subgroup. Since G cannot contain a strongly embedded subgroup as in the proof of Corollary 3.15, G = M as G is simple. □

3.2 Thompson Rank Formula

In this section we give an application of the Thompson rank formula and prove the following theorem.

Theorem 3.18 Let C_{1}, C_{2} be the nonisomorphic centralizers in PSp_{4}(K) of two involutions where K is an algebraically closed field of characteristic not 2. Let G be a group of finite Morley rank that contains involutions i and j such that C_{G}(i) ≅ C_{1} and C_{G}(j) ≅ C_{2}. Set k := max{rk(Z(C_{G}(u_{1}))), rk(Z(C_{G}(u_{2})))}. Then rk G = 10 k.

The proof of this theorem is quite easy and involves only computations in C_{1} and C_{2}. We proceed in the following way. First we describe C_{1} and C_{2}. We show that G contains exactly two conjugacy classes of involutions. This information allows us to construct a function f as in Proposition 1.2, and finally to compute

r_{w} := rk({(κ, λ) : κ ∈ i^{G} ∩ C_{G}(w), λ ∈ j^{G} ∩ C_{G}(w) and f(κλ) = w})

for w = i, j. This gives us the Morley rank of G by Proposition 1.2.

3.2.1 Structure of G

We may assume that C_{G}(i) = (L_{1} ∗ L_{2})⟨u⟩ where L_{n} ≅ SL_{2}(K) for n = 1, 2, u is an involution in G such that L_{1}^{u} = L_{2}, and furthermore L_{1} ∩ L_{2} = Z(L_{1}) = Z(L_{2}) = ⟨i⟩. Set L := L_{1}L_{2}. Then C_{G}(j) ≅ (GL_{2}(K)/⟨−I⟩) ⋊ ⟨w⟩ where −I = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) and w is an involution in G such that w acts as an inverse-transpose automorphism on GL_{2}(K), i.e.

( \begin{matrix}a & b \\ c & d \end{matrix} )^{w} = (ad − bc)^{−1} ( \begin{matrix}d & −c \\ −b & a \end{matrix} )

for any ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(K). For the structure of centralizers of involutions in this case compare [57]. General information about centralizers of groups of Lie type is available in [37].

Since both i and j are contained in the connected component of a Sylow 2-subgroup of G and as Sylow 2-subgroups are conjugate, we may assume that i, j ∈ S^{∘}, where S is a Sylow 2-subgroup of G. Let D := ⟨i, j⟩ and T := C_{G}(D)^{∘}. Since D contains a central involution by [18, ex. 12, p. 14], S ≤ C_{G}(i).

To apply the Thompson rank formula we need precise information about the distribution of conjugacy classes of involutions in C_{G}(i) and C_{G}(j). Let us first consider C_{G}(i). As all elements of order 4 are conjugate in SL_{2}(K), I(L) = j^{L} ∪ i as in [57]. Furthermore I(Lu) = u^{L_{1}} ∪ iu^{L_{1}}, as lu ∈ I(G) for l ∈ L if and only if l^{u} = l or l^{u} = il, i.e. if and only if l = l_{1}l_{1}^{u} or l = il_{1}l_{1}^{u} for l_{1} ∈ L_{1}.

Set now R := C_{G}(j)^{∘}. Elements of R can be represented by elements of GL_{2}(K) where the two matrices A and −A are identified. Then I(R) = j ∪ i^{R} ∪ (ij)^{R}, depending on the determinant of the corresponding matrix. Let v ∈ R such that v corresponds to V := ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). Then A^{V} = det(A)A^{w} for all A ∈ GL_{2}(K). As furthermore for A := ( \begin{matrix}a & b \\ c & d \end{matrix})

AA^{V} = ( \begin{matrix}−det(A) + b(b − c) & a(c − b) \\ d(b − c) & −det(A) + c(c − b) \end{matrix} ) ,

gw is an involution for g ∈ R, if and only if g is represented by one of the following matrices:

( \begin{matrix}a & b \\ b & d \end{matrix} ) or ( \begin{matrix}0 & b \\ −b & 0 \end{matrix} )

where we can choose a, b, d ∈ K such that the determinants are unequal to 0. It is now easy to see that I(Rw) = w^{R} ∪ (vw)^{R}.

Lemma 3.19 C_{G}(i) contains four conjugacy classes of involutions represented by {i, j, u, iu} and C_{G}(j) contains five conjugacy classes of involutions represented by {j, i, ij, w, vw}. Furthermore i^{G} ∩ C_{G}(i) = i ∪ (iu)^{L_{1}}, j^{G} ∩ C_{G}(i) = j^{L} ∪ u^{L_{1}}, j^{G} ∩ C_{G}(j) = j ∪ (ij)^{R} ∪ w^{R} and i^{G} ∩ C_{G}(j) = i^{R} ∪ (vw)^{R}. Especially G contains exactly two conjugacy classes of involutions.

Proof. We have already finished the proof of the first part of the lemma. As C_{G}(D) = T ⋊ ⟨v, w⟩ where ⟨v, w⟩ is a four-subgroup and w inverts T, w ∈ L and w is conjugate to j in C_{G}(i). Let w = w_{1}w_{2}, where w_{1} ∈ L_{1}T, w_{2} ∈ L_{2}T. As w_{1}, w_{2} ∈ N_{G}(T), w_{2}^{u} = w_{1}t for some t ∈ T and j^{w_{1}} = ij. We may assume that u ∈ R. For if u ∉ R, then uw = uw^{u} = uw_{1}^{u}w_{2}^{u} = w_{1}uw_{1}t ∈ R and (uw)^{w_{1}} ∈ uT ∩ R^{w_{1}} = uT ∩ C_{G}(ij)^{∘}, i.e. u ∈ C_{G}(ij)^{∘}. Especially u is conjugate to i or ij in C_{G}(j) and the same is true for iu. Let D_{1} = ⟨u, j⟩. Then D_{1} is conjugate to D in C_{G}(j)^{∘} and u and ju belong to different conjugacy classes. This implies that ju and iu are conjugate. Hence det(J) = −1, where J is the matrix corresponding to i. As det(V) = 1, v has to be conjugate to ij in R and vi conjugate to i. We may hence assume that u is conjugate to j and ui to i. On the other hand vw ∈ C_{G}(i)\L. This implies that vw and vwi are not conjugate. Thus vwi ∈ C_{G}(j)\R is conjugate to w and hence to j and vw is conjugate to i. □

Lemma 3.20 The following rank equations are valid.

• k = rk(C_{G}(i, j, u)).

• rk(C_{G}(i)) = 6k, rk(C_{G}(j)) = 4k.

• rk(u^{C_{G}(i)}) = rk((iu)^{C_{G}(i)}) = 3k and rk(j^{C_{G}(i)}) = 2k.

• rk(i^{C_{G}(j)}) = rk((ij)^{C_{G}(j)}) = 2k and rk(w^{C_{G}(j)}) = 3k, rk((vw)^{C_{G}(j)}) = k.

Proof. Since Z(L) is finite, k = rk(Z(R)) by definition. However Z(R) = C_{G}(i, j, u)^{∘}. Projection gives us that a torus T_{1} in L_{1} has Morley rank k. Let T_{1} be contained in a Borel subgroup B of L_{1}. Then rk(L_{1}) = rk(T_{1}) + 2 rk(U_{1}), where U_{1} is the unipotent radical of B by the Bruhat decomposition of L_{1} [42, 28.3]. Since T_{1} acts transitively on U_{1}\{1} with finite 1-point stabilizers rk(L_{1}) = 3k. Hence rk(C_{G}(i)) = rk(L_{1}) + rk(L_{1}^{u}) = 6k and rk(C_{G}(D)) = 2k. R is the direct product of Z(R) with PSL_{2}(K) where the torus in PSL_{2}(K) has Morley rank k again. Thus rk(C_{G}(j)) = k + 3k = 4k using similar arguments. The remaining equalities are easily seen. □

We are now in a position to show that there is a definable function f from i^{G}j^{G} to I(G) such that f(x^{g}) = f(x)^{g} for all g ∈ G and

f(x) ∈ {g ∈ G| g^{y} = g^{−1} for all y ∈ I(C_{G}^{∗}(x))}.

The basic idea is that the definable closure of such an element x contains at least one involution by [18, 10.2] and all we have to do is to distinguish one in a definable way. Let x ∈ i^{G}j^{G}. If x ∈ I(G) set f(x) = x. If x ∉ I(G), let

H_{x} := {g ∈ Z(C_{G}(x))| g^{y} = g^{−1} for all y ∈ I(C_{G}^{∗}(x)\C_{G}(x))}.

H_{x} is an abelian group. Furthermore d(x) ≤ ⟨x⟩H_{x}^{∘}, since x ∈ H_{x} and D_{x} := ⟨x⟩H_{x}^{∘} contains an involution. If H_{x}^{∘} contains involutions, let E_{x} be the maximal elementary abelian 2-subgroup of H_{x}^{∘}. Let S_{0} be the Sylow 2-subgroup of H_{x}^{∘}. Since S_{0} is connected by [18, 9.29], E_{x} is conjugate to a subgroup of D. Assume now that H_{x}^{∘} is a 2^{⊥}-group. Let k := |D_{x}/H_{x}^{∘}|. Since H_{x}^{∘} is 2-divisible as a 2^{⊥}- group by [18, ex. 11, p. 72], there exists h ∈ H_{x}^{∘} such that x^{k} = h^{2}. Since k is even, i_{x} := x^{k/2}h^{−1} is the unique involution in D_{x}. Furthermore i_{x} is definable as there exists m ∈ ℕ such that |H_{x}/H_{x}^{∘}| < m for all x ∈ i^{G}j^{G}\I(G) by [18, B.5].

• If I(H_{x}^{∘}) ≠ ∅ set f(x) = C_{G}(N_{G}(E_{x})) ∩ I(H_{x}^{∘}).

• If I(H_{x}^{∘}) = ∅, set f(x) = i_{x} = I(D_{x}).

f is definable. We need to show that f is well-defined. Assume that H_{x}^{∘} contains exactly one involution λ. Then

f(x) = C_{G}(N_{G}(E_{x})) ∩ I(H_{x}^{∘}) = C_{G}(N_{G}(⟨λ⟩)) ∩ {λ} = λ.

If H_{x} contains more than one involution, E_{x} is conjugate to D and we only need to compute C_{D}(N_{G}(D)). However C_{D}(N_{G}(D)) = ⟨i⟩ and f is well-defined in both cases. Let g ∈ G. Then H_{x}^{g} = H_{x^{g}} and D_{x}^{g} = D_{x^{g}}. Furthermore

• f(x)^{g} = x^{g} = f(x^{g}) in case that x ∈ I(G),

• f(x)^{g} = C_{G}(N_{G}(E_{x}^{g})) ∩ I((H_{x}^{∘})^{g}) = f(x^{g}) in case that I(H_{x}^{∘}) ≠ ∅,

• f(x)^{g} = I(D_{x})^{g} = I(D_{x^{g}}) = f(x^{g}) in case that I(H_{x}^{∘}) = ∅.

Thus f(x)^{g} = f(x^{g}) for all g ∈ G.

To finish the proof of our theorem, the only thing left to do is to compute r_{i} and r_{j}. Up to now we have the following approximations:

r_{j} ≤ rk(i^{G} ∩ C_{G}(j)) + rk(j^{G} ∩ C_{G}(j)) = 5 k.

Furthermore

r_{i} ≤ rk(i^{G} ∩ C_{G}(i)) + rk(j^{G} ∩ C_{G}(i)) = 4 k

and

rk(G) = max{4 k + r_{i}, 6 k + r_{j}} ≤ 11 k.

3.2.2 Computing r_{i} and r_{j}

To compute r_{i} and r_{j}, it is sufficient to consider the function f on its restriction to C_{G}(i) and C_{G}(j) respectively. Let κ ∈ i^{G}, λ ∈ j^{G}. If x = κλ ∈ C_{G}(s) for s = i, j is not an involution, then

H_{x} = {g ∈ Z(C_{G}(x)) : g^{κ} = g^{−1}} ≤ C_{G}(s),

since s ∈ C_{G}(x) and C_{G}^{∗}(x) = κC_{G}(x) = λC_{G}(x).

We are going to compute r_{j} first because it is a better candidate for an actual maximum. Since f(x) = x if x ∈ I(G) and furthermore f(x^{g}) = f(x)^{g} it is enough to consider the following four cases:

(i) x = i(ij)^{g} for all g ∈ R.

(ii) x = iw^{g} for all g ∈ R.

(iii) x = (vw)^{g}(ij) for all g ∈ R.

(iv) x = (vw)^{g}w = g^{−1}vg^{w} for all g ∈ R.

The easiest case is case (iv). If g corresponds to A ∈ GL_{2}(K), then x corresponds to A^{−1}V A^{w} = det(A)^{−1}V. Set b := det(A)^{−1} and let t ∈ Z(R) such that x = vt. Then d(x) = ⟨v⟩d(t^{2}) and x has order either even or infinite. Furthermore d(x) ≤ C_{G}(D).

If b = 1, then x = v and f(x) = v. Otherwise x ∉ I(G). As E := ⟨i, j, v⟩ ≤ C_{G}(x), H_{x} ≤ Z(C_{G}(x)) ≤ C_{G}(E) = Z(R)⟨v⟩. As w inverts x and T and centralizes v, H_{x} = Z(C_{G}(x)) and H_{x}^{∘} ≤ Z(R) ≅ K^{∗}. Let s ∈ H_{x} be any involution. Then C_{G}(x) ≤ C_{G}(s), where C_{G}(s) is either isomorphic to C_{1} or C_{2}. This implies that Z(C_{G}(x)) is isomorphic to an infinite Zariski closed subgroup of the algebraic group PSp_{4}(K). Thus H_{x}^{∘} = Z(R) and f(x) = j for all x = (vw)^{g}w^{h}, where g, h ∈ R and det(A) ≠ 1 for A ∈ GL_{2}(K) corresponding to g.

Especially r_{j} ≥ k + 3 k = 4 k by Lemma 3.20 and 10 k ≤ rk(G) ≤ 11 k. Note that this implies by virtue of Lemma 3.20 that the only case which is left to compute is case (ii).

Let Z := {z ∈ R| z^{i} = z^{−1}}. Then H_{x}^{∘} ≤ C_{Z}(x). Let z ∈ Z. Then either z = j, z = ij or z corresponds to a matrix in

\{( \begin{matrix}u & v \\ w & t \end{matrix}) | ut − vw = ϵ, u = ϵt where ϵ = 1, −1\}

Let x = rw for r ∈ R such that r corresponds to R := ( \begin{matrix}a & b \\ −b & −d \end{matrix} ). Note that if either a = d = 0 or b = 0 then x is an involution. In this case f(x) = x. Let from now on ab ≠ 0 or bd ≠ 0. As z^{−1}xz = x if and only if z^{−1}xwz^{w} = xw, we can do our computations completely within R.

ad ≠ 0: Set ρ := ad^{−1}. Let first z ∈ Z correspond to ( \begin{matrix}u & v \\ w & t \end{matrix}) where ut − vw = 1. Then either z = ij or u = t. Then z^{−1}xz = x iff the following equations are satisfied:

−at^{2} + dv^{2} = −ϵa  −b + awt − duv = −ϵb
−aw^{2} + du^{2} = ϵd  b + awt − duv = ϵb

where ϵ = 1 or ϵ = −1. Since however b ≠ 0 the case ϵ = −1 can never hold. Furthermore ij ∉ C_{Z}(x) and we may assume that u = t. Hence u(aw − dv) = 0. If u = 0 then v^{2} = −ρ. On the other hand v = ρw if u ≠ 0.

Let now z ∈ Z correspond to ( \begin{matrix}u & v \\ w & t \end{matrix}) where ut − vw = −1. Then either z = i, z = j or u = −t. Then z^{−1}xz = x iff the following slightly modified equations are satisfied:

−at^{2} + dv^{2} = ϵa  b + awt − duv = ϵb
−aw^{2} + du^{2} = −ϵd  −b + awt − duv = −ϵb

where ϵ = 1 or ϵ = −1. Since b ≠ 0 the case ϵ = −1 again cannot hold. As i ∉ C_{Z}(x), j ∈ C_{Z}(x) we may assume that u = −t. Then either u = t = 0 and v^{2} = ρ or v = −ρw. If however v = −ρw then

−1 + ρw^{2} = u^{2} = −1 + vw = −1 − ρw^{2}

and v = 0 = w. Contradiction. Hence H_{x}^{∘} corresponds to \{( \begin{matrix}u & ρw \\ w & u \end{matrix}) | u^{2} −ρw^{2} = 1\} and f(x) corresponds to ( \begin{matrix}0 & −w^{−1} \\ w & 0 \end{matrix} ) where w^{2} = −ρ^{−1}. a = 0: Using similar arguments H_{x}^{∘} is a unipotent subgroup normalized by T and containing x^{2}. As x^{2} corresponds to ( \begin{matrix}1 & 0 \\ 2db^{−1} & 1 \end{matrix} ) the involution in D_{x} is gw, where g corresponds to ( \begin{matrix}0 & b \\ −b & 0 \end{matrix} ). Thus f(x) = gw.

d = 0: f(x) = gw as well.

As f(x) = j never occurs in this case, r_{j} = 4 k and rk(G) = 10 k. □

Groups of Small Prüfer 2-rank

In this chapter we are dealing with the classification of tame simple K^{∗}-groups of odd type where the normal 2-rank of the group is small. Due to Conjecture 4, the following should hold.

Conjecture 6 An infinite simple tame K^{∗}-group G of odd type and normal 2rank n(G) ≤ 2 is isomorphic to one of the groups PSL_{2}(K), PSL_{3}(K), PSp_{4}(K), G_{2}(K) over an algebraically closed field K of characteristic not 2.

The basic idea is to construct a BN-pair of G out of information about the centralizers of involutions in G. Two subgroups B and N of a group G are said to be a BN-pair of G if the following conditions hold.

BN1. G = ⟨B, N⟩ and T := B ∩ N ⊲ N.

BN2. W := N/T is generated by a nonempty set S of involutions.

BN3. v_{i}Bv ⊆ BvB ∪ Bv_{i}vB for all v_{i} ∈ N such that v_{i}T ∈ S and all v ∈ N. BN4. vBv ≠ B for all v ∈ N such that vT ∈ S. The group W is called the Weyl group of G. The cardinality of the set S of

generators of W, which may be infinite, is called the Tits rank of the BN-pair. A conjugate P of a proper subgroup of G that contains B is called parabolic subgroup. Finally a BN-pair is called split BN-pair if B = U ⋊ T for some normal subgroup U of B. The standard example of a BN-pair, is the pair of subgroups (B, N) of a semisimple algebraic group over an algebraically closed field, where B is a Borel subgroup, T ≤ B a maximal torus and N := N_{G}(T). For further information about BN-pairs see [42].

As an immediate consequence of the results about quasiunipotent groups in Chapter 2 a very important theorem by DeBonis and Nesin [33] can be generalized to the characteristic 0 case. We need the following version.

Theorem 4.1 Let G be an infinite, simple K^{∗}-group of finite Morley rank that does not interpret a bad field and which has a BN-pair (B, N) of Tits rank 2. Assume further that B = UT is solvable where U ⊲ B is the maximal quasiunipotent subgroup of B and T := B ∩ N. Then G ≃ PSL_{3}(K), PSp_{4}(K) or G_{2}(K) for some algebraically closed field K.

Virtually no changes in the proof of the corresponding result in [33] are needed. Sketch of the proof. As G has Tits rank 2, G contains two parabolic sub-

groups P_{1} and P_{2}. For i = 1, 2, set R_{i} = ⋂_{g∈P_{i}} B^{g}. Then P_{i}/R_{i} is called a Levi factor of G. By [33, Theorem 1] the two Levi factors of G are infinite simple algebraic groups over algebraically closed fields, since otherwise there would exist a definable normal subgroup V of U such that U/V ≅ K^{∗} for some algebraically closed field K by [33, Section 6]. This cannot be the case since U/V is quasiunipotent by Fact 2.4. Since B is solvable the proofs of [33, Lemma 22 and Proposition 23] give us that the Levi factors are actually 3-transitive permutation groups isomorphic to PSL_{2}(K_{i}) for some algebraically closed fields K_{i} (i = 1, 2) acting on the corresponding projective lines. The rest of the proof can be copied from [33] noting that in the proof of [33, Lemma 24 i] “p-subgroup” has to be replaced by “quasiunipotent subgroup”. □

We are still far from a complete classification of tame K^{∗}-groups of small Prüfer 2-rank. In the following we will restrict ourselves to the case where pr(G) = 2 and the centralizers of four-subgroups are minimal in the following sense.

Proposition 4.2 Let G be a simple K^{∗}-group of odd type that does not interpret any bad field. Assume that pr(G) = 2 and let D be the four-subgroup which is contained in the connected component of a Sylow 2-subgroup S of G. Set O_{D} := ⋂_{l∈D^{∗}} O(C_{G}(l)). If T_{0} := C_{G}(D)^{∘}/O_{D} ≅ K^{∗} × K^{∗} for some algebraically closed field of characteristic not 2, then C_{G}(i)^{∘}/O(C_{G}(i)) is isomorphic to one of the following groups for any i ∈ D^{∗}.

(i) K^{∗} × K^{∗}.

(ii) GL_{2}(K)

(iii) PSL_{2}(K) × K^{∗} ≅ GL_{2}(K)/⟨−I⟩ where −I = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ).

(iv) SL_{2}(K) ∗ SL_{2}(K) where the two copies of SL_{2}(K) intersect non-trivially.

If C_{G}(i)^{∘}/O(C_{G}(i)) is isomorphic to one of the groups in (iii) or (iv), then O(C_{G}(l)) = 1 for all l ∈ D^{∗}.

Proof. Let i ∈ D^{∗}, then C := C_{G}(i)^{∘}/O(C_{G}(i)) is a central product of an abelian divisible group T and a semisimple group H all of whose components are simple algebraic groups over algebraically closed fields of characteristic different from 2 by Fact 3.9. Furthermore pr(C) = 2 by Lemma 3.11 and pr(H) = pr(C)−pr(T) by Lemma 3.12. Thus H is the central product of simple algebraic groups of Prüfer 2-rank less than or equal to 2 by [42, 27.5].

Set C_{k} := C_{G}(k) and O_{k} := O(C_{k}) for all k ∈ I(G). Let θ be the signalizer functor defined by θ(s) := O_{s} for all s ∈ I(G). Then O_{i} ∩ C_{j} = O_{j} ∩ C_{i} = O_{j} ∩ O_{i} = O_{D} for all j ∈ D^{∗}, j ≠ i. Hence

C_{i} ∩ C_{j}/O_{D} ≅ (C_{i} ∩ C_{j})O_{i}/O_{i} ≅ (C_{i} ∩ C_{j})O_{j}/O_{j}.

Thus by [3, 2.52] (C_{i} ∩ C_{j})/O_{D} ≅ C_{C_{i}/O_{i}}(jO_{i}) ≅ C_{C_{j}/O_{j}}(iO_{j}). This implies that T ≤ C_{C}(jO_{i}) ≅ C_{G}(D)/O_{D} = T_{0} and either T = 1, T ≅ K^{∗} or T ≅ T_{0}, since T is connected and G does not interpret a bad field.

Thus (i) holds, if pr(T) = 2, as in this case H = 1 and T ≅ T_{0}. If pr(T) = 1, then T ≅ K^{∗} and pr(H) = 1. Hence H is of type A_{1}(K), i.e. H ≅ SL_{2}(K) or H ≅ PSL_{2}(K). If C = H × T, then H ≅ PSL_{2}(K), as C_{C}(jO_{i}) ≅ T_{0} for any involution j ∈ H. Thus C ≅ GL_{2}(K)/⟨−I⟩ in this case. If on the other hand C = H ∗ T, where H ∩ T is non-trivial, then H ≅ SL_{2}(K) and C ≅ GL_{2}(K).

If finally pr(H) = 2, then T is trivial by Lemma 3.12 and H is of type A_{1}(K) × A_{1}(K), A_{2}(K), C_{2}(K) or G_{2}(K). However, H has to contain a central involution. Thus H cannot be of type A_{2}(K) or G_{2}(K) and it cannot be isomorphic to PSp_{4}(K). As furthermore C_{C}(jO_{i}) ≅ T_{0} for any j ∈ D\⟨i⟩, H cannot be isomorphic to Sp_{4}(K) either. Hence H is of type A_{1}(K) × A_{1}(K), contains the central involution i and C_{C}(jO_{i}) ≅ T_{0} for any other involution j. Especially C ≅ SL_{2}(K) ∗ SL_{2}(K), where the two copies of SL_{2}(K) intersect in ⟨i⟩.

If C is isomorphic to the group in (iii) or (iv), then C_{G}(i)^{∘} contains an elementary abelian subgroup E of order 8 which contains D by Lemma 3.10 as C does. Thus O(C_{G}(i)) = 1 by Corollary 3.16. □

We expect that all simple tame K^{∗}-groups of odd type and Prüfer 2-rank 2 fulfill the assumptions of Proposition 4.2.

Conjecture 7 Let G be a simple tame K^{∗}-group of odd type and Prüfer 2-rank 2. Let i ∈ G be any involution and C := C_{G}(i)^{∘}/O(C_{G}(i)).

(1) If C ≅ GL_{2}(K), then G ≅ PSL_{3}(K)

(2) If C ≅ PSL_{2}(K) × K^{∗}, then G ≅ PSp_{4}(K).

(3) If C ≅ SL_{2}(K) ∗ SL_{2}(K), where the two copies of SL_{2}(K) intersect nontrivially, then G ≅ PSp_{4}(K) or G ≅ G_{2}(K).

4.1. PSL_{3}(K) 55

Furthermore one of these three cases holds.

We are going to prove cases (2) and (3) of the conjecture. We furthermore have a partial result of case (1), if O(C_{G}(i)) = 1. To prove the complete case (1) seems to be extremely challenging and one will probably need new methods. It should furthermore be difficult to exclude case (i) from Proposition 4.2, as we do not have any information about normalizers of maximal tori and conjugacy classes of involutions in this case.

4.1 PSL_{3}(K)

The aim of this section is to prove the following result which is a partial analog to the result in finite group theory by Brauer [22] and a first step in proving case (1) of Conjecture 7. The main difficulty will probably be to show that cores of centralizers of involutions of a group G are trivial, if G does not contain an elementary abelian subgroups of order 8.

Theorem 4.3 Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that the centralizers of involutions of G are isomorphic to GL_{2}(K) for some algebraically closed field K of characteristic not 2. Then G is isomorphic to PSL_{3}(K), if there exists an involution i and a quasiunipotent subgroup U ≤ C_{G}(i) which is maximal in C_{G}(i) but not maximal in G.

We strongly believe, that G is always isomorphic to PSL_{3}(K) and this is the result one would need for the classification program. At the moment we just have the following partial information.

Proposition 4.4 Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that the centralizers of involutions of G are isomorphic to GL_{2}(K) for some algebraically closed field K of characteristic not 2 and that a maximal quasiunipotent subgroup U ≤ C_{G}(i) is maximal in G for some i ∈ I(G). Set f := rk(Z(C_{G}(i)).

(i) C_{G}(x)^{∘} is abelian for all 1 ≠ x ∈ G\I(G).

(ii) Either G = ⋃_{i∈G} C_{G}(i) and rk(G) = 8f or there exists a maximal quasiunipotent group V which is not conjugate to U such that rk(G) = rk(U) + 7f + r for some 0 ≤ r ≤ f and furthermore 8f < rk(G) < 11f in this case.

4.1.1 Basic results

Let G be a group as in Theorem 4.4. Let S be a Sylow 2-subgroup of G. By [18, 6.22] S contains a central involution k and hence S ≤ C_{k}. Let D be a foursubgroup in S^{∘}. Since C_{G}(l) contains only two conjugacy classes of involutions for every l ∈ D^{∗}, all involutions in D must be conjugate and N_{G}(D)/C_{G}(D) ≅ S_{3}. D is characteristic in S^{∘} and Sylow 2-subgroups of groups of finite Morley rank are conjugate. Thus four-subgroups in the connected component of a Sylow 2subgroup must be conjugate to D and hence all four-subgroups of G are conjugate to each other.

Proposition 4.5 G = ⟨C_{G}(l)|l ∈ D^{∗}⟩ and Z(C_{G}(i))∩Z(C_{G}(j)) = 1 for i, j ∈ D^{∗} and i ≠ j.

Proof. G = ⟨C_{G}(l)|l ∈ D^{∗}⟩ by Corollary 3.17. Since j and ij are conjugate in C_{G}(i), Z(C_{G}(i)) ∩ Z(C_{G}(j)) = Z(C_{G}(i)) ∩ Z(C_{G}(ij)) = Z(G) = 1. □

Let T := C_{G}(D) and U a quasiunipotent subgroup of G that is normalized by T. Then U = ⟨C_{U}(l)^{∘}|l ∈ D^{∗}⟩ due to [19, 4.6] and U is called n-dimensional where n = |{l ∈ D^{∗}|C_{U}(l) ≠ 1}|. There are exactly 6 one-dimensional unipotent subgroups which are normalized by T, since there are exactly two in C_{G}(l) for each l ∈ D^{∗}. Furthermore N_{G}(T) acts transitively on them. Assume that D = ⟨i_{0}, i_{1}⟩ and let X ≤ C_{G}(i_{0}) be a one-dimensional quasiunipotent subgroup which is normalized by T. Let i_{2} := i_{0}i_{1} and write \overline{x} := xT for all x ∈ N_{G}(T).

Lemma 4.6 (i) W := N_{G}(T)/T ≅ S_{3} is a finite group of order 6, which is generated by \overline{w}_{1} and \overline{w}_{2}, where w_{n} is an involution in N_{G}(T)\T ∩ C_{G}(i_{n}) for n = 1, 2. The other elements of W are \overline{1}, \overline{w}_{0} = \overline{w}_{1}^{\overline{w}_{2}} = \overline{w}_{2}^{\overline{w}_{1}}, \overline{y} = \overline{w}_{2}\overline{w}_{1} and \overline{y}^{2} = \overline{y}^{−1} = \overline{w}_{1}\overline{w}_{2}.

(ii) w_{0}T Xw_{0} ⊆ T Xw_{0}X.

(iii) ⟨T, X, X^{w_{0}}⟩ = C_{G}(i_{0}).

Proof. (i) follows directly from the definitions. As C_{G}(i_{0}) is a reductive algebraic group of rank 1, it has a BN-pair (B_{1}, N_{1}), where B_{1} := T X and N_{1} = ⟨w_{0}, T ⟩. Hence w_{0}T Xw_{0} ⊆ T X ∪ T Xw_{0}X = T Xw_{0}X and C_{G}(i_{0}) = ⟨B, N⟩ = ⟨T, X, X^{w_{0}}⟩ which proves (ii) and (iii). □

Lemma 4.7 Let X_{1} := X, X_{2} := X_{1}^{w_{0}} and X_{n}^{(λ)} := X_{n}^{y^{λ}} for any n = 1, 2 and λ ∈ ℤ. Then

(i) X_{n}^{(λ)} = X_{n}^{(λ+3)} for all n = 1, 2 and λ ∈ ℤ.

(ii) (X_{1}^{(λ)})^{w_{κ}} = X_{2}^{(2λ−κ)} for κ, λ = 0, 1, 2.

(iii) i_{λ} centralizes X_{n}^{(λ)} and inverts X_{n}^{(λ−1)} and X_{n}^{(λ+1)} for λ = 0, 1, 2 and n = 1, 2.

Proof. Since X is normalized by T and N_{G}(T)/T ≅ S_{3}, (i) follows. Furthermore \overline{y} is inverted by \overline{w}_{λ} for λ = 0, 1, 2, \overline{y^{λ}} = \overline{w_{λ}w_{0}} and hence

\overline{y^{κ}w_{λ}y^{λ−2κ}w_{0}} = \overline{y^{3κ−λ}w_{λ}w_{0}} = \overline{y^{−λ}y^{λ}} = \overline{1}

for λ, κ = 0, 1, 2 which proves (ii). Elementary computations in GL_{2}(K) show finally that i_{0} centralizes X_{n} and that i_{1}, i_{2} invert X_{n} for n = 1, 2. □

By Proposition 2.18 there exists a maximal quasiunipotent subgroup U of G that contains X_{1} and is normalized by T. There are only four possible cases.

Lemma 4.8 Let U be a maximal quasiunipotent subgroup of G that contains X_{1} and is normalized by T. Then one of the following holds.

(1) U is 3-dimensional.

(a) Z(U) is 3-dimensional. Then U = X_{1}X_{1}^{(1)}X_{1}^{(2)}

(b) Z(U) is 1-dimensional. Without loss of generality Z(U) = X_{1} and U = X_{1}⟨X_{2}^{(1)}, X_{2}^{(2)}⟩.

(2) U is 2-dimensional. We may assume that U = X_{1} × X_{2}^{(2)}.

(3) U is 1-dimensional and U = X_{1}.

Proof. U = ⟨C_{U}(l)^{∘}|l ∈ D^{∗}⟩ due to [19, 4.6]. Thus U is generated by at k one-dimensional quasiunipotent subgroups for 1 ≤ k ≤ 3 as T acts transitively on X_{1}\{1}. As U is nilpotent, Z(U) is infinite and Z(U)^{∘} = ⟨C_{Z(U)}(l)^{∘}|l ∈ D^{∗}⟩ by [19, 4.6] again. As furthermore T acts transitively on X\{1}, either C_{Z(U)}(l)^{∘} = C_{U}(l)^{∘} or C_{Z(U)}(l)^{∘} = 1 for all l ∈ D^{∗}. We may assume that X_{1} ≤ Z(U) by conjugating with an element from N_{G}(T).

If U is 1-dimensional, then obviously U = X_{1}. Let thus U be at least 2dimensional. If X_{1}^{(λ)} ≤ U for some λ = 1, 2, then X_{1} = X_{1}^{λ−λ} ≤ C_{G}(X_{1}^{−λ}) and X_{1}^{(k)} ≤ C_{G}(X_{1}) for all k = 0, 1, 2. Hence (X_{2}^{k}) ∩ U = 1 for all k = 0, 1, 2, since otherwise w_{k} ∈ C_{G}(i_{k}) ≤ N_{G}(X_{1}) by Lemma 4.6. Thus U = X_{1}X_{1}^{(1)}X_{1}^{(2)} in this case and Z(U) is 3-dimensional.

Assume therefore that X_{1} < U ≤ X_{1}⟨X_{2}^{(1)}, X_{2}^{(2)}⟩. If U is 2-dimensional, then U = X_{1} × X_{2}^{(k)} for some k = 1, 2 and we are in case (2). If U is 3-dimensional, U = X_{1}⟨X_{2}^{(1)}, X_{2}^{(2)}⟩. These are all possible cases. □

Lemma 4.9 If X_{1} centralizes Y := ⟨X_{2}^{(1)}, X_{2}^{(2)}⟩, then Y^{w_{k}} ⊆ T Y w_{k}X_{2}^{(k)} for k = 1, 2.

Proof. Let k ∈ {1, 2}. Since X_{1} is connected there exists n ∈ ℕ such that Y = (X_{2}^{(k)}X_{2}^{(−k)})^{n} by Zil’bers indecomposability theorem [18, 5.28]. Set Y_{m} = (X_{2}^{(k)}X_{2}^{(−k)})^{m} for all 1 ≤ m ≤ n. We proceed by induction on m. Then by Lemma 4.7 and Lemma 4.6

(Y_{1})^{w_{k}} = X_{1}^{(2k−k)}X_{1}^{−2k−k} = X_{1}^{(k)}X_{1} ⊆ T X_{2}^{(k)}w_{k}X_{2}^{(k)}X_{1}.

Since X_{1} centralizes X_{2}^{(k)} by assumption

Y_{1}^{w_{k}} ⊆ T X_{2}^{(k)}X_{1}^{w_{k}}w_{k}X_{2}^{(k)} = T Y_{1}w_{k}X_{2}^{(k)}.

Furthermore by induction hypothesis

Y_{m+1}^{w_{k}} = Y_{m}X_{2}^{(k)}X_{2}^{(−k)} ⊆ T (Y_{m}w_{k}X_{2}^{(k)})(X_{2}^{(k)}X_{2}^{(−k)}w_{k}X_{2}^{(k)})

and hence

Y_{m+1}^{w_{k}} ⊆ T Y_{m}Y_{1}^{w_{k}}X_{2}^{(k)} ⊆ T Y_{m+1}w_{k}X_{2}^{(k)}

which gives us the claim. □

Theorem 4.10 Assume that there exists a maximal quasiunipotent Q of G which is normalized by T and 3-dimensional and non-abelian. Then G is a split BN-pair of Tits rank 2, where B = N_{G}(Q)^{∘}, N := N_{G}(T) and \overline{w_{1}}, \overline{w_{2}} are generators of the Weyl group.

Proof. We may assume by Lemma 4.8 that Q = X_{1}Y, where X_{1} centralizes Y = ⟨X_{2}^{(1)}, X_{2}^{(2)}⟩. Then B = Q ⋊ T by Proposition 2.20 and B ≤ N_{G}(Z(Q)) = N_{G}(X_{1}). We first show BN1. G = ⟨C_{G}(k)^{∘}| k ∈ D^{∗}⟩ by Proposition 4.5. As ⟨N, X_{1}⟩ ≥ C_{G}(i)^{∘} by Lemma 4.6 and N operates transitively on D^{∗}, G = ⟨B, N⟩. Furthermore B ∩ N = T ⊲ N, since B = Q ⋊ T, as desired. BN2 follows from the fact that ⟨\overline{w}_{1}, \overline{w}_{2}⟩ = N_{G}(T)/T by Lemma 4.7. Let S := \{\overline{w}_{1}, \overline{w}_{2}\} and W := N/T.

We prove BN3. We have to show that vBw ⊆ BvwB ∪ BvB for all v, w ∈ N such that \overline{v} ∈ W and \overline{w} ∈ S. By Lemma 4.9 and Lemma 4.7

w_{1}B = T X_{1}^{w_{1}}Y^{w_{1}}w_{1} ⊆ T X_{2}^{(2)}Y w_{1}X_{2}^{(1)}w_{1} = T X_{2}^{(2)}Y X_{1}^{(1)} ⊆ BX_{1}^{(1)}

and

w_{2}B = T X_{1}^{w_{2}}Y^{w_{2}}w_{2} ⊆ T X_{2}^{(1)}Y X_{1}^{(2)} ⊆ BX_{1}^{(2)}.

Hence

w_{1}Bw_{1} ⊆ Bw_{1}(X_{1}^{(1)})^{w_{1}} = Bw_{1}X_{2}^{(1)} ⊆ Bw_{1}B,
w_{1}By^{2} ⊆ By^{2}(X_{1}^{(1)})^{y^{2}} = By^{2}X_{1} ⊆ By^{2}B,
w_{1}Bw_{0} ⊆ Bw_{0}(X_{1}^{(1)})^{w_{0}} = Bw_{0}X_{2}^{(2)} ⊆ Bw_{0}B

as well as

w_{2}Bw_{2} ⊆ Bw_{2}(X_{1}^{(2)})^{w_{2}} = Bw_{2}X_{2}^{(2)} ⊆ Bw_{2}B,
w_{2}By ⊆ By(X_{1}^{(2)})^{y} = ByX_{1} ⊆ By^{2}B,
w_{2}Bw_{0} ⊆ Bw_{0}(X_{1}^{(2)})^{w_{0}} = Bw_{0}X_{2}^{(1)} ⊆ Bw_{0}B

proving BN3 in these cases. For the other cases just note that

w_{1}By = (w_{1}Bw_{1})w_{0} ⊆ B(w_{1}Bw_{0}) ⊆ Bw_{0}B = Bw_{1}yB

and

w_{1}Bw_{2} = (w_{1}Bw_{1})y^{2} ⊆ B(w_{1}By^{2}) ⊆ By^{2}B = Bw_{1}w_{2}B

as well as

w_{2}By^{2} = (w_{2}Bw_{2})w_{0} ⊆ B(w_{2}Bw_{0}) ⊆ Bw_{0}B = Bw_{2}y^{2}B

and

w_{2}Bw_{1} = (w_{2}Bw_{2})y ⊆ B(w_{2}By) ⊆ ByB = Bw_{2}w_{1}B

yielding the claim as wB1 ⊆ Bw1B for all w ∈ N.

We finally prove BN4. Assume that B = B^{w_{k}} for some k = 1, 2. Then (X_{2}^{(k)})^{w_{k}} = X_{1}^{(k)} ≤ B ≤ N_{G}(X_{1}) and w_{k} ∈ C_{G}(i_{k}) ≤ N_{G}(X_{1}). Contradiction. Hence B ≠ B^{w_{k}} for k = 1, 2 and we have proved everything. □

4.1.2 3- and 2-dimensional case

Assume from now on that all maximal quasiunipotent subgroup of G that are normalized by T are abelian. We are first going to show that U cannot be 2dimensional.

We call t ∈ T regular point of T iff C_{G}(t) = T and denote the set of regular points by R(T). An element g ∈ G is called regular point iff it is conjugate to an element of R(T). Set furthermore T [w_{k}] := {t ∈ T | t^{w_{k}} = t^{−1}} for all k = 0, 1, 2.

Lemma 4.11 R(T) is a generic subset of T. In particular there are 3 different kinds of points in T.

(a) The set of regular points R(T).

(b) The set of central points X := ⋃_{l∈D^{∗}} Z(C_{G}(l)).

(c) The finite group C_{T} (y), where y ∈ N_{G}(T), such that y^{3} ∈ T.

Finally, C_{G}(t) = T ⟨y⟩ for t ∈ C_{T} (y).

Proof. Let t ∈ T \X. Then C_{G}(t)^{∘} = ⟨C_{C_{G}(t)}(l)^{∘}|l ∈ D^{∗}⟩ by [19, 5.14]. Since t ∉ X, C_{G}(t) ∩ C_{G}(l) = T for l ∈ D^{∗} and hence C_{G}(t)^{∘} = T. Let x ∈ C_{G}(t)\T. Then x ∈ N_{G}(T). However x ∉ I(N_{G}(T)), since for x ∈ I(N_{G}(T)): C_{G}(x) ∩ T ⊆ X. Thus t ∈ C_{T} (y) ∪ R(T) and C_{G}(t) = T ⟨y⟩ for t ∈ C_{T} (y).

We finally show that C_{T} (y) is a finite group. Let t ∈ C_{T} (y). We may assume that y = w_{2}w_{1}. Then

(tt^{w_{2}})^{y} = tt^{w_{1}} ∈ Z(C_{G}(i_{2}))^{y} ∩ Z(C_{G}(i_{1})) = Z ∩ Z(C_{G}(i_{1})) = 1

by Proposition 4.5. Hence t^{w_{2}} = t^{w_{1}} = t^{−1} and C_{T} (y) ≤ T [w_{1}]. As T [w_{1}] ≅ K^{∗} and G does not interpret a bad field, C_{T} (y) is either finite or equals T [w_{1}]. However C_{T} (y) < T [w_{1}] since i_{1} ∉ C_{T} (y) and the claim follows. □

Lemma 4.12 Let H ≤ C_{G}(i_{0}) be a definable subgroup that contains T. Then either H = T, H = X_{1}T, H = X_{2}T or H = C_{G}(i_{0}).

Proof. First assume that H is solvable. Then T ≤ H is contained in a Borel subgroup – in the sense of algebraic group theory – of C_{G}(i_{0}) by [42, 23.1]. There are exactly two of them, namely X_{1}T and X_{2}T. Since T operates transitively on X_{n}\{1} for n = 1, 2 either H = T or H = X_{n}T for n = 1, 2. If H is non-solvable then H^{∘}/σ is isomorphic to PSL_{2}(K) for an algebraically closed field K where σ is the solvable radical of H^{∘} by [2]. Since σ is solvable and normalized by H, σ = Z(C_{G}(i_{0})) which forces H to be equal to C_{G}(i_{0}). □

Proposition 4.13 There is a maximal quasiunipotent subgroup V of G that is normalized by T such that C_{G}(X_{1}) = V ⋊ Z(C_{G}(i_{0})).

Proof. Set Z := Z(C_{G}(i_{0})). We show first that C_{G}(X_{1})^{∘} = V ⋊ Z. Let V be a maximal quasiunipotent subgroup that is normalized by T and contains X_{1}. V exists by Proposition 2.18. V is – as maximal quasiunipotent subgroup normalized by T – abelian and hence contained in C_{G}(X_{1}). On the other hand N_{G}(X_{1})^{∘} = ⟨N_{k}| k = 0, 1, 2⟩ by [19, 5.14], where N_{k} := C_{N_{G}(X_{1})}(i_{k})^{∘} for k = 0, 1, 2. Let k ∈ {0, 1, 2}. Since X_{1}^{w_{k}} = X_{2}^{(−k)} by Lemma 4.7, T ≤ N_{k} < C_{G}(k). Thus either N_{k} = T or N_{k} = T X_{n}^{(k)} for n = 1 or n = 2 by Lemma 4.12. Especially N_{G}(X_{1})^{∘} ≤ T X_{1}⟨X_{n}^{(1)}, X_{n}^{(2)}⟩ for n = 1, 2 by Lemma 4.8.

We show that N_{G}(X_{1}^{∘}) is solvable. Assume that N_{G}(X_{1})^{∘} is not solvable. Then N_{G}(X_{1})^{∘} = T X_{1}⟨X_{2}^{(1)}, X_{2}^{(2)}⟩ by Lemma 4.8. Let σ be the solvable radical of N_{G}(X_{1})^{∘}. Then X_{1} ≤ σ. \overline{N} := N_{G}(X_{1})^{∘}/σ is the direct product of simple algebraic groups over algebraically closed fields by [2], i.e. \overline{N} ≅ P SL(2, K), as G does not interpret any bad field. Let S be the Sylow 2-subgroup of T. Then Sσ/σ ≅ S/(σ ∩S) is a Sylow 2-subgroup of \overline{N} by Lemma 3.11. As pr(T) = 2 and pr(\overline{N}) = 1, this implies that σ ∩S is infinite and hence that T_{0} := (σ ∩T)^{∘} ≅ K^{∗}, as T does not interpret a bad field. σ^{∘} = X_{1} ⋊ T_{0} ≤ C_{G}(i_{0})^{∘} is normalized by X_{2}^{(k)} for k = 1, 2. Thus T_{0}^{g} ≤ C_{G}(i_{0})^{∘} for any g ∈ N_{G}(X_{1})^{∘} and hence i_{0}^{X_{2}^{(1)}} ∪ i_{0}^{X_{2}^{(2)}} ⊆ C_{G}(T_{0}). This implies by Lemma 4.11 that T_{0} ≤ Z(C_{G}(i_{1})) ∩ Z(C_{G}(i_{2})). Contradiction, as Z(C_{G}(i_{1})) ∩ Z(C_{G}(i_{2})) = 1 by Proposition 4.5.

Hence N_{G}(X_{1})^{∘} is solvable and V is normal in N_{G}(X_{1})^{∘} by Proposition 2.16. Thus N_{G}(X_{1})^{∘} ≤ N_{G}(V)^{∘} = V ⋊ T by Proposition 2.20 and N_{G}(X_{1})^{∘} = V ⋊ T. Especially C_{G}(X_{1})^{∘} = V ⋊ (T ∩ C_{G}(X_{1})^{∘}) = V ⋊ Z.

We finally claim that C_{G}(X_{1}) is connected. Let x ∈ C_{G}(X_{1}). Then X_{1} ≤ C_{G}(i_{0}^{x}) and hence

(∗) i_{0}^{x} ∈ Z(C_{G}(i_{0}^{x})) ≤ C_{G}(X_{1})^{∘} = V ⋊ Z.

Since Z ∩Z(C_{G}(i_{k})) = 1 for k = 1, 2 by Proposition 4.5, Z acts freely on X_{n}^{(k)}\{1} for n, k = 1, 2. Thus X_{n}^{(k)} ⋊ Z = X_{n}^{(k)} ∪ ⋃_{u∈X_{n}^{(k)}} Z^{u} for n, k = 1, 2 by [18, ex.11, p.98]. As V is abelian, this gives us that V ⋊ Z = V ∪ ⋃_{u∈V} X_{1}Z^{u}. By (∗) i_{0}^{x} ∈ V Z and as i_{0}^{x} is an involution it cannot be contained in V. This means that i_{0}^{x} ∈ X_{1}Z^{v} for some v ∈ V. However, X_{1} is a 2^{⊥}-group that is centralized by Z^{v} and thus i_{0}^{x} ∈ Z^{v} = Z(C_{G}(i_{0}^{v})). Hence i_{0}^{x} = i_{0}^{v} and vx^{−1} ∈ C_{G}(i_{0}) ∩ C_{G}(X_{1}) = ZX_{1} ≤ C_{G}(X_{1})^{∘}. Especially x ∈ C_{G}(X_{1})^{∘} and the claim is proved. □

Corollary 4.14 X_{1} is contained in exactly one maximal quasiunipotent subgroup, namely Q(C_{G}(X_{1})).

Proof. V := Q(C_{G}(X_{1})) is a maximal quasiunipotent subgroup that contains X_{1} by Proposition 4.13. Let W ≥ X_{1} be another maximal quasiunipotent subgroup of G. As W is nilpotent, Z(W) is infinite by [18, 6.2]. Then 1 ≠ Z(W)^{∘} ≤ Q(C_{G}(X_{1})) = V and since V is abelian, V, W ≤ C_{G}(Z(W)^{∘}). As C_{G}(Z(W)^{∘}) is a K-group, V and W are conjugate by Proposition 2.16. Thus W is abelian as well and W = Z(W)^{∘} ≤ V. By the maximality of W, V = W. □

Theorem 4.15 Let U be a maximal quasiunipotent subgroup of G that is normalized by T. Then U is not 2-dimensional.

Proof. Assume that U is 2-dimensional. Thus without loss of generality U = X_{1} × X_{2}^{(2)} by Lemma 4.8 and U is normalized by w_{1} ∈ C_{G}(i_{1}). Let V = C_{U}(w_{1}). Then V = {uu^{w_{1}}| u ∈ X_{1}}. Furthermore since U is the direct product of X_{1}, X_{2}^{(2)} there is a definable bijection ϕ : X_{1} ⟶ V where ϕ(u) = uu^{w_{1}}. Hence V is connected and rk(V) = rk(X_{1}). Thus V is a maximal quasiunipotent subgroup of C_{G}(w_{1}) and

w_{1} ∈ C_{G}(V) = U ⋊ Z(C_{G}(w_{1})) ≤ N_{G}(U)^{∘}

by Proposition 4.13. However N_{G}(U)^{∘} = U ⋊ T by Proposition 2.20. Thus w_{1} = u_{0}u_{2}t for u_{0} ∈ X_{1}, u_{2} ∈ X_{2}^{(2)}, t ∈ T and i_{2} = i_{0}^{w_{1}} = i_{0}^{u_{2}t}. Since u_{2}t ∈ C_{G}(i_{2}) we conclude that i_{0} = i_{2}. Contradiction. □

It is slightly more difficult to show that U is not 3-dimensional.

Lemma 4.16 Let i ∈ I(G) be an involution of G. Then

(i) N_{G}(C_{G}(i)) = C_{G}(i),

(ii) C_{G}(z) = C_{G}(i) for any z ∈ Z(C_{G}(i)) and

(iii) the only strongly real element in Z(C_{G}(i)) is i.

Proof. Let i ∈ I(G). If x ∈ N_{G}(C_{G}(i)) then C_{G}(i) = C_{G}(i^{x}) and since I(Z(C_{G}(i))) = {i}, i = i^{x} and x ∈ C_{G}(i). Let furthermore z ∈ Z(C_{G}(i)). Then C_{G}(z)^{∘} = ⟨C_{C_{G}(z)}(l)^{∘}| l ∈ D^{∗}⟩ by [19, 5.14]. Let l ∈ D^{∗}\{i} then z ∉ Z(C_{G}(l)) due to Proposition 4.5 and hence (C_{G}(z) ∩ C_{G}(l))^{∘} = T ≤ C_{G}(i) by Lemma 4.11. Thus C_{G}(z)^{∘} = C_{G}(i) and (i) implies that C_{G}(z) = C_{G}(i). Let finally x ∈ Z(C_{G}(i)) be strongly real. Then there exists an involution w ∈ I(G) such that x^{w} = x^{−1}, i.e. w normalises C_{G}(x) = C_{G}(i) by (ii). Hence w ∈ C_{G}(i) by (i) and x = x^{−1} is an involution. □

Proposition 4.17 Let u ∈ X_{1} be a unipotent element and z ∈ Z(C_{G}(i_{0})). Then C_{G}(zu)^{∘} is solvable and C_{G}(u)^{∘} = C_{G}(X_{1})

Proof. Let N := C_{G}(zu)^{∘} and Z := Z(C_{G}(i_{0})). Then Z ≤ N. Furthermore N cannot contain a four-subgroup D, since C_{G}(D) does not contain unipotent elements. Hence any Sylow 2-subgroups of N is connected and contains a single involution and N has Prüfer 2-rank 1. N contains thus only one conjugacy class of involutions as the Sylow 2-subgroups of N are conjugate. Let σ be the solvable radical of N and S a Sylow 2-subgroup of σ. By the Frattini argument [18, 10.12] N = N_{N}(S)σ. Assume that S ≠ 1. Then I(N) ≤ σ and we may assume that i_{0} ∈ S. Thus N_{N}(S) = C_{N}(i_{0}) = X_{1} ⋊ Z is solvable and N = σ.

We assume from now on that σ does not contain involutions. Let \overline{N} := N/σ. As pr(\overline{N}) = 1 by Lemma 3.11, \overline{N} ≅ PSL_{2}(K) for some algebraically closed field K of characteristic not 2 by [2]. As σ is a 2^{⊥}-group, \overline{ZX_{1}} = \overline{C_{N}(i_{0})} = C\overline{N}(\overline{ı}_{0}) by [3, 2.52]. The centralizers of involutions in PSL_{2}(K) where K is an algebraically closed field of characteristic not 2 are finite extensions of tori and thus X_{1} ≤ σ. As σ is solvable it contains a unique maximal quasiunipotent subgroup Q(σ). Let V be a maximal quasiunipotent subgroup of G that contains Q(σ). Then X_{1} ≤ V^{g} for all g ∈ N as X_{1} ≤ Q(σ). Thus V^{g} ≤ C_{G}(X_{1}) for all g ∈ N as V is abelian. However, C_{G}(X_{1}) = V ⋊ Z by Proposition 4.13 and Corollary 4.14. Thus V = V^{g} for all g ∈ N. Hence N ≤ N_{G}(V)^{∘} = V ⋊ T by Proposition 2.20 and N is solvable. Contradiction.

Let z = 1, then N = C_{G}(u)^{∘} ≥ C_{G}(X_{1}) = V ⋊Z. As N is solvable, V = Q(N) by Proposition 2.16. Thus N ≤ N_{G}(V)^{∘} = V ⋊ Z by Proposition 2.20 which implies the claim. □

Remark 4.18 Let g ∈ C_{G}(i) for some i ∈ I(G), then g is conjugate to an element in T or Z(C_{G}(i_{0}))X_{1}.

Proof. As G has one conjugacy class of involutions, we may assume that g ∈ C_{G}(i_{0}). As C_{G}(i_{0}) is isomorphic to a connected algebraic group all elements of C_{G}(i_{0}) are conjugate to an element in the Borel-subgroup B = X_{1}T of C_{G}(i_{0}). However B = Z(C_{G}(i_{0}))X_{1} ∪ ⋃_{u∈X_{1}} T^{u} of C_{G}(i_{0}) and we are done. □

Corollary 4.19 Let g ∈ G. Then either C_{G}(g) = C_{G}(i) for some i ∈ I(G) or C_{G}(g)^{∘} is solvable.

Proof. If C_{G}(g) is a 2^{⊥}-group, then C_{G}(g)^{∘} is nilpotent by [18, B1]. Hence we may assume that g ∈ C_{G}(i) for some i ∈ I(G). Then g is conjugate to an element in T or Z(C_{G}(i_{0})X_{1} by Remark 4.18. Thus either g ∈ Z(C_{G}(k)) for some k ∈ I(T) and C_{G}(g) = C_{G}(i) by Lemma 4.16 or C_{G}(g)^{∘} is solvable by Proposition 4.17 and Lemma 4.11 □

Lemma 4.20 Let N be a nilpotent 2^{⊥}-group of finite Morley rank and w a definable involutive automorphism. Let V ≤ N be an abelian group such that V ∩ V^{w} = 1. Then rk(C_{N}(w)) ≥ rk(V).

Proof. Without loss of generality N = ⟨V, V^{w}⟩. The proof proceeds by induction on the nilpotency class of N.

If N is abelian, W := {vv^{w}| v ∈ V } ⊆ C_{N}(w). Then N is the direct product of V and V^{w} and we get a definable bijection ϕ : X_{1} ⟶ V by ϕ(u) = uu^{w_{1}}. Hence W is connected and rk(V) = rk(W) ≤ rk(C_{N}(w)).

Let now Z := Z(N) and denote by \overline{X} := XZ/Z for all X ≤ N. Since V is abelian, V Z is abelian as well and we can apply the first part of the proof to V Z ∩ V^{w} ≤ V Z ∩ V^{w}Z. Hence

(∗) rk(C_{V Z∩V^{w}Z}(w)) ≥ rk(V Z ∩ V^{w}).

Furthermore since \overline{V} ∩ \overline{V}^{w} ⊲ \overline{N} and (\overline{V} /(\overline{V} ∩ \overline{V}^{w})) ∩ (\overline{V}^{w}/(\overline{V} ∩ \overline{V}^{w})) = 1, we get by induction hypothesis and Lascar’s equality

(∗∗) rk(C_{\overline{N}/(\overline{V} ∩\overline{V}^{w})}(w)) ≥ rk(\overline{V} /(\overline{V} ∩ \overline{V}^{w})) = rk(V Z/(V Z ∩ V^{w}Z)).

Since N is a 2^{⊥}-group [3, 2.44] implies that

rk(C\overline{N}(w)) = rk(\overline{C_{N}(w)}) = rk(C_{N}(w)) − rk(C_{Z}(w))

and using the same argument again

rk(C_{\overline{N}/(\overline{V} ∩\overline{V}^{w})}(w)) = rk(C\overline{N}(w)) − rk(C_{\overline{V} ∩\overline{V}^{w}}(w))
= rk(C_{N}(w)) − rk(C_{V Z∩V^{w}Z}(w)).

Therefore

rk(C_{N}(w)) = rk(C_{\overline{N}/(\overline{V} ∩\overline{V}^{w})}(w)) + rk(C_{V Z∩V^{w}Z}(w))
≥ rk(V Z) − rk(V Z ∩ V^{w}Z) + rk(V Z ∩ V^{w})

by (∗) and (∗∗). As V Z ∩ V^{w}Z = (V Z ∩ V^{w})Z, [18, ex.18, p.66] implies that

rk(V Z ∩ V^{w}Z) = rk(V Z ∩ V^{w}) + rk(Z) − rk(V^{w} ∩ Z)
= rk(V Z ∩ V^{w}) + rk(V^{w}Z) − rk(V)

and rk(C_{N}(w)) ≥ rk(V Z) + rk(V) − rk(V^{w}Z) = rk(V). □

Theorem 4.21 Let X_{1} ≤ U be a maximal quasiunipotent subgroup of G that is normalized by T. Then U is not abelian and 3-dimensional.

Proof. Assume that U is abelian and 3-dimensional. Thus U is normalized by y := w_{2}w_{1} as U = X_{1}×X_{1}^{(1)}×X_{1}^{(2)} by Lemma 4.8. Set N := C_{G}(y)^{∘}. N is solvable by Corollary 4.19 since y ∈ Sr(G)\I(G) and hence cannot be contained in the center of C_{G}(λ) for some λ ∈ I(G) by Lemma 4.16. Let V_{1} = C_{U}(y), V_{2} = V_{1}^{w_{1}}. Obviously V_{1} = {uu^{y}u^{y^{2}}| u ∈ X_{1}}. Furthermore since U is the direct product of X_{1}, X_{1}^{(1)}, X_{1}^{(2)} we get a definable bijection ϕ : X_{1} ⟶ V_{1} via ϕ(u) = uu^{y}u^{y^{2}} and hence V_{1} is quasiunipotent with rk(V_{1}) = rk(X_{1}). Since V_{1}, V_{2} ≤ Q(N) by Proposition 2.16 and V_{1} ∩ V_{2} ≤ U ∩ U^{w_{1}} = 1, rk(C_{Q(N)}(w_{1})) ≥ rk(V_{1}) by Lemma 4.20. As all quasiunipotent subgroups of centralizers of involutions are conjugate rk(C_{Q(N)}(w_{1})) = rk(V_{1}) and V_{3} := C_{Q(N)}(w_{1}) ≤ Q(N) is a maximal quasiunipotent subgroup of C_{G}(w_{1}). Let W := Q(C_{G}(V_{3})) be the unique maximal quasiunipotent subgroup that contains V_{3} as in Corollary 4.14. W is conjugate to U and thus abelian. Furthermore Q(N) ≤ W as V_{3} ≤ Q(N) and hence V_{1}, V_{2} ≤ W. This implies that V_{2} ≤ C_{G}(V_{1}).

As C_{G}(V_{1}) is solvable by Corollary 4.19, it contains a unique maximal quasiunipotent subgroup which has to be U = W by the maximality of U and W. Hence V_{2} ≤ U. Contradiction. □

This finishes the proof of Theorem 4.3 by Theorem 4.1 and Theorem 4.15.

4.1.3 1-dimensional case

Assume from now on, that all maximal quasiunipotent subgroup of G which are normalized by T are one-dimensional. We won’t be able to exclude this case completely. However, there is a great deal of information available.

Proposition 4.22 N_{G}(X_{1}) = X_{1} ⋊ T

Proof. N_{G}(X_{1})^{∘} = C_{G}(X_{1}) = X_{1} ×Z(C_{G}(i_{0})) by Proposition 4.13 and Proposition 2.20. Thus C_{G}(X_{1}) is a connected normal abelian subgroup of N_{G}(X_{1}). Furthermore C_{G}(X_{1}) ∩ C_{G}(i_{1}) = ⟨i_{0}⟩. As i_{1} inverts X_{1} by Lemma 4.7, i_{1}^{n} inverts X_{1} for all elements n ∈ N_{G}(X_{1}). Thus i_{1}^{N_{G}(X_{1})} ⊆ I(i_{1}C_{G}(X_{1})) = i_{1}^{X_{1}} ∪ i_{0}i_{1}^{X_{1}}. Especially [i_{1}, N_{G}(X_{1})] ⊆ ⟨i_{0}⟩X_{1} ≤ C_{G}(X_{1}). Hence N_{G}(X_{1}) = C_{G}(X_{1})C_{N_{G}(X_{1})}(i_{1}) by [18, ex. 10, p. 98] and C_{N_{G}(X_{1})}(i_{1})^{∘} ≤ C_{G}(X_{1}). Thus C_{N_{G}(X_{1})}(i_{1}) = T by Lemma 4.12 and the claim follows. □

Corollary 4.23 Let x ∈ C_{G}(X_{1})\Z(C_{G}(i_{0})). Then C_{G}(x) = C_{G}(X_{1}).

Proof. As C_{G}(X_{1}) is abelian, C_{G}(X_{1}) ≤ C_{G}(x). Furthermore C_{G}(x)^{∘} is solvable by Proposition 4.17 and hence C_{G}(x)^{∘} ≤ N_{G}(X_{1}) = X_{1} ⋊ T by Proposition 2.16 and Proposition 4.22. This implies that C_{G}(x) ≤ N_{G}(X_{1}) as well and C_{G}(x) = (X_{1} ⋊ T) ∩ C_{G}(x) = X_{1} ⋊ (T ∩ C_{G}(x)) = X_{1} ⋊ Z(C_{G}(i_{0})) = C_{G}(X_{1}). □

Lemma 4.24 If x ∈ C_{G}(i_{0})\{1}, then either (i) C_{G}(x) < C_{G}(i_{0}) is a connected abelian group,

(ii) C_{G}(x) = (T ⟨y⟩)^{g} for some g ∈ C_{G}(i_{0}) or

(iii) C_{G}(x) = C_{G}(k) for some k ∈ I(C_{G}(i_{0})).

Proof. By Remark 4.18 we only have to prove the claim for elements of Z(C_{G}(i_{0}))X_{1} and T. Let x ∈ Z(C_{G}(i_{0}))X_{1}\Z(C_{G}(i_{0})). Then C_{G}(x) = X_{1} ⋊ Z(C_{G}(i_{0})) is a connected abelian group by Proposition 4.23. If x ∈ T, then the claim follows by Lemma 4.11. □

Let H be a group of finite Morley rank. A definable subset X ⊆ H is called generic if rk(X) = rk(H).

Proposition 4.25 The definable set C := ⋃_{k∈I(G)} C_{G}(k) is generic in G.

Proof. Consider the definable map ϕ : G × C_{G}(i_{0}) → C with ϕ(g, c) = c^{g}. Let c ∈ C_{G}(i_{0}) and compute the Morley rank of the fiber of c.

ϕ^{−1}(c) = {(h, x) ∈ G × C_{G}(i_{0})| x^{h} = c}
= {(h, x)| x := c^{h^{−1}} ∈ C_{G}(i_{0}), h ∈ G}.

Therefore

(∗) rk(ϕ^{−1}(c)) = rk({h ∈ G| i_{0}^{h} ∈ C_{G}(c)}) = rk(I(C_{G}(c)) + rk(C_{G}(i_{0}))

since i_{0}^{g_{1}} = i_{0}^{g_{2}} if and only if g_{1}C_{G}(i_{0}) = g_{2}C_{G}(i_{0}). Consider the definable subset Y := {x ∈ C| rk(ϕ^{−1}(x)) = rk(C_{G}(i_{0}))}.

Claim: ϕ^{−1}(Y) is generic in G × C_{G}(i_{0}). By (∗) it is enough to show, that W := {x ∈ C_{G}(i_{0})| rk(I(C_{G}(x))) = 0} is a generic subset of C_{G}(i_{0}). As ⋃_{x∈C_{G}(i_{0})} R(T)^{x} ⊆ W and R(T) ∩ R(T)^{x} = ∅ for all x ∈ C_{G}(i_{0})\N_{G}(T), we have

rk(W) ≥ rk( ⋃_{x∈C_{G}(i_{0})} R(T)^{x}) = rk(C_{G}(i_{0})) + rk(R(T) − rk(T)) = rk(C_{G}(i_{0}))

by Lemma 4.11. □

By additivity of Morley rank this implies that

rk(G) + rk(C_{G}(i_{0})) = rk(ϕ^{−1}(Y)) = rk(Y) + rk(C_{G}(i_{0})).

Thus rk(C) ≥ rk(Y) = rk(G) and C is generic in G. □ □

Remark 4.26 The proof of Lemma 4.25 actually shows that the set of regular points is generic in G.

Let x ∈ G. We call x visible iff x ∈ ⋃_{i∈I(G)} C_{G}(i) and invisible iff x is not visible. We call a definable connected subgroup H ≤ G visible, iff H ≤ C_{G}(i) for an involution i ∈ I(G) and invisible, if H\{1} consists of invisible elements. Notice, that a connected definable subgroup consisting of visible elements need not be visible. Inv(G) denotes the definable set of all invisible elements. We are going to show that all invisible elements are strongly real.

Proposition 4.27 (i) Let x ∈ G be visible. Then C_{G}(x)^{∘} is an infinite visible group.

(ii) Let x ∈ G be invisible. Then C_{G}(x)^{∘} is an infinite invisible group.

(iii) Let U be an infinite invisible group. Either N_{G}(U) contains an involution i, in which case U is abelian and inverted by i, or N_{G}(U)^{∘} is invisible.

(iv) Let U be invisible. If there exists a strongly real element ij ∈ U for some i, j ∈ I(G), then U is abelian and inverted by i.

(v) Two maximal invisible groups are either equal or have trivial intersection.

(vi) rk( ⋃_{g∈G} U^{g}) = rk(G) − rk(N_{G}(U)/U) for any maximal invisible group U.

(vii) Invisible elements are strongly real, i.e. Inv(G) ⊆ Sr(G).

Proof. (i) follows directly from Lemma 4.24.

(ii) Let x ∈ G be invisible and K := C_{G}(x)^{∘}. Then x^{G} consists of invisible elements and rk(x^{G}) = rk(G) − rk(K). Since ⋃_{i∈I(G)} C_{G}(i) is generic in G by Proposition 4.25, rk(K) > 0 and K is infinite. Furthermore K is as a connected 2^{⊥}-group nilpotent by [19, 5.8]. Assume that K contains a non-trivial visible element g. As Z(K) is infinite by [18, 6.2], 1 ≠ Z(K)^{∘} ≤ C_{G}(g)^{∘} is visible by (i). Especially K ≤ C_{G}(Z(K)^{∘}) is visible by (i) again. As K is thus a connected 2^{⊥}- subgroup of C_{G}(s) for some s ∈ I(G), K is unipotent and C_{G}(K) is connected by Proposition 4.23. Especially C_{G}(K) is visible by (i). Contradiction to x ∈ C_{G}(K).

(iii) Let U be an infinite invisible group. Assume that N_{G}(U) contains an involution w. Since C_{U}(w) = 1, w inverts U by [18, ex.15, p.79] and U is an abelian group consisting of strongly real elements. Otherwise N_{G}(U)^{∘} is a connected 2^{⊥}- group and therefore nilpotent by by [19, 5.8]. Then U contains infinitely many central elements of N_{G}(U)^{∘} by [18, ex.5, p.98] and A := (U ∩ Z(N_{G}(U)^{∘}))^{∘} is an infinite invisible group. Furthermore N_{G}(U)^{∘} ≤ C_{G}(A)^{∘} which proves the claim by (ii).

(iv) Let i, j ∈ I(G) such that ij ∈ U. Then 1 ≠ Z(U)^{∘} ≤ C_{G}(ij)^{∘} by [18, 6.2]. As C_{G}(ij)^{∘} is invisible by (ii) and normalized by i, C_{G}(ij)^{∘} is inverted by i and abelian by (iii). Thus there exists an involution k ∈ I(G) such that ik ∈ Z(U)^{∘}. Especially U ≤ C_{G}(ik)^{∘} is abelian and inverted by i as well.

(v) Let A and B be two maximal invisible groups and assume that 1 ≠ g ∈ A∩B. Assume that A is abelian. Then A = C_{G}(g)^{∘} by (i) and the maximality of A. On the other hand 1 ≠ Z(B)^{∘} ≤ C_{G}(g)^{∘} by [18, 6.2] and B = C_{G}(Z(B)^{∘})^{∘} = A by (ii) and maximality of A and B again.

We may hence assume by (iv) that A and B do not contain strongly real elements. We are going to show that A = B. As before 1 ≠ Z(A)^{∘} ≤ C_{G}(g)^{∘} and hence Z(C_{G}(g)^{∘})^{∘} ≤ C_{G}(Z(A)^{∘})^{∘} = A by (i) and maximality of A. Using the same argument Z(C_{G}(g)^{∘})^{∘} ≤ B and A∩B is infinite by [18, 6.2]. If A ≠ B, then A > (A ∩ B)^{∘} and (A ∩ B)^{∘} is a definable invisible subgroup of infinite index. Hence N_{A}((A ∩ B)^{∘})/(A ∩ B)^{∘} is infinite by [18, 6.3]. Let B_{1} ≥ N_{G}((A ∩ B)^{∘})^{∘} be a maximal invisible subgroup. B_{1} exists by (iii) as A does not contain strongly real elements. Then rk(A∩B_{1}) > rk(A∩B). If A > (A∩B_{1})^{∘} then there exists a maximal invisible group B_{2} ≥ N_{G}((A∩B_{1})^{∘})^{∘} such that rk(A∩B_{2}) > rk(A∩B_{1}). Iterating this process we eventually arrive at a maximal invisible subgroup B_{n} ≥ N_{G}((A ∩ B_{n−1})^{∘})^{∘} such that A = (A ∩ B_{n})^{∘} by ascending chain condition on connected subgroups. Hence A = B_{n} by maximality of A. Assume that B_{n−1} > (A ∩ B_{n−1})^{∘}. Then (A ∩ B_{n−1})^{∘} has infinite index in N_{B_{n−1}}((A ∩ B_{n−1})^{∘})^{∘} ≤ A ∩ B_{n−1} by [18, 6.3]. This would imply that rk((A ∩ B_{n−1})^{∘}) < rk(A ∩ B_{n−1}). Contradiction. Thus B_{n−1} = A already. We can repeat the argument until we arrive at A = B.

(vi) As U ∩ U^{g} = 1 if and only if g ∈ G\N_{G}(U) by (v),

rk( ⋃_{g∈G} U^{g}) = rk(G) − rk(N_{G}(U)) + rk(U)
= rk(G) − rk(N_{G}(U)/U).

by the additivity of Morley rank and Lascar’s equality.

(vii) Assume that G contains an invisible element x which is not strongly real. Then there exists by (ii) an infinite maximal invisible group A ≥ C_{G}(x)^{∘}, such that A = N_{G}(A)^{∘} by (iii), as x is not strongly real. Thus ⋃_{g∈G} A^{g} is generic in G by (vi). Contradiction to Proposition 4.25. □

Corollary 4.28 Let U ≤ G be a maximal invisible group and u ∈ U. Then C_{G}(u) is abelian, inverted by i and I(C_{G}^{∗}(u)) = iC_{G}(u) = i^{C_{G}(u)}. Furthermore U = C_{G}(u)^{∘}.

Proof. Let U be a maximal invisible group. U consists of strongly real elements by Proposition 4.27 (vii). Thus U is abelian and inverted by an involution i by Proposition 4.27 (iv). Let u ∈ U be a nontrivial element.

Assume that C_{G}(i) ∩ C_{G}(u) is not trivial. Then, conjugating with a suitable element of G, there exists a nontrivial element g ∈ C_{G}(i_{0}) ∩ C_{G}(v) for some invisible element v that is inverted by i_{0}. As v ∈ C_{G}(g), C_{G}(g) is conjugate in C_{G}(i_{0}) to T ⟨y⟩ by Lemma 4.24. However T ⟨y⟩ does not contain an element that is inverted by i_{0}. Contradiction.

Thus i acts fixed-point free on C_{G}(u), C_{G}(u) is abelian, inverted by i and I(C_{G}^{∗}(u)) = iC_{G}(u) = i^{C_{G}(u)} by [18, 10.5]. As finally U is abelian U ≤ C_{G}(u)^{∘}. However, C_{G}(u)^{∘} is invisible by Proposition 4.27 (ii) and thus U = C_{G}(u)^{∘} by maximality of U. □

Corollary 4.29 Let u ∈ G be invisible. Then C_{G}(u)\{1} consists of invisible elements and C_{G}(u)^{∘} is a maximal invisible group.

Proof. As C_{G}(u)^{∘} is an infinite invisible group by Proposition 4.27 (ii), u is contained in a maximal invisible group U ≥ C_{G}(u)^{∘}. Thus U = C_{G}(u)^{∘} and C_{G}(u) is abelian by Corollary 4.28. Let v ∈ C_{G}(u)\{1}. Then U ≤ C_{G}(v)^{∘} and v cannot be visible by Proposition 4.27 (i). Hence v is invisible and the claim follows. □

We are going to estimate the Morley rank of G. Let f := rk(Z(C_{G}(i_{0}))).

Lemma 4.30 rk(C_{G}(i_{0})) = 4f.

Proof. Since C_{G}(i_{0}) ≅ GL_{2}(K) for an algebraically closed field K, C_{G}(i_{0}) has a Bruhat decomposition C_{G}(i_{0}) = B ∪ Bw_{0}B, where B = T X_{1} by [42, 28.3]. Thus

rk(C_{G}(i)) = rk(BwB) = rk(X_{1}^{w}X_{1}T).

By Proposition 4.5 T = Z(C_{G}(i_{0}))Z(C_{G}(i_{1})) and thus rk(T) = 2f. Furthermore Z(C_{G}(i_{1})) acts transitively on X_{1}\{1} and rk(X_{1}) = f. Hence

rk(X_{1}^{w}X_{1}T) = 2rk(X_{1}) + rk(T) = 4f

which proves the claim □

We need a precise description of strongly real elements. Let

X_{s} := ⋃_{g∈G} T [w_{0}]^{g},
X_{u} := ⋃_{g∈G}(⟨i_{0}⟩X_{1}\{i_{0}})^{g}.

Lemma 4.31 If x ∈ Sr(C_{G}(i_{0})), then either I(C_{G}^{∗}(x)) ⊆ C_{G}(l) for some l ∈ I(C_{G}(i_{0})) or x ∈ C_{T} (y). Furthermore I(G) ∪ X_{s} ∪ X_{u} is the set of strongly real visible elements of G.

Proof. Let x ∈ C_{G}(i_{0}). By Remark 4.18 we may assume that either x ∈ T or x ∈ Z(C_{G}(i_{0}))X_{1}. If x ∈ T, then x is an involution or C_{G}(x)^{∘} = T by Lemma 4.16. If x ∈ I(T), then C_{G}^{∗}(x) = C_{G}(x). Assume that x is not an involution and let i invert x. Then i, ix ∈ I(N_{G}(T)) and x ∈ ⋃_{1≤n≤3} T [w_{n}] by Lemma 4.64. If x ∈ Z(C_{G}(i_{0}))X_{1} then either x ∈ Z(C_{G}(i_{0})) and x = i_{0} by Lemma 4.16 or C_{G}(x) = Z(C_{G}(i_{0}))X_{1} by Lemma 4.23. If i inverts x thus i ∈ N_{G}(X_{1}) = T X_{1} by Lemma 4.22. Hence x ∈ ⟨i_{0}⟩X_{1}.

The second part of the Lemma follows as a visible element x ∈ Sr(G) is a strongly real element of C_{G}(s) for some involution s ∈ I(G). □.

Lemma 4.32 rk(X_{s}) = rk(G) − f and rk(X_{u}) = rk(G) − 2f.

Proof. First note that rk(T [w_{0}]) = rk(T) − rk(Z(C_{G}(i_{0})) = f. Consider ψ : G × T [w_{0}] → X_{s} where ψ(g, t) = t^{g}. If x ∈ T [w_{0}], then

ψ^{−1}(x) := {(t, g) ∈ T [w_{0}] × G| t^{g} = x},

and

rk(ψ^{−1}(x)) = rk({g ∈ G| x^{g^{−1}} ∈ T [w_{0}]})
= rk({g ∈ G| x ∈ T [w_{0}] ∩ T [w_{0}]^{g}}).

However if x ∈ T [w_{0}] ∩ T [w_{0}]^{g} then T [w_{0}], T [w_{0}]^{g} ≤ C_{G}(x)^{∘}. If x ∉ ⟨i_{0}⟩, then C_{G}(x)^{∘} = T by Lemma 4.11. This implies that C_{G}(T [w_{0}]) = T = T^{g^{−1}} and hence g ∈ N_{G}(T). Thus

rk(ψ^{−1}(x)) = rk(N_{G}(T)) = rk(T) = 2f

and rk(G) + rk(T [w_{0}]) = rk(X_{s}) + 2f by additivity of Morley rank. Hence rk(X_{s}) = rk(G) − f.

Finally (⟨i_{0}⟩X_{1}\{i_{0}}) ∩ (⟨i_{0}⟩X_{1}\{i_{0}})^{g} ≠ 1 if and only if g ∈ N_{G}(X_{1}) by Lemma 4.23. Thus

rk(X_{s}) = rk(G) − (rk(N_{G}(X_{1})) + rk(X_{1})) = rk(G) − rk(T) = rk(G) − 2f

as N_{G}(X_{1}) = X_{1} ⋊ T by Proposition 4.22. □

Proposition 4.33 rk(G) ≥ 8f, rk(I(i_{0}X_{s})) = 4f and rk(I(i_{0}X_{u})) = 3f.

Proof. We are going to use the standard technique for rank estimates from the proof of Fact 1.1. Consider φ : I(G) × I(G) ⟶ G where φ((λ, κ)) = λκ for all λ, κ ∈ I(G). Then X_{s} ∪ X_{u} ⊆ φ(I(G) × I(G)) and

(∗) rk(I(G)) = rk(i_{0}^{G}) = rk(G) − rk(C_{G}(i_{0})) = rk(G) − 4f

by Lemma 4.30. Let t ∈ T [w_{0}]\{1, i_{0}}. As

rk(φ^{−1}(t)) = rk({λ ∈ I(G)|t^{λ} = t^{−1}}) = rk(I(C_{G}^{∗}(t)\C_{G}(t)))
= rk(I(w_{0}T)) = rk(T [w_{0}]) = f,

rk(φ^{−1}(X_{s})) = rk(X_{s}) + rk(T [w_{0}]) = rk(G) by Lemma 4.32. By additivity of Morley rank we can hence conclude that

rk(G) ≤ rk(I(G) × I(G)) = 2 rk(G) − 8f

by (∗). Thus rk(G) ≥ 8f. Furthermore

φ^{−1}(X_{s}) = {(κ, λ) ∈ I(G) × I(G)|λ ∈ κX_{s}}

and thus rk(φ^{−1}(X_{s})) = rk(I(G)) + rk(I(i_{0}X_{s})). As rk(φ^{−1}(X_{s})) = rk(G) the second part of the claim follows by (∗) again.

Finally let u ∈ X_{u} be a nontrivial element. Then

rk(φ^{−1}(X_{u})) = rk(X_{u}) + rk(I(C_{G}^{∗}(u)\C_{G}(u)))
= rk(X_{u}) + rk(X_{1}) = rk(G) − f

by Lemma 4.23 and Lemma 4.32. As rk(φ^{−1}(X_{s})) = rk(I(G)) + rk(I(i_{0}X_{u})) the claim follows by (∗). □

Corollary 4.34 Let i ∈ I(G). Then rk({j ∈ I(G)| ij is visible }) = 4f.

Proof. If ij is visible for ij ∈ I(G), then ij ∈ I(G) ∪ X_{s} ∪ X_{f} by Lemma 4.31. If ij ∈ I(G), then j ∈ I(C_{G}(i)). Since rk(I(C_{G}(i))) = 2f, the claim follows by Proposition 4.33. □

It is more difficult to get an upper bound for rk(G).

Lemma 4.35 Let i, j ∈ I(G).

(i) If ij is invisible, then C_{G}(i) ∩ C_{G}(j) = 1.

(ii) If ij is visible, then C_{G}(i) ∩ C_{G}(j) contains an involution.

Proof. If ij is invisible, then C_{G}(ij)\{1} consists of invisible elements by Corollary 4.29. As C_{G}(i) ∩ C_{G}(j) ≤ C_{G}(ij) thus C_{G}(i) ∩ C_{G}(j) = 1.

Assume now ij is visible. Then either i, j ∈ I(C_{G}^{∗}(ij)) ⊆ C_{G}(s) for some involution s ∈ I(G) or ij is conjugate to an element in C_{T} (y) by Lemma 4.31. Assume that ij is centralized by y. Then j ∈ N_{G}(C_{G}(ij)^{∘}) = N_{G}(T) by Lemma 4.11. Hence j = i(ij) ∈ iT ∩N_{G}(T) ≤ C_{G}(l) for some involution l ∈ D^{∗} by Lemma 4.64. Thus there always exists an involution s ∈ I(G) such that i, j ∈ C_{G}(s) and the claim follows. □

Lemma 4.36 Let H ≤ T be an infinite definable subgroup such that H ≅ K^{∗}. Then either H = T [w_{0}], H = Z(C_{G}(i_{0})) or T = HH^{w_{0}} and rk(H) = f.

Proof. Since H, H^{w_{0}} ≤ T, either H = H^{w_{0}} or H ∩ H^{w_{0}} is finite as G does not interpret a bad field. Assume that w_{0} normalises H. If C_{H}(w_{0}) is finite, then w_{0} inverts H and H = T [w_{0}]. Thus rk(H) = f in this case. If C_{H}(w_{0}) is infinite, then C_{H}(w_{0}) = H = Z(C_{G}(i_{0})) since G does not interpret a bad field and rk(H) = f again. If on the other hand w_{0} does not normalize H, H ∩ H^{w_{0}} is finite and rk(HH^{w_{0}}) = 2 rk(H). Furthermore C_{HH^{w_{0}}}(w_{0}) = {hh^{w_{0}}| h ∈ H} = Z(C_{G}(i_{0})). Especially as H ∩ H^{w_{0}} is finite, rk(H) = f again and T = HH^{w_{0}} is this case. □

Proposition 4.37 Let U be maximal invisible group and i an involution that inverts U. Then

(i) N_{G}(U) = C_{G}(U) ⋊ H where H = C_{G}(i) ∩ N_{G}(U). Furthermore H = C_{G}(i) ∩ C_{G}(j)C_{G}(U) for all j ∈ I(N_{G}(U)).

(ii) H ∩ H^{u} = 1 for all u ∈ C_{G}(U).

(iii) rk(⋃_{g∈G} U^{g}) = rk(G) − rk(H) and H is infinite.

(iv) H^{∘} ≅ K^{∗}, H^{∘} < C_{G}(D) for some four-subgroup D ≤ C_{G}(i) and U contains a subgroup A ≅ K^{+} on which H acts transitively. Especially rk(H) = f.

(v) N_{G}(U)^{∘} = U ⋊ H^{∘} is a 2-step solvable centerless group and (N_{G}(U)^{∘})′ = U is either a p-subgroup of bounded exponent, if char(K) = p, or a torsion-free divisible group, if char(K) = 0.

(vi) N_{G}(U)^{∘} = U ∪ ⋃_{c∈U} H^{c} and Uh = h^{U} for all h ∈ H\{1}. Especially H ∩ H^{u} = 1 for all u ∈ U.

(vii) If M ≤ N_{G}(U)^{∘} is such that M ∩ U = 1, then M is contained in (H^{∘})^{x} for some x ∈ U.

Proof. We first show (i). Since any involution that normalizes C_{G}(U) already inverts it,

(∗) i^{N_{G}(U)} ⊆ I(N_{G}(U)) = iC_{G}(U) = i^{C_{G}(U)}

by Corollary 4.28. Hence N_{G}(U) ⊆ C_{G}(k)C_{G}(U) for all k ∈ I(N_{G}(U)). Furthermore [N_{G}(U), i] ⊆ C_{G}(U) and N_{G}(U) = C_{G}(U) ⋊ C_{N_{G}(U)}(i) by [18, ex. 9, p. 98]. Let H := C_{N_{G}(U)}(i). Then H ⊆ C_{G}(i) ∩ C_{G}(j)C_{G}(U) for all j ∈ I(N_{G}(U)). On the other hand C_{G}(i)C_{G}(U) ∩ C_{G}(j)C_{G}(U) ⊆ N_{G}(U) and hence N_{G}(U) = C_{G}(i)C_{G}(U) ∩ C_{G}(j)C_{G}(U) and (i) follows.

(ii) is a direct consequence of Corollary 4.35, as H ∩ H^{u} ≤ C_{G}(i) ∩ C_{G}(i^{u}) and ii^{u} ∈ C_{G}(U) is invisible by Corollary 4.29.

(iii) follows since by Corollary 4.28 rk(C_{G}(U)) = rk(U) and thus by Proposition 4.27

rk(⋃_{g∈G} U^{g}) = rk(G) − rk(N_{G}(U)/U) = rk(G) − rk(H).

Especially rk(H) > 0 by Lemma 4.25.

To prove (iv), let B ≤ H a Borel subgroup of H and A ≤ U a B-minimal group. Then B′ centralizes A by [18, 9.8]. However, C_{H}(x) ≤ C_{G}(i) ∩ C_{G}(U) = 1 for all x ∈ U\{1} by Corollary 4.29. Hence B′ = 1 and B is abelian. Since H is a K-group all Borel subgroups are conjugate by Proposition 1.6. Hence H does not contain connected solvable non-nilpotent definable subgroups. Thus H^{∘} ≤ C_{G}(i) ≅ GL_{2}(K) has to be solvable already and hence abelian. As C_{H}(A) = 1, A ≅ K^{+} and H^{∘} ≅ K^{∗} for some algebraically closed field K by [18, 9.1], since G does not interpret a bad field. As H ≤ C_{G}(i), H^{∘} < C_{G}(D) for some four-subgroup D ≤ C_{G}(i). Furthermore H operates transitively on A\{1}, i.e. rk(A) = rk(H) = f by Lemma 4.36.

We show (v). Since H^{∘} is abelian, (N_{G}(U)^{∘})′ ≤ U and N_{G}(U)^{∘} is a 2-step solvable centerless group. As furthermore U = [i, U] = (N_{G}(U)^{∘})′ by equation (∗), U is quasiunipotent by [18, ex. 3, p. 148]. U is thus the direct sum of a definable subgroup of bounded exponent B and a definable divisible torsion-free subgroup D, both of which are characteristic in U by Lemma 2.3. Since however H^{∘}-minimal subgroups of U are isomorphic to K^{+}, either B = 1 and U is a divisible torsion-free group or D = 1 and U is a p-group, where p = char(K).

(vi) follows from [18, ex. 8, ex. 11, p. 98] as U ∩ C_{G}(h) = 1 for all h ∈ H\{1} by Corollary 4.29.

We finally show (vii). M is abelian, since [M, M] ≤ G′ ∩ M = U ∩ M = 1. Thus M ≤ C_{G}(m) for all m ∈ M. Set V := UM ∩ H^{∘} and let v ∈ V. Since C_{N_{G}(U)^{∘}}(v) = H^{∘}, C_{UM}(v) ≤ V. On the other hand v ∈ Um for some m ∈ M and thus m ∈ vU = v^{U} by (vi). Thus there exists u ∈ U such that v = m^{u} and M^{u} ≤ C_{UM}(m)^{u} = C_{UM}(v) ≤ V ≤ H^{∘}. Especially M ≤ (H^{∘})^{u^{−1}}. □

Proposition 4.38 Either G consists of visible elements and rk(G) = 8f or there exist a maximal invisible subgroup U such that

7f + rk(U) ≤ rk(G) < 8f + rk(U).

Furthermore

rk(G) = rk(U) + 7f + rk(B/ ≡)

where

B := {g ∈ Inv(G)| rk(C_{G}(g)) = rk(U)}

and b_{1} ≡ b_{2} if there exists g ∈ G such that b_{1} ∈ C_{G}(b_{2})^{g}.

Proof. We are again using the technique from the proof of Fact 1.1. Let φ : I(G) × I(G) ⟶ G be defined by φ((x, y)) = xy. Then 2 rk(I(G)) = max_{1≤n≤k}{B_{n} + a_{n}} by [1, 6.1], where

B_{n} := {g ∈ φ(I(G) × I(G))| rk(φ^{−1}(g)) = a_{n}}

and 0 ≤ a_{n} ≤ 2 rk(I(G)) . Now rk(φ^{−1}(1)) = rk(I(G)),

rk(φ^{−1}(g)) = rk(I(C_{G}(g)) = 2f if g ∈ I(G)

and

rk(φ^{−1}(g)) = rk(I(C_{G}^{∗}(g)\C_{G}(g))) for g ∈ Sr(G)\I(G).

If g is invisible then rk(φ^{−1}(g)) = rk(C_{G}(g)) as C_{G}(g)^{∘} is a maximal invisible group by Corollary 4.29 which is inverted by an involution i ∈ I(G). If g ∈ Sr(G)\I(G) is visible, then g ∈ X_{s} ∪X_{u} by Lemma 4.31 and thus rk(φ^{−1}(g)) = f as in Proposition 4.33. Let m := rk(I(G)). Hence, if we set a_{n} := nf for 1 ≤ n ≤ k := 2f^{−1}I(G), then

B_{1} ⊆ X_{s} ∪ X_{u} ∪ Inv(G), B_{2} ⊆ I(G) ∪ Inv(G), B_{m} ⊆ 1 ∪ Inv(G)

and B_{n} ⊆ Inv(G) for all 3 ≤ n ≤ k, n ≠ m. By Proposition 4.37 either rk(Inv(G) ∩ B_{n}) = 0 or rk(Inv(G) ∩ B_{n}) ≥ rk(G) − f for all 1 ≤ n ≤ k. If Inv(G) = ∅ then 2 rk(I(G)) = f + (rk(G) − f) by Lemma 4.32. In this case rk(G) = 8f. If Inv(G) ≠ ∅ then 2 rk(I(G)) = rk(U) + rk(B) where

B := {g ∈ Inv(G)| rk(C_{G}(g)) = rk(U)}.

Especially rk(G) = rk(U) + 8f + (rk(B) − rk(G)). By Lemma 4.25 and Proposition 4.37 rk(G) − f ≤ rk(B) < rk(G). Define an equivalence relation ≡ on B via b_{1} ≡ b_{2} if b_{1} ∈ ⋃_{g∈G} C_{G}(b_{2})^{g}. Since rk[b]_{≡} = rk(G) − f for all b ∈ B by Proposition 4.37

rk(B/ ≡) = rk(B) − rk(G) + f = rk(G) − rk(U) − 7f < f

and rk(G) = rk(U) + 7f + rk(B/ ≡). □

Lemma 4.39 Let i ∈ I(G). Then {j ∈ I(G)| ij is invisible} is either empty or generic in I(G).

Proof. Assume there exists j ∈ I(G) such that ij is invisible. Then U := C_{G}(ij)^{∘} is a maximal invisible group by Corollary 4.29. We may assume by Proposition 4.38 that rk(G) = 7f + rk(U) + rk(B/ ≡) and thus

rk(I(G)) = rk(G) − rk(C_{G}(i)) = 3f + rk(U) + rk(B/ ≡).

We are going to show that J := {j ∈ I(G)| ij ∈ B} is generic in I(G). To do this we define an equivalence relation ≃ on J via j_{1} ≃ j_{2} if there exists g ∈ G such that C_{G}(ij_{1}) = C_{G}(ij_{2})^{g}. We claim that

i^{C_{G}(ij_{1})C_{G}(i)} ⊆ [j_{1}]_{≃}.

for all j_{1} ∈ J. For let c ∈ C_{G}(ij_{1}), x ∈ C_{G}(i), then ii^{cx} = (ii^{c})^{x} = (c^{−i}c)^{x} = c^{2x} ∈ C_{G}(ij_{1})^{x} by Corollary 4.28 and thus C_{G}(ii^{cx}) = C_{G}(ij_{1})^{x} as both groups are abelian.

We compute the Morley rank of [j]_{≃}. Consider ϕ : UC_{G}(i) ⟶ i^{UC_{G}(i)} where ϕ(x) = i^{x}. Let u_{n} ∈ U, c_{n} ∈ C_{G}(i) for n = 1, 2. Then i^{u_{1}c_{1}} = i^{u_{2}c_{2}} if and only if

(∗) ii^{u_{1}c_{1}c_{2}^{−1}} = ii^{u_{2}} ∈ U ∩ U^{c_{1}c_{2}^{−1}}.

However (∗) is only the case if c_{1}c_{2}^{−1} ∈ N_{G}(U) ∩ C_{G}(i). Set h := c_{1}c_{2}^{−1}. Then i^{u_{1}hu_{2}^{−1}} = i by assumption and therefore i = i^{u_{1}u_{2}^{−h^{−1}}h} = i. This implies that u_{1}u_{2}^{−h^{−1}} ∈ C_{G}(i) ∩ U = 1 and u_{1}^{h} = u_{2}. Especially rk(ϕ^{−1}(x)) = rk(N_{G}(U) ∩ C_{G}(i)) = f for all x ∈ UC_{G}(i) by Proposition 4.37. Hence by additivity of Morley rank

rk(i^{UC_{G}(i)}) = rk(UC_{G}(i)) − f = rk(U) + rk(C_{G}(i)) − f
= rk(I(G)) − rk(B/ ≡).

This implies that rk(J) ≥ rk(J/ ≃)+(rk(I(G))−rk(B/ ≡)) and it is only left to show that rk(J/ ≃) ≥ rk(B/ ≡). Define ψ : J ⟶ B/ ≡ by ψ(j) = [ij]_{≡}. Then ψ(j_{1}) = ψ(j_{2}) iff ij_{2} ∈ C_{G}(ij_{1})^{g} for some g ∈ G iff j_{1} ≃ j_{2}. Thus rk(ψ^{−1}([ij]_{≡})) = rk([j]_{≃}) and rk(J/ ≃) = rk(B/ ≡) since ψ is surjective. □

Let U be a maximal invisible subgroup as in Proposition 4.38. Let N := N_{G}(U)^{∘} and H a complement of U in N. Define as in [18, 14.5] for all w ∈ I(G\N)

H[w] := {n ∈ N| n^{w} = n^{−1}}
X := {w ∈ I(G\N)| H[w] ≠ 1}
X′ := {w ∈ I(G\N)| H[w] = 1}

Lemma 4.40 Let w ∈ I(G\N). Then H[w] is an abelian subgroup of N and is contained in some complement H of U.

Proof. As U ∩ U^{w} = 1 by Corollary 4.28, U^{w} ∩ N is contained in H^{u} for some u ∈ U by Proposition 4.37. Thus U^{w} ∩ N = 1 since U^{w} is invisible and H^{u} ≤ C_{G}(k) for an involution k ∈ N. Thus (N ∩ N^{w}) ∩ U = 1 as well and H[w] ⊆ N ∩ N^{w} is contained in a complement of U in N by Proposition 4.37. This implies that H[w] consists of commuting elements and H[w] is hence a subgroup of N. □

Corollary 4.41 Let w ∈ I(G\N). Then H[w] is either finite or a complement of U in N.

Proof. H[w] ≤ H for some complement H of U by Lemma 4.40. As H ≅ K^{∗} by Proposition 4.37 and G does not interpret a bad field, either H[w] is finite or already equals H. □

Proposition 4.42 rk(X′) = rk(I(G)) and rk(U) ≤ 3f.

Proof. I(G) = I(N) ∪ X ∪ X′ and

rk(I(N)) = rk(U) < rk(U) + 3f + r = rk(G) − 4f = rk(I(G)).

4.2. PSP_{4}(K) AND G_{2}(K) 77

To show, that X′ is generic in I(G), it is therefore enough to prove that rk(X) < rk(I(G)). By Lemma 4.40 there exists a complement of U in N which contains H[w]. Let I_{u} := {w ∈ X| H[w] ≤ H^{u}} for all u ∈ U. Then rk(X) = ⋃_{u∈U} I_{u} ≤ rk(U) + rk(I_{1}) as H[w^{u}] = H[w]^{u} for all u ∈ U. However w ∈ I_{1} only if w inverts an element t ∈ H. Thus rk(I_{1}) ≤ rk(I(C_{G}^{∗}(t))) ≤ 2f by Lemma 4.24 and rk(X) ≤ rk(U) + 2f < rk(I(G)). Hence

rk(G) − 4f = rk(I(G)) = rk(X′) ≤ rk(G/N) = rk(G) − (rk(U) + f)

since for w_{1}, w_{2} ∈ X′: w_{1}N = w_{2}N if and only if w_{1}w_{2} ∈ H[w_{1}] = 1. This implies that rk(U) ≤ 3f. □

Corollary 4.43 Either G = ⋃_{g∈G} C_{G}(i)^{g} and rk(G) = 8f or there exists an invisible element and 8f < rk(G) < 11f.

Proof. Assume that G does not contain invisible elements. Then rk(G) = 8f by Proposition 4.38. Assume now that G contains an invisible element u which is inverted by i ∈ I(G). Then {j ∈ I(G)| ij is invisible} is generic in I(G) by Lemma 4.39. On the other hand rk({j ∈ I(G)| ij is visible}) = 4f by Lemma 4.34. As G is connected and thus deg(I(G)) = deg(i^{G}) = 1, rk(I(G)) > 4f in this case. Furthermore rk(I(G)) = rk(G) − rk(C_{G}(i)) = 3f + rk(U) + rk(B/ ≡) by Proposition 4.38. Hence rk(U) + rk(B/ ≡) > f and rk(G) > 8f. As however rk(G) < 8f + rk(U) by Proposition 4.37 and rk(U) ≤ 3f by Proposition 4.42, rk(G) < 11f and everything is proven. □

4.2 PSp_{4}(K) and G_{2}(K)

The aim of this section is to prove cases (2) and (3) of Conjecture 7.

Theorem 4.44 Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that G contains an involution i, such that either

(i) C_{G}(i)^{∘}/O(C_{G}(i)) ≅ SL_{2}(K) ∗ SL_{2}(K) where the two copies of SL(2, K) intersect non-trivially or

(ii) C_{G}(i)^{∘}/O(C_{G}(i)) ≅ PSL(2, K) × K^{∗}

for an algebraically closed field K of characteristic neither 2 nor 3. Then G ≅ PSp_{4}(K) or G ≅ G_{2}(K).

4.2.1 Basic Results

We are first going to show, that case (ii) of Theorem 4.44 implies case (i).

Lemma 4.45 Let G be a K^{∗}-group of finite Morley rank. Assume that there exist two definable subgroups N, H ≤ G such that N ⊲ H and \overline{H} := H/N is a proper simple section of G allowing no graph automorphisms. Then N_{G}(H) = C_{N_{G}(H)}(\overline{H})H.

Proof. Let g ∈ N_{G}(H) and set R := d(g). R ≤ N_{G}(H) acts on \overline{H}. As G is a K^{∗}-group and H is a proper simple section, H is a simple algebraic group over an algebraically closed field. Consider the semidirect product \overline{H} ⋊ R/C_{R}(\overline{H}). Then, viewing R/C_{R}(\overline{H}) as a subgroup of Aut(\overline{H}), R/C_{R}(\overline{H}) ≤ Inn(\overline{H})Γ, where Γ are the graph automorphisms of \overline{H} by [18, 8.4]. As Γ = 1, either g ∈ C_{N_{G}(H)}(\overline{H}) or there exists an h ∈ H such that x^{g} ∈ x^{h}N for all x ∈ H and gh^{−1} ∈ C_{N_{G}(H)}(\overline{H}). Thus g ∈ C_{N_{G}(H)}(\overline{H})H. □

Lemma 4.46 Let H be a group of finite Morley rank, such that H^{∘} is a reductive algebraic group and let \overline{H}^{∘} := H^{∘}/Z(H^{∘}). Then C_{C_{H}(Z(H^{∘}))}(H^{∘}) = C_{C_{H}(Z(H^{∘}))}(\overline{H}^{∘}).

Proof. Let g ∈ C_{H}(\overline{H}^{∘}) ∩ C_{H}(Z(H^{∘})). As H^{∘} is reductive, \overline{H}^{∘} is a simple group by [42, 27.5] and m := o(\overline{g}) < |H/H^{∘}|. Set z_{x} := x^{−1}x^{g} ∈ Z(H^{∘}) for all x ∈ H^{∘}. Then, as x^{g} = xz_{x}, x = x^{g^{m}} = xz_{x}^{m} and o(z_{x})|m for all x ∈ H^{∘}. On the other hand (x^{m})^{g} = (xz_{x})^{m} = x^{m}z_{x}^{m} and g ∈ C_{H}(x^{m}) for all x ∈ H^{∘}. Especially C_{P} (g) = P, for any maximal torus P of H^{∘}, since maximal tori are divisible. As finally H^{∘} is a reductive algebraic group, H^{∘} = ⟨P^{h}| h ∈ H^{∘}⟩ ≤ C_{H}(g) by [42, ex. 12, p. 162] and g ∈ C_{H}(H^{∘}). □

Theorem 4.47 Let G be a simple K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that G contains an involution i, such that C_{G}(i)^{∘}/O(C_{G}(i)) ≅ PSL_{2}(K) × K^{∗} where K is an algebraically closed field of characteristic not 2. Then G contains an involution j, such that C_{G}(j)^{∘} ≅ SL_{2}(K) ∗ SL_{2}(K).

Proof. Let S be Sylow 2-subgroup of G that contains i and D ≤ S^{∘} a foursubgroup. We may assume that i ∈ D by Proposition 4.2. Set T := C_{G}(D)^{∘}. Hence O(C_{G}(t)) = 1 for all t ∈ D^{∗} by Proposition 4.2 again. Furthermore G = ⟨C_{G}(k)^{∘}| k ∈ D_{1}^{∗}⟩ for any four-subgroup D_{1} ≤ G by Corollary 3.17. Set T := C_{G}(D)^{∘} and let u ∈ (C_{G}(D) ∩ C_{G}(i)^{∘})\T. Then C_{T} (u) = Z(C_{G}(i)^{∘}).

Assume that C_{G}(j)^{∘} ≅ GL_{2}(K) for some j ∈ D^{∗}. Then there exists an involution w ∈ C_{G}(j)^{∘} ∩ N_{G}(D) such that i^{w} = ij. Then uu^{w} = w^{u}w ∈ C_{G}(D) ∩ C_{G}(j)^{∘} = T. Furthermore, as u centralises Z(C_{G}(i)^{∘}), u^{w} centralises Z(C_{G}(i)^{∘}) as well. On the other hand C_{T} (u^{w}) = C_{T} (u)^{w} = Z(C_{G}(ij)^{∘}). Contradiction as Z(C_{G}(ij)^{∘}) ≠ Z(C_{G}(i)^{∘}).

Assume that C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗} for some j ∈ D^{∗}, j ≠ i. Let v ∈ C_{C_{G}(j)^{∘}}(D)\T be an involution. As C_{T} (v)^{∘} = Z(C_{G}(j)^{∘}), v ∈ C_{G}(i)\C_{G}(i)^{∘}. Let C := C_{G}(i)^{∘} and \overline{C} := C/Z(C). By Lemma 4.45 C_{G}(i) = C_{C_{G}(i)}(\overline{C})C.

v normalises Z(C) ≅ K^{∗} and G does not interpret a bad field. Thus v either inverts Z(C) or centralizes it by [18, 10.5]. Since C_{T} (v) = Z(C_{G}(j)^{∘}), the second case cannot occur and v inverts Z(C). Let x ∈ C, such that xv ∈ C_{C_{G}(i)}(\overline{C}). Then x ∈ N_{G}(D) ∩ C = T ⟨u⟩ and xv inverts Z(C). As u inverts Z(C_{G}(j)^{∘}) as above, yv inverts T for all y ∈ uT. Hence x ∈ T, since xv ∈ C_{C_{G}(i)}(\overline{C}). Furthermore we may assume that x ∈ Z(C_{G}(j)^{∘}), as T = Z(C_{G}(j)^{∘})Z(C). Thus (xv)^{2} = x^{2} ∈ C_{C}(\overline{C}) = Z(C) ∩ Z(C_{G}(j)^{∘}) = 1 by Lemma 4.46 and x ∈ ⟨j⟩. Then [xv, C] ⊆ Z(C) and C = Z(C) ∗ C_{C}(xv) by [18, ex. 10, p. 98] where Z(C) ∩ C_{C}(xv) = ⟨i⟩. Set H := C_{C}(xv)^{∘}. As C ≅ PSL_{2}(K) × K^{∗}, H ≅ PSL_{2}(K) and C = Z(C) × H. As x ∈ ⟨j⟩, xv ∈ I(C_{G}(D)). Thus D_{1} := ⟨i, xv⟩ is a four-subgroup and G := ⟨C_{G}(k)| k ∈ D_{1}^{∗}⟩. Furthermore H ≤ C_{G}(D_{1}). As H is a normal subgroup of C_{G}(i)^{∘} and G is simple, H cannot be a normal subgroup of both C_{G}(ux) and C_{G}(uxi). As ux and uxi are conjugate to involutions in D by Lemma 3.19, this implies that C_{G}(k) ≅ SL(2, K) ∗ SL(2, K) for k = ux or k = uxi by Proposition 4.2.

As finally G = ⟨C_{G}(k)^{∘}| k ∈ D^{∗}⟩, it is impossible that C_{G}(k)^{∘} is abelian for all k ∈ D\⟨i⟩. This implies the claim by Proposition 4.2. □

Thus Theorem 4.44 basically consists of two parts. The first one is a partial analogue of the result of Wong [57] in finite group theory. It will appear in [8] and be proven in Section 4.2.

Theorem 4.48 Let C_{1} and C_{2} be the non-isomorphic centralizers in PSp_{4}(K) of two involutions where K is an algebraically closed field of characteristic not 2. Let G be a K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that G contains two involutions i and j such that C_{G}(i) ≅ C_{1} and C_{G}(j) ≅ C_{2}. Then G ≅ PSp_{4}(K).

The corresponding result for G_{2}(K) is the following theorem, which will be proven in Section 4.3.

Theorem 4.49 Let C be the centralizer in G_{2}(K) of an involutions where K is an algebraically closed field of characteristic not 2. Let G be a K^{∗}-group of finite Morley rank that does not interpret a bad field. Assume that G contains one conjugacy class i^{G} of involutions such that C_{G}(i) ≅ C. Then G ≅ G_{2}(K) if char(K) ≠ 3.

The rest of the subsection is devoted to the proof that Theorems 4.48 and 4.49 imply Theorem 4.44.

Let G be a group as in Theorem 4.44. By Theorem 4.47 there exists an involution i ∈ G, such that C_{G}(i)^{∘}/O(C_{G}(i)) ≅ L_{1} ∗ L_{2} where L_{n} ≅ SL_{2}(K) for an algebraically closed field K of characteristic neither 2 nor 3 and n = 1, 2. We may assume that i ∈ C_{G}(i)^{∘} by Lemma 4.2. Thus L_{1} ∩L_{2} = ⟨\overline{ı}⟩. Let S be a Sylow 2-subgroup of C_{G}(i), D a four-subgroup such that D ≤ S^{∘} and set T := C_{G}(D)^{∘}. Then D contains a central involution of S by [18, ex. 12, p. 14] which means that i is central in S, since j and ij are conjugate in S for j ∈ D\⟨i⟩.

Proposition 4.50 G = ⟨C_{G}(k)^{∘}| k ∈ D_{1}^{∗}⟩ for any four-subgroup D_{1} ≤ G and O(C_{G}(k)) = 1 for all k ∈ D^{∗}.

Proof. O(C_{G}(t)) = 1 for all t ∈ D^{∗} by Proposition 4.2 and G = ⟨C_{G}(k)^{∘}| k ∈ D_{1}^{∗}⟩ for any four-subgroup D_{1} ≤ G by Corollary 3.17. □

Corollary 4.51 C_{G}(k)^{∘} is nonabelian for all k ∈ D^{∗}.

Proof. Assume that C_{G}(j)^{∘} is abelian for some j ∈ D^{∗}. Then C_{G}(j)^{∘} ≤ C_{G}(i)^{∘} since i ∈ C_{G}(j)^{∘}. As furthermore j and ij are conjugate, G = C_{G}(i)^{∘} by Proposition 4.50. Contradiction. □

Corollary 4.52 Let j ∈ D^{∗}. If i and j are not conjugate in G, then C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗}.

Proof. Let j ∈ D^{∗} be not conjugate to i. Then j is conjugate to ij in C_{G}(i)^{∘}. Now C_{G}(j)^{∘} is not abelian by Corollary 4.51. Furthermore C_{G}(D)^{∘} ≅ K^{∗}×K^{∗} and there are only three possibilities by Proposition 4.2, namely C_{G}(j)^{∘} ≅ GL_{2}(K), C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗} or C_{G}(j)^{∘} ≅ SL_{2}(K) ∗ SL_{2}(K). The first and third case cannot occur since i and ij are not conjugate, which proves the claim. □

We have to distinguish two different cases:

(i) C_{G}(i) is connected.

(ii) C_{G}(i) is not connected.

We are going to show that G ≅ G_{2}(K) in the first case and G ≅ PSp_{4}(K) in the second case. Assume from now on that D = ⟨i, j⟩.

Lemma 4.53 If C_{G}(i) is connected, then G contains one conjugacy class of involutions.

Proof. S ≤ C_{G}(i), since i ∈ Z(S). As Sylow 2-subgroups of G are conjugate, and C_{G}(i) contains only two conjugacy classes of involutions i and j^{C_{G}(i)} by Lemma 3.19, it is enough to show that i and j are conjugate.

Assume that j is not conjugate to i and let w ∈ N_{G}(T) such that D⟨w⟩ is an elementary abelian 2-subgroup of order 8. Then C_{G}(D) = T ⟨w⟩ and w inverts T. As i and ij are not conjugate in G, C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗} by Corollary 4.52. Let u ∈ (N_{G}(T) ∩ C_{G}(j)^{∘})\T be an involution. Then u ∈ C_{G}(D), i.e. since C_{G}(D) = T ⟨w⟩, u = tw for some t ∈ T. Thus u has to invert T. Contradiction as Z(C_{G}(j)^{∘}) ≤ T is infinite. □

Corollary 4.54 If C_{G}(i) is connected, then G ≅ G_{2}(K).

Proof. If C_{G}(i) is connected, then G contains one conjugacy class of involutions by Lemma 4.53. Since furthermore the centralisers of involutions in G_{2}(K) are isomorphic to SL_{2}(K) ∗ SL_{2}(K), the claim follows by Theorem 4.49. □

Lemma 4.55 N_{G}(L_{1}) = C_{G}(i)^{∘}C_{G}(C_{G}(i)^{∘}).

Proof. As C_{G}(i)^{∘}C_{G}(C_{G}(i)^{∘}) ≤ N_{G}(L_{1}), we only need to prove the reverse inclusion. Let g ∈ N_{G}(L_{1}). As Z(L_{1}) = ⟨i⟩, g ∈ C_{G}(i) and g ∈ N_{G}(L_{2}) as well by [18, 7.1]. By Lemma 4.45 and Lemma 4.46, g ∈ C_{G}(L_{1})L_{1} ∩ C_{G}(L_{2})L_{2}. Thus there exists l_{n} ∈ L_{n} for n = 1, 2 such that gl_{n} ∈ C_{G}(L_{n}). As L_{1} and L_{2} commute, gl_{1}l_{2} ∈ C_{G}(L_{1}) ∩ C_{G}(L_{2}) and hence g ∈ C_{G}(C_{G}(i)^{∘})C_{G}(i)^{∘}. □

Set K_{s} := C_{G}(C_{G}(s)^{∘}) for all involutions s ∈ G.

Lemma 4.56 Let s ∈ I(G). Then K_{s} ∩ C_{G}(s)^{∘} = Z(C_{G}(s)^{∘}). Furthermore

(i) K_{i} is a finite group,

(ii) I(K_{i}) = i,

(iii) K_{i} ∩ K_{s} = 1 for all s ∈ D^{∗} such that i ≠ s.

Proof. As Z(C_{G}(i)^{∘}) = ⟨i⟩, K_{i} ≤ C_{G}(i) and K_{i} ∩ C_{G}(i)^{∘} = ⟨i⟩. Hence K_{i}^{∘} ≤ C_{G}(i)^{∘} ∩ K_{i} = ⟨i⟩ and K_{i} is a finite group, proving (i).

To show (ii) let t ∈ I(K) be an involution. Then O(C_{G}(t)) = 1 by Corollary 3.16. Assume that C_{G}(t)^{∘} > C_{G}(i)^{∘}. Then C_{G}(t)^{∘} ≅ G_{2} or C_{G}(t)^{∘} ≅ PSp_{4}(K) by Fact 3.9 as pr(C_{G}(t)^{∘}) = pr(G) = 2. However this would imply that C_{G}(j)^{∘}, C_{G}(ij)^{∘} ≤ C_{G}(t)^{∘} by Proposition 4.2 and G = C_{G}(t)^{∘} by Proposition 4.50. Contradiction. Thus C_{G}(t)^{∘} = C_{G}(i)^{∘} = C_{G}(it)^{∘} and t = i, as otherwise ⟨i, t⟩ is a four-subgroup and G = C_{G}(i)^{∘} by Proposition 4.50.

Let finally s ∈ D^{∗} such that s ≠ i. As C_{G}(i)^{∘}, C_{G}(s)^{∘} ≤ C_{G}(K_{i} ∩ K_{s}) and si is conjugate to s in C_{G}(i)^{∘}, [19, 5.14] and Proposition 4.50 imply that C_{G}(K_{s} ∩ K_{t})^{∘} = G. As G is simple, K_{i} ∩ K_{s} = 1. □

Corollary 4.57 C_{G}(i) = (C_{G}(i)^{∘} ∗ K_{i}) ⋊ ⟨v⟩ for any v ∈ G such that L_{1}^{v} = L_{2}. In this case v^{2} ∈ C_{G}(i)^{∘}K_{i} and we can choose v ∈ N_{G}(T) to be a 2-element.

Proof. Assume that there exists an element v ∈ G such that L_{1}^{v} = L_{2}. As Z(L_{1}) = Z(L_{2}) = ⟨i⟩, v ∈ C_{G}(i)\C_{G}(i)^{∘}, L_{2}^{v} = L_{1} by [18, 7.1] and v^{2} ∈ C_{G}(i)^{∘}K_{i} by Lemma 4.55.

Let x ∈ C_{G}(i). There are two possibilities: Either x ∈ N_{G}(L_{1}) = C_{G}(i)^{∘} ∗ K_{i} by Lemma 4.55 and Lemma 4.56 or L_{1}^{x} = L_{2} and L_{2}^{x} = L_{1} by [18, 7.1]. In the second case xv ∈ N_{G}(L_{1}) = C_{G}(i)^{∘}K_{i} by Lemma 4.55 again. Thus x ∈ (C_{G}(i)^{∘} ∗ K_{i}) ⋊ ⟨v⟩ in all cases.

Let finally d(v) = V × F, where V is a connected divisible group and F = ⟨f⟩ a finite cylic group by [18, ex. 10, p. 93]. Then V ≤ C_{G}(i)^{∘} and f^{2} ∈ N_{G}(L_{1}). Thus f has even order. Let o(f) = 2^{k}m where k ≥ 1 and m ∈ ℕ is odd. Then f^{m} is a 2-element such that L_{1}^{f^{m}} = L_{2}. Furthermore f^{m} is contained in a Sylow 2-subgroup of C_{G}(i) and as they are all conjugate to each other, we may assume that f^{m} ∈ N_{G}(T). □

Lemma 4.58 If all involutions in D are conjugate, then K_{i} = ⟨i⟩.

Proof. Assume that all involutions in D are conjugate. Then there exists an element y ∈ N_{G}(T), such that i^{y} = j by [18, 10.22]. Let w ∈ N_{G}(T) ∩ C_{G}(i)^{∘} such that E := D⟨w⟩ is an elementary abelian 2-subgroup of order 8. w inverts T and all other involutions that invert T are contained in wC_{G}(T) = wT K_{i} by Corollary 4.57. However, I(wT K_{i}) = wT by Lemma 4.56 and w^{y} ∈ wT. K_{i} acts hence on C_{G}(j)^{∘} centralising T ⟨w⟩. Thus K_{i} ≤ C_{G}(j)^{∘}K_{j} by Corollary 4.57. Let k ∈ K_{i} and x ∈ C_{G}(j)^{∘}, k_{1} ∈ K_{j} such that k = xk_{1}. Then x = kk_{1}^{−1} ∈ C_{G}(E) ∩ C_{G}(j)^{∘} = E. Thus K_{j} ≤ EK_{i}. Furthermore K_{ij} = K_{j}^{w_{1}} ≤ EK_{i}, where w_{1} ∈ L_{1} ∩ N_{G}(E) is an involution conjugating j and ij. Let now k := |K_{i}|. As all involutions in D are conjugate, k = |K_{l}| for all l ∈ D^{∗}. Furthermore |EK_{i}| = 4k as E ∩ K_{i} = ⟨i⟩. On the other hand K_{i}, K_{j}, K_{ij} ≤ EK_{i} and all three subgroups normalise each other. Hence k^{3} ≤ 4k by Lemma 4.56 and k = 2. Thus K = ⟨i⟩, if all involutions in D are conjugate. □

Proposition 4.59 If all involutions in D are conjugate, then C_{G}(i) is connected.

Proof. Suppose that all involutions in D are conjugate. Then K_{i} = ⟨i⟩ by Lemma 4.58. Assume that C_{G}(i) is not connected. Then there exists a 2-element v ∈ N_{G}(T) such that C_{G}(i) = C_{G}(i)^{∘} ⋊ ⟨v⟩ where L_{1}^{v} = L_{2} by Lemma 4.57. As v^{2} ∈ C_{G}(i)^{∘} is a 2-element, it is contained in a maximal torus of C_{G}(i)^{∘}. As maximal tori are divisible, there exists an element t ∈ C_{G}(i)^{∘}, such that t^{2} = v^{2}. Assume that t = t_{1}t_{2}^{v}, where t_{1}, t_{2} ∈ L_{1}. Then

(∗) 1 = v^{2}t^{−2} = v^{2}(t_{2}^{−2v}t_{1}^{−2}) = vt_{2}^{−2}vt_{1}^{−2}.

On the other hand v^{2} ∈ C_{G}(v) and thus t_{1}^{2}t_{2}^{2v} = (t_{1}^{2}t_{2}^{2v})^{v} = t_{2}^{2v^{2}}t_{1}^{2v}. This implies that

t_{2}^{−2v^{2}}t_{1}^{2} = t_{1}^{2v}t_{2}^{−2v} ∈ L_{1} ∩ L_{2} = ⟨i⟩

Hence either t_{1}^{2} = t_{2}^{2} or t_{1}^{2} = it_{2}^{2}. Set s := t_{1}^{2}. If t^{2} = ss^{v}, then (∗) implies that vs^{−1} is an involution. If t^{2} = iss^{v}, then (vs^{−1})^{2} = i. By replacing v with a conjugate of vs^{−1}, we may assume that v ∈ N_{G}(T), such that v^{2} ∈ ⟨i⟩.

As all involutions in D are conjugate and j is conjugate to ij in C_{G}(i), there exists an element y ∈ N_{G}(T), such that i^{y^{2}} = j^{y} = ij by [18, 10.22], where y^{3} ∈ T. Furthermore v ∈ C_{G}(D) and v does not invert T. This implies that v ∈ C_{G}(j)\C_{G}(j)^{∘} and v ∈ C_{G}(ij)\C_{G}(ij)^{∘}. Thus there exist elements x_{1}, x_{2} ∈ C_{G}(D) such that v^{y} = vx_{1} and v^{y^{2}} = vx_{2}, where x_{1} ∈ C_{G}(j)^{∘} and x_{2} ∈ C_{G}(ij)^{∘} by Corollary 4.57.

Let P := C_{T} (v)^{∘}. Then P = {ll^{v}|l ∈ T ∩ L_{1}} and C_{T} (v) = P ⟨i⟩. On the other hand x_{n} either inverts T or centralizes it for some n = 1, 2. If x_{n} would centralize T for n = 1, 2, then P = C_{T} (vx_{n})^{∘} = (C_{T} (v)^{y^{n}})^{∘} = P^{y^{n}}. Contradiction as P contains a unique involution from D. Thus x_{1} and x_{2} invert T.

Let T [v] := {t ∈ T |t^{v} = t^{−1}}. Then T [v] ≤ T and T [v]^{∘} = {ll^{−v}|l ∈ T ∩ L_{1}}. Obviously T = P T [v]. As x_{n} inverts T for n = 1, 2, C_{T} (x^{y^{n}}) = C_{T} (vx_{n}) = T [v]. Thus T [v]^{∘} = (C_{T} (v^{y})^{∘})^{y} = (T [v]^{∘})^{y}. Contradiction again, as T [v]^{∘} contains a unique involution from D. Hence C_{G}(i) = C_{G}(i)^{∘} is connected. □

Proposition 4.60 If C_{G}(i) is not connected, then C_{G}(i) = C_{G}(i)^{∘} ⋊ ⟨u⟩, where u ∈ C_{G}(D) is an involution such that L_{1}^{u} = L_{2}.

Proof. As C_{G}(i) is not connected, i and j are not conjugate in G by Proposition 4.59 and C := C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗} by Corollary 4.52. Let u ∈ (N_{G}(T) ∩ C)\T be an involution. Then u ∈ C_{G}(D) but u ∉ C_{G}(i)^{∘}, as u does not invert T. Thus C_{G}(i) = C_{G}(i)^{∘}K_{i}⟨u⟩ by Lemma 4.57 and we need to show that K_{i} = ⟨i⟩.

As K_{i} centralizes Z(C), K_{i} ≤ K_{j}C by Lemma 4.45 and Lemma 4.46. We show that C_{K_{i}}(u) = ⟨i⟩. u acts on K_{i}. Let k ∈ C_{K_{i}}(u). Then k centralizes the elementary abelian subgroup E_{1} := ⟨i, j, u⟩. As K_{i} ≤ K_{j}C and C_{C}(E_{1}) = Z(C)⟨u⟩, we must have k ∈ K_{j}⟨u⟩ by Lemma 4.56. Thus k^{2} ∈ K_{j} ∩ K_{i} = 1 by Lemma 4.56 and k ∈ ⟨i⟩ by Lemma 4.56 again. Especially C_{K_{i}}(u) = ⟨i⟩.

As |K_{i}|/|C_{K_{i}}(u)| = |S|, where S = {[k, u]| k ∈ K_{i}} ⊆ K_{i} consists of elements inverted by u by [18, ex. 17, p. 7], 2|S| = |K_{i}|. Let s ∈ S. Then u^{s} ∈ C and u^{s} = s^{u}us = us^{2}. Furthermore s centralizes T and thus u^{s} ∈ N_{G}(T)\T which implies that there exists a t ∈ T such that us^{2} = u^{s} = ut. Hence s^{2} = t ∈ K_{i} ∩ T = ⟨i⟩. Assume that s^{2} = i. Then u^{s} = ui. As ⟨j, u⟩ is conjugate to D in C, u and uj belong to different conjugacy classes. However, there are at most two conjugacy classes of involutions in uC_{G}(i)^{∘} by Lemma 3.19, namely u^{T} and (ui)^{T}. Contradiction. Hence s^{2} = 1 by Lemma 4.56 and s ∈ ⟨i⟩. Then |K_{i}| = 2|S| ≤ 4.

We finally prove that |S| = 1. Assume that |S| = 2. Then |K_{i}| = 4 and there exists an element k ∈ K_{i} of order 4 by Lemma 4.56. As C_{K_{i}}(u) = ⟨i⟩, k^{u} = k^{3} = k^{−1}. Especially u^{k} = k^{u}uk = uk^{2} = ui. Contradiction as before and we are done. □

Proposition 4.61 If C_{G}(i) is not connected, C_{G}(j) ≅ (GL_{2}(K)/⟨−I⟩) ⋊ ⟨w⟩ where −I = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) and w ∈ C_{G}(D) ∩ C_{G}(i)^{∘} is an involution that acts as an inverse-transpose automorphism on GL_{2}(K), i.e.

( \begin{matrix}a & b \\ c & d \end{matrix} )^{w} = (ad − bc)^{−1} ( \begin{matrix}d & −c \\ −b & a \end{matrix} )

for any ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(K).

Proof. As C_{G}(i) is not connected, C_{G}(j)^{∘} ≅ PSL_{2}(K) × K^{∗} ≅ GL_{2}(K)/⟨−I⟩ by Proposition 4.59 and Corollary 4.52. Set C := C_{G}(j)^{∘}.

Let w ∈ C_{G}(D) ∩ C_{G}(i)^{∘} be an involution that inverts T. We show that C_{G}(j) = C ⋊ ⟨w⟩. As w inverts T, w ∉ C and C_{G}(j) ≥ C ⋊ ⟨w⟩. To prove the other inclusion let \overline{C} := C/Z(C). By Lemma 4.45 C_{G}(j) = C_{C_{G}(j)}(\overline{C})C. Let h ∈ C_{C_{G}(j)}(\overline{C}). Then T^{h} ≤ T Z(C) = T and h ∈ N_{G}(T). As h ∈ C_{G}(j) and i and ij are not conjugate, h ∈ C_{G}(D) = T ⟨u, w⟩. Thus, as T ⟨u⟩ ≤ C, C_{G}(j) = C⟨w⟩ = C_{C_{G}(j)}(\overline{C})C.

As w inverts T, w ∉ C_{C_{G}(j)}(\overline{C}). Thus there exists v ∈ C_{G}(D) such that wv ∈ C_{C_{G}(j)}(\overline{C}). w inverts T and v ∈ C, hence wv inverts Z(C). Furthermore [wv, C] ⊆ Z(C) and C = Z(C) ∗ C_{C}(wv) by [18, ex. 10, p. 98] where Z(C) ∩ C_{C}(wv) = ⟨j⟩. Set H := C_{C}(wv)^{∘}. As C ≅ PSL_{2}(K) × K^{∗}, H ≅ P SL_{2}(K) and C = Z(C) × H. We may assume that v ∈ H.

Let x ∈ C. Then there exists z ∈ Z(C) and h ∈ H such that x = zh. Furthermore x^{wv} = z^{−1}h, and x^{w} = z^{−1}h^{v}. Thus h^{w} = h^{v} for all h ∈ H. Set T_{1} = H ∩ T. As w inverts T_{1}, v has to invert T_{1} as well. Hence v ∈ I(N_{G}(T_{1})\T_{1}) and after maybe conjugating w with an element from T, we may assume that v corresponds to ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ). This implies the claim. (Compare Section 3.2.) □

Proof of Theorem 4.44. Let G be as in Theorem 4.44. If C_{G}(i) is connected, then G ≅ G_{2}(K) by Corollary 4.54. If on the other hand C_{G}(i) is not connected, C_{G}(i) and C_{G}(j) are isomorphic to centralizers of involutions in PSp_{4}(K) by Proposition 4.60 and Proposition 4.61. (Compare Section 3.2. again.) Thus G ≅ PSp_{4}(K) by Theorem 4.48. □

4.2.2 PSp_{4}(K)

This section is devoted to the proof of Theorem 4.48. Let G be as in Theorem 4.48. Then C_{G}(i) = (L_{1} ∗ L_{2})⟨u⟩ where L_{n} ≅ SL_{2}(K) for n = 1, 2, u is an involution in G such that L_{1}^{u} = L_{2}, and furthermore L_{1} ∩ L_{2} = Z(L_{1}) = Z(L_{2}) = ⟨i⟩. We may furthermore assume as in the proof of Theorem 3.18 that there exists a Sylow 2-subgroup S of G, such that i, j ∈ S^{∘} and S ≤ C_{G}(i). Let D := ⟨i, j⟩ and T := C_{G}(D)^{∘}.

We are going to construct a BN-pair of G, where N is N_{G}(T) and B := N_{G}(Q)^{∘} for a maximal quasiunipotent subgroup Q of G. Let U be a maximal quasiunipotent subgroup of L_{1} that is normalized by T and f := rk(Z(C_{G}(j)^{∘})).

Lemma 4.62 W := UU^{u} is properly contained in a maximal quasiunipotent subgroup of G which is normalized by T.

Proof. W is a maximal quasiunipotent subgroup of C_{G}(i). Assume that W is already a maximal quasiunipotent subgroup of G. Let V := C_{W} (u). Then rk(V) ≥ f by Lemma 4.20 and V is normalized by T. We may assume by Lemma 3.19 that u ∈ C_{G}(j)^{∘} and that u is conjugate to ij in C_{G}(j)^{∘}. Thus V is a maximal quasiunipotent subgroup of C_{G}(u) and rk(V) = f. Consider now N_{G}(V)^{∘} and let D_{1} be a four-subgroup of C_{G}(u) such that C_{G}(D_{1})^{∘} normalises V. Then N_{G}(V)^{∘} = ⟨C_{N_{G}(V)}(l)^{∘}| l ∈ D_{1}^{∗}⟩ by [19, 5.14]. Since W normalises V, N_{G}(V)^{∘} cannot be solvable because otherwise u ∈ N_{G}(V)^{∘} ≤ N_{G}(W)^{∘} = T W ≤ C_{G}(i)^{∘} by Proposition 2.20 and Proposition 2.16 since W normalises V. Let B be a Borel subgroup of N_{G}(V) such that W ≤ B. B contains a maximal normal quasiunipotent subgroup by Proposition 2.16 which must be equal to W by assumption. This implies that B ≤ N_{G}(W)^{∘} ∩ N_{G}(V)^{∘} = T W ∩ N_{G}(V)^{∘} = Z(C_{G}(u)^{∘})W by Proposition 2.20 again. Especially pr(B) = 1. On the other hand there exists a Borel subgroup B_{1} of N_{G}(V)^{∘} such that C_{G}(D_{1})^{∘} ≤ B_{1}. Then pr(B_{1}) = 2. Contradiction as Borel subgroups of K-groups are conjugate by Proposition 1.6. □

Proposition 4.63 There exists a maximal quasiunipotent subgroup X of C_{G}(j) which is normalized by T, such that Q := W ⟨X, X^{w_{2}}⟩ is a maximal quasiunipotent subgroup of G, where w_{2} is an involution in (L_{2}T ∩ N_{G}(T))\T.

Proof. Consider C_{G}(U)^{∘}. C_{G}(U) has Prüfer 2-rank 1, since C_{G}(T)^{∘} = T. Thus, since L_{2} ≤ C_{G}(U)^{∘}, C_{G}(U)^{∘} is not solvable and C_{G}(U)^{∘}/σ is isomorphic to PSL(2, K) where σ is the solvable radical of C_{G}(U)^{∘} by [2] and Lemma 3.11. Let Q be a maximal quasiunipotent subgroup of G which contains W = UU^{u}. By Lemma 4.62 W < Q. By Proposition 2.18 we can choose Q to be normalized by T. Assume that Q∩σ = U. Then Qσ/σ ≅ Q/U > W/U ≅ U^{u} is a quasiunipotent subgroup of C_{G}(U)^{∘}/σ. Contradiction. Hence Q ∩ σ > U. On the other hand Q ∩ σ is the maximal quasiunipotent subgroup of σ and thus normal in C_{G}(U)^{∘}. In particular (Q ∩ σ)^{w_{2}} = Q ∩ σ. Furthermore Q(σ) = ⟨C_{Q(σ)}(l)| l ∈ D^{∗}⟩ by [19, 4.6], i.e. Q ∩ σ = U⟨C_{Q∩σ}(j)^{∘}, C_{Q∩σ}(ij)^{∘}⟩. As j^{w_{2}} = ij and Q ∩ σ > U, there is a maximal quasiunipotent subgroup X of C_{G}(j) such that Q ∩ σ = U⟨X, X^{w_{2}}⟩ and X is normalized by T since Q ∩ σ is. Hence C_{G}(U)^{∘} contains a maximal quasiunipotent subgroup Q of G such that Q = UU^{u}⟨X, X^{w_{2}}⟩. □

Let \overline{x} := xT for all x ∈ N_{G}(T)

Lemma 4.64 (i) W := N_{G}(T)/T is a finite group of order 8, which is generated by \overline{u} and \overline{w}_{2}. The other elements of W are \overline{1}, \overline{w_{1}} = \overline{w}\overline{u} ; 2, \overline{w} = \overline{w}\overline{u} ; 2\overline{w}_{2} =

\overline{w}_{2}\overline{w}\overline{u} ; 2, \overline{uw} = \overline{wu} = \overline{u}^{\overline{w}^{2}}, \overline{u}\overline{w_{2}} and \overline{w}_{2}\overline{u} = \overline{uw_{1}}.

(ii) uT Xu ⊆ T XuX and w_{2}T U^{u}w_{2} ⊆ T U^{u}w_{2}U^{u}.

(iii) C_{G}(j)^{∘} = ⟨T, X, u⟩ and C_{G}(i)^{∘} = ⟨T, W, w_{1}, w_{2}⟩.

(iv) X^{w_{2}u} = X^{w_{2}} and U^{w_{2}} = U.

Proof. (i) follows directly from the definitions noting that N_{G}(T) ≤ C_{G}(i). C_{G}(j)^{∘}, being a reductive algebraic group of rank 1, has a BN-pair (B_{1}, N_{1}),

where B_{1} := T X and N_{1} = ⟨u, T ⟩. Hence uT Xu ⊆ T X ∪ T XuX = T XuX and C_{G}(j)^{∘} = ⟨B_{1}, N_{1}⟩. L_{2}T has a BN-pair (B_{2}, N_{2}), where B_{2} := T U^{u} ∩ L_{2} and N_{2} = ⟨w_{2}, T ⟩. This gives us that w_{2}T U^{u}w_{2} ⊆ T U^{u}w_{2}U^{u} and L_{2}T = ⟨B_{2}, N_{2}⟩. This implies (ii) and (iii) as (L_{1}T)^{u} = L_{2}T.

Furthermore C_{G}(j)^{∘} contains exactly two quasiunipotent subgroups that are normalized by T and swapped by u and w. In particular X^{uw} = X and X^{w_{2}u} = X^{uw_{1}(w_{2}w_{2})} = X^{w_{2}}. Finally, U ≤ L_{1} is normalized by w_{2} as w_{2} ∈ L_{2}T, proving (iv). □

Proposition 4.65 There exists a quasiunipotent subgroup Q of G which is normalized by T such that

(i) Q = Y ⋊ X, where Y is normalized by u and

(ii) Q = Z ⋊ U^{u}, where Z is normalized by w_{2}.

Furthermore Q = ⟨W, X, X^{w_{2}}⟩.

Proof. We have chosen ⟨X, X^{w_{2}}⟩ ≤ C_{G}(U) by Proposition 4.63. Thus U centralizes X^{w_{2}} and U^{u} centralizes X^{w_{2}u} = X^{w_{2}} by Lemma 4.64. Set Y := W X^{w_{2}}. Then Y is a quasiunipotent group that is normalized by u by Fact 2.4. Furthermore Q := Y X is a maximal quasiunipotent subgroup of G by Proposition 4.63. Thus Y has infinite index in N_{Q}(Y) by [18, 6.3], as Q > Y. As N_{Q}(Y) is normalized by T, N_{Q}(Y)^{∘} = ⟨C_{N_{Q}(Y)}(l)^{∘}| l ∈ D^{∗}⟩ by [19, 4.6]. Hence X ≤ N_{Q}(Y) and Q = Y ⋊ X.

Set Z := U⟨X, X^{w_{2}}⟩. Then Z ≤ Q is normalized by w_{2} by Lemma 4.64. Furthermore U^{u} normalizes Z by the proof of Proposition 4.63 and Q = Z ⋊ U^{u}. □

We are now in the position to construct a BN-pair of our group G.

Theorem 4.66 G is a split BN-pair of Tits rank 2, where B := N_{G}(Q)^{∘}, N := N_{G}(T) and \overline{u}, \overline{w}_{2} are the generators of the Weyl group.

Note that this proves our result, Theorem 4.48, by Theorem 4.1.

Proof. By Corollary 3.17, G = ⟨C_{G}(k)^{∘}| k ∈ D^{∗}⟩. Since ⟨N, UU^{u}⟩ ≥ C_{G}(i)^{∘}, ⟨N, X⟩ ≥ C_{G}(j)^{∘} and ⟨N, X^{w_{2}}⟩ ≥ C_{G}(ij)^{∘} by Lemma 4.64, it follows that G = ⟨B, N⟩. As B = QT by Proposition 2.20, B ∩ N = T ⊲ N and BN1 is valid. Furthermore we have ⟨\overline{u}, \overline{w}_{2}⟩ = N_{G}(T)/T by Lemma 4.64 proving BN2. Let S := \{\overline{u}, \overline{w}_{2}\} and W := N/T.

To prove BN3 we will show that vBw ⊆ BvwB ∪ BvB for all v, w ∈ N such that \overline{v} ∈ W and \overline{w} ∈ S. This is obviously equivalent to the original condition. By Lemma 4.64 and Proposition 4.65

uBu = uT XY u ⊆ T XuXY ⊆ BuB

w_{2}Bu = w_{2}T XY u = T X^{w_{2}}w_{2}uY ⊆ Bw_{2}uB

uwBu = uwT XY u = T X(uw)uXY ⊆ B(uw)uB

uw_{2}Bu = uw_{2}T XY u = T X^{w_{2}}(uw_{2})uY ⊆ B(uw_{2})uB.

Thus

w_{1}Bu = (uw_{2})uBu ⊆ uw_{2}BuB ⊆ Buw_{2}uB = Bw_{1}B
wBu = (wu)uBu ⊆ wuBuB ⊆ BwB
w_{2}uBu ⊆ w_{2}BuB ⊆ Bw_{2}uB.

Now 1Bw ⊆ B1wB for all w ∈ N and there are only seven cases left. By Lemma 4.64 and Proposition 4.65 again

uBw_{2} = uT U^{u}Zw_{2} = T Uuw_{2}Z ⊆ Buw_{2}B
w_{2}Bw_{2} = w_{2}T U^{u}Zw_{2} ⊆ T U^{u}w_{2}U^{u}Z ⊆ Bw_{2}B
w_{1}Bw_{2} = w_{1}T U^{u}Zw_{2} = T U^{u}w_{1}w_{2}Z ⊆ Bw_{1}w_{2}B
w_{2}uBw_{2} = w_{2}uT U^{u}Zw_{2} = T Uw_{2}uw_{2}Z ⊆ Bw_{2}uw_{2}B

Hence

wBw_{2} = w_{1}w_{2}Bw_{2} ⊆ w_{1}Bw_{2}B ⊆ Bw_{1}w_{2}B = BwB
uwBw_{2} = (w_{2}u)w_{2}Bw_{2} ⊆ w_{2}uBw_{2}B ⊆ Bw_{2}uw_{2}B = BuwB
uw_{2}Bw_{2} ⊆ uBw_{2}B ⊆ Buw_{2}B

and BN3 is valid. By Proposition 4.65 finally Q = XY = U^{u}Z, Q^{u} = X^{u}Y ≠ Q and Q^{w_{2}} = U^{uw_{2}}Z ≠ Q which proves BN4 and gives us the theorem. □

4.2.3 G_{2}(K)

In this section we prove Theorem 4.49. Let G be as in Theorem 4.49, D := ⟨i_{0}, i_{1}⟩ a four-subgroup which is contained in the connected component of a Sylow 2subgroup of G and set i_{2} = i_{0}i_{1}. Let furthermore T := C_{G}(D)^{∘}. Then C_{G}(i_{0}) contains four quasiunipotent subgroups which are normalized by T and isomorphic to K^{+}. Let X and Y be two of them that centralize each other and let v be an involution in N_{C_{G}(i_{0})}(T)\T that normalizes X. Write \overline{x} := xT for all x ∈ N_{G}(T).

Lemma 4.67 Let W := N_{G}(T)/T. Then |W | = 12 and W is generated by two involutions \overline{w} and \overline{v}, where w ∈ I(C_{G}(i_{1})), such that i_{0}^{wv} = i_{1} = i_{2}^{vw} and o(wv) = 6. The center of W is ⟨\overline{z}⟩ where z := (wv)^{3}. Set y := (wv)^{4} = (wv)z and w_{k+1} := w^{y^{k}}, v_{k} := v^{y^{k}} for k ∈ ℕ. Then \overline{w}_{k} = \overline{w_{k+3}} and \overline{v}_{k} = \overline{v_{k+3}}. The elements in W are hence

\overline{1}, \overline{z} = \overline{(wv)^{3}}, \overline{wv} = \overline{yz}, \overline{(wv)^{2}} = \overline{y^{−1}}, \overline{(wv)^{4}} = \overline{y}, \overline{(wv)^{5}} = \overline{vw} = \overline{y^{−1}z}

and

\overline{w}_{k+1} = \overline{w^{(vw)^{2k}}} = \overline{y^{k}w} and \overline{v}_{k} = \overline{v^{(vw)^{2k}}} = \overline{y^{k}v}

for k = 0, 1, 2.

Proof. As G contains one conjugacy class of involutions, all involutions of D are conjugate in N_{G}(T) by [18, 10.22]. N_{G}(D)/C_{G}(D) ≅ S_{3}, N_{G}(T) = N_{G}(D) and |C_{G}(D)/C_{G}(T)| = 2, which implies that W is the dihedral group of order 12. □

Lemma 4.68 Let w_{0}, v_{0} as in Lemma 4.67. Then w_{0}T Xw_{0} ⊆ T Xw_{0}X and v_{0}T Y v_{0} ⊆ T Y v_{0}Y. Furthermore C_{G}(i_{0})^{∘} = ⟨T, X, Y, v_{0}, w_{0}⟩.

Proof. L := ⟨X, T, X^{w_{0}}⟩ is a reductive algebraic group of rank 1. L has thus a BN-pair (B_{1}, N_{1}), where B_{1} := T X and N_{1} = ⟨w_{0}, T ⟩. Hence w_{0}T Xw_{0} ⊆ T X ∪T Xw_{0}X = T Xw_{0}X and L := ⟨T, X, w_{o}⟩. The same argument for ⟨Y, T, Y_{0}⟩ yields the remaining part of the lemma. □

Lemma 4.69 Let X_{1} := X, Y_{1} := Y, X_{2} := X_{1}^{w_{0}}, Y_{2} := Y_{1}^{v_{0}} and X_{n}^{(λ)} := X_{n}^{y^{λ}} as well as Y_{n}^{(λ)} := Y_{n}^{y^{λ}} for any n = 1, 2 and λ ∈ ℤ. Then

(i) X_{n}^{(λ)} = X_{n}^{(λ+3)} and Y_{n}^{(λ)} = Y_{n}^{(λ+3)} for all n = 1, 2 and λ ∈ ℤ.

(ii) v_{λ} centralizes X_{n}^{(λ)} and w_{λ} centralizes Y_{n}^{(λ)} for λ = 0, 1, 2 and n = 1, 2.

(iii) (X_{1}^{(κ)})^{w_{λ}} = X_{2}^{(2κ−λ)} and (X_{1}^{(κ)})^{v_{λ}} = X_{1}^{(2κ−λ)} for κ, λ = 0, 1, 2.

(iv) (Y_{1}^{(κ)})^{w_{λ}} = Y_{1}^{(2κ−λ)} and (Y_{1}^{(κ)})^{v_{λ}} = Y_{2}^{(2κ−λ)} for κ, λ = 0, 1, 2.

Proof. Since \overline{y^{3}} = \overline{1}, (i) follows. We have furthermore chosen w_{0} such that w_{0} centralizes Y_{1}. Hence v_{0} has to centralize X_{1} which yields (ii). To prove (iii) note that \overline{y^{λ}} = \overline{w_{λ}w_{0}} = \overline{v_{λ}v_{0}} and hence

\overline{y^{κ}w_{λ}y^{λ−2κ}w_{0}} = \overline{y^{3κ−λ}w_{λ}w_{0}} = \overline{y^{−λ}y^{λ}} = \overline{1}

and

\overline{y^{κ}v_{λ}y^{λ−2κ}v_{0}} = \overline{y^{3κ−λ}v_{λ}v_{0}} = \overline{1}

for λ, κ ∈ 0, 1, 2. Since however X_{1}^{w_{0}} = X_{2}, X_{1}^{v_{0}} = X_{1}, Y_{1}^{w_{0}} = Y_{1} and Y_{1}^{v_{0}} = Y_{2} by (ii), (iii) and (iv) follow. □

Lemma 4.70 Let G be a connected K-group of finite Morley rank such that the solvable radical σ of G is finite. Then G is a central product of quasi-simple algebraic groups over algebraically closed fields.

Proof. Since any definable action of a definable connected group on a finite set is trivial σ = Z(G). Furthermore G/σ is isomorphic to the direct product of simple algebraic groups over algebraically closed fields by [2]. Then G/σ = (G/σ)′ = G′σ/σ ≅ G′/(G′ ∩ σ) and rk(G) = rk(G′). As G is connected, G = G′ and G is semisimple. Assume that G is quasi-simple. Then G is an algebraic group by [4]. Let now G/σ ≅ A_{1} × ⋯ × A_{m} for some m ∈ ℕ, where A_{i} are simple algebraic groups over algebraically closed fields for 1 ≤ i ≤ m. Let G_{i} be the preimages of A_{i} in G for 1 ≤ i ≤ m. Then G = G_{1}^{∘}....G_{m}^{∘}. Now [G_{i}^{∘}, G_{j}^{∘}] is a connected subgroup of σ and hence trivial for any 1 ≤ i < j ≤ m. Furthermore [G_{i}^{∘}, G_{i}^{∘}] = G_{i}^{∘} for any 1 ≤ i ≤ m as above. Thus G is a central product of m quasi-simple algebraic groups by the first case. □

Proposition 4.71 If char(K) ≠ 3 then there exists an element t of order 3, such that C_{G}(t)^{∘} ≅ SL_{3}(K). Actually C_{G}(t)^{∘} ≅ SL_{3}(K) for exactly two elements of order 3 in T.

Proof. Assume that char(K) is not 3. Since y^{3} ∈ T, y = y′t, where we can choose y′ to be a 3-element by [18, ex. 11, p. 93]. Let P be a Sylow 3-subgroup of N_{G}(T). Since N_{G}(T) is solvable, P is nilpotent-by-finite by [18, 6.20]. Hence there exists an element t ∈ T ∩ Z(P) by [18, ex. 12, p. 14]. It follows that t is centralized by y. Thus (t^{w_{0}}t)^{y^{2}} = (t^{w_{0}}t)^{w_{2}w_{0}} = t^{yw_{0}}t = t^{w_{0}}t and t^{w_{0}}t ∈ C_{T} (y). Hence either t^{w_{0}} = t^{−1} or t^{w_{0}}t ∈ C_{T} (y) is an element of order 3 that is inverted by w_{0}. We may assume that t is inverted by w_{0}. Now C_{G}(t)^{∘} = ⟨C_{C_{G}(t)}(l)^{∘}| l ∈ D^{∗}⟩ and C_{C_{G}(i_{0})}(t) = C_{C_{G}(i_{1})}(t)^{y^{2}} = C_{C_{G}(i_{2})}(t)^{y}. However, C_{C_{G}(i_{0})}(t)^{∘} ≅ GL_{2}(K).

We show that the solvable radical σ of C_{G}(t)^{∘} is finite. σ is normalized by T and hence σ^{∘} = ⟨C_{σ}(l)^{∘}|l ∈ D⟩ by [19, 4.12]. However, C_{σ}(l)^{∘} ≤ Z(C_{C_{G}(l)}(t)^{∘}) for all l ∈ D^{∗} as C_{σ}(l)^{∘} is contained in the solvable radical of C_{C_{G}(l)}(t)^{∘}. Thus σ^{∘} ≤ ⋂_{l∈D^{∗}} Z(C_{C_{G}(l)}(t)^{∘}) ≤ T. One the other hand Z(C_{C_{G}(i_{0})}(t)^{∘}) ≅ K^{∗} and as G does not interpret a bad field, σ is either finite or i_{0} ∈ Z(C_{C_{G}(i_{0})}(t)^{∘}) = Z(C_{C_{G}(i_{1})}(t)^{∘}). The second case cannot occur since C_{G}(D)^{∘} = T and σ finite.

Thus C_{G}(t)^{∘} is a central product of quasi-simple algebraic groups by Lemma 4.70 and C_{G}(t)^{∘} ≅ SL_{3}(K). Furthermore T ≤ C_{G}(t)^{∘} contains eight elements of order 3 and while y centralizes exactly t and t^{−1}, yw operates transitively on the remaining six. □

Now we can construct a BN-pair as in [16].

Proposition 4.72 If char(K) ≠ 3 then there exists a maximal quasiunipotent group Q in G which is normalized by T such that

(i) Q = V ⋊ Y_{1} where V is normalized by v_{0} and

(ii) Q = M ⋊ X_{2}^{(1)} where M is normalized by w_{1}.

Furthermore Q = ⟨X_{1}, X_{2}^{(1)}, X_{2}^{(2)}, Y_{1}, Y_{2}^{(1)}, Y_{1}^{(2)}⟩.

Proof. Since char(K) is not 3, there exists an element t ∈ T of order 3, such that C_{G}(t)^{∘} ≅ SL_{3}(K). We can hence assume that C_{G}(t)^{∘} contains the maximal quasiunipotent subgroup U := X_{1}⟨X_{2}^{(1)}, X_{2}^{(2)}⟩, where X_{1} is central in U. (Compare Section 4.2). Consider C_{G}(X_{1}). We proceed as in Lemma 4.63. U ≤ C_{G}(X_{1}) and L := ⟨T, Y_{1}, Y_{2}⟩ ≤ C_{G}(X_{1}). Thus C_{G}(X_{1})^{∘} is not solvable. Furthermore C_{G}(X_{1})^{∘} has Prüfer 2-rank 1, since C_{G}(T)^{∘} = T. Thus C_{G}(X_{1})^{∘}/σ ≅ PSL_{2}(K) by [2] and Lemma 3.11, where σ is the solvable radical of C_{G}(X_{1}). On the other hand Lσ/σ ≅ L/(σ ∩L) ≅ L/Z(L) ≅ PSL_{2}(K). Hence C_{G}(X_{1})^{∘} = Lσ^{∘} and U ≤ σ. Set V := Q(σ). Then V^{v_{0}} = V.

Assume that U = V. Then Y_{1}U is a quasiunipotent group by Fact 2.4. Especially Y_{1}U is nilpotent and W := Y_{1}X_{1} < Y_{1}U has infinite index in N := N_{Y_{1}U}(W)^{∘} by [18, 6.3]. N is a quasiunipotent group that is normalized by T and hence N = ⟨C_{N}(l)^{∘}| l ∈ D^{∗}⟩ by [19, 4.6]. Furthermore C_{N}(i_{0}) = W and C_{N}(i_{k}) is either trivial or equals X_{2}^{(k)} for k = 1, 2 as T normalises N and acts transitively on X_{2}\{1}. We may assume that X_{2}^{(1)} ≤ N.

Set P := W X_{1}^{(2)} ⋊ T. Then P ≤ N_{G}(W). Let w ∈ W \{1} such that C_{G}(w) ∩ T = ⟨i_{0}⟩. As T acts transitively on W \{1}, for any element x ∈ P, there exists an element t_{x} ∈ T such that w^{x} = w^{t_{x}}. Thus P ⊆ C_{P} (w)T. As i_{1} and i_{2} invert W, C_{P} (w) is a solvable group normalized by D. Thus C_{P} (w)^{∘} = ⟨C_{C_{P} (w)}(l)^{∘}| l ∈ D^{∗}⟩ by [18, 4.6]. Furthermore (C_{P} (w) ∩ C_{G}(i_{0}))^{∘} = W, (C_{P} (w) ∩ C_{G}(i_{1}))^{∘} ≤ X_{1}^{2} and (C_{P} (w) ∩ C_{G}(i_{2}))^{∘} = 1. Thus Q := C_{P} (w)^{∘} ≤ W X_{1}^{(2)} is a quasiunipotent subgroup of P containing W. Since there is a definable surjective map from C_{P} (w) × T onto N_{P} (W)

rk(W X_{1}^{(2)}) + rk(T) = rk(N_{P} (W)) ≤ rk(C_{P} (w)) + rk(T) = rk(Q) + rk(T).

Thus Q = W X_{1}^{(2)} and N_{P} (W) = C_{P} (w)^{∘} ⋊ T. Especially Q ≤ C_{P} (w^{t}) for all t ∈ T and hence Q ≤ C_{G}(W) as T acts transitively on W. Thus X_{2}^{(1)} ≤ C_{G}(W) and X_{2}^{(1)} centralizes Y_{1}. This implies that Y_{1}^{(1)} = Y_{1}^{w_{2}} ≤ C_{G}(X_{2}^{(1)})^{w_{2}} = C_{G}(X_{1}). As C_{G}(X_{1})^{∘} = Lσ this implies that Y_{1}^{(1)} ≤ V. Contradiction.

Hence U < V and either ⟨Y_{2}^{(1)}, (Y_{2}^{(1)})^{v_{0}}⟩ ≤ V or ⟨Y_{1}^{(1)}, (Y_{1}^{(1)})^{v_{0}}⟩ ≤ V. We may assume that V = ⟨Y_{2}^{(1)}, Y_{1}^{(2)}, U⟩. Set Q := Y_{1}V. Q is obviously a maximal quasiunipotent subgroup of G and V is normalized by v_{0} by construction. Set Y := ⟨Y_{1}, Y_{2}^{(1)}, Y_{1}^{(2)}⟩. Then Y is invariant under w_{1} and M := Q ∩ Q^{w_{1}} = ⟨Y, X_{1}X_{2}^{(2)}⟩. Furthermore M is normalized by X_{2}^{(1)}, since N_{Q}(M)^{∘} = Q by [18, 6.3] as in the previous paragraph. □

Theorem 4.73 If char(K) ≠ 3, G is a split BN-pair of Tits rank 2, where B := N_{G}(Q)^{∘}, N := N_{G}(T) and \overline{w}_{2}, \overline{v}_{0} are the generators of the Weyl group.

Note that this proves our result, Theorem 4.49, by Fact 4.1.

Proof. G = ⟨C_{G}(k)| k ∈ D^{∗}⟩ by Corollary 3.17 and ⟨N, X_{1}Y_{1}⟩ ≥ C_{G}(i_{0}) by Lemma 4.9. Thus G = ⟨B, N⟩ as i_{1}, i_{2} are conjugate to i_{0} in N. BN1 now follows since by Proposition 2.20 B = QT and thus B ∩ N = T ⊲ N. Furthermore ⟨\overline{v}_{0}, \overline{w}_{1}⟩ = N_{G}(T)/T by Lemma 4.64 proving BN2. Let S := \{\overline{v}_{0}, \overline{w}_{1}\} and W := N/T.

To prove BN3 we will show that vBw ⊆ BvwB ∪ BvB for all v, w ∈ N such that \overline{v} ∈ W and \overline{w} ∈ S as in Theorem 4.66. By Lemma 4.7 and Proposition 4.72

1Bv_{0} ⊆ B1v_{0}B

v_{0}Bv_{0} = v_{0}T Y_{1}V v_{0} ⊆ T Y_{1}v_{0}Y_{1}V ⊆ Bv_{0}B

v_{2}Bv_{0} = v_{2}T Y_{1}V v_{0} = T Y_{2}^{(1)}v_{2}v_{0}V ⊆ Bv_{2}v_{0}B

w_{0}Bv_{0} = w_{0}T Y_{1}V v_{0} = T Y_{1}w_{0}v_{0}V ⊆ Bw_{0}v_{0}B

w_{1}Bv_{0} = w_{1}T Y_{1}V v_{0} = T Y_{1}^{(2)}w_{1}v_{0}V ⊆ Bw_{1}v_{0}B

y^{2}Bv_{0} = y^{2}T Y_{1}V v_{0} = T Y_{1}^{(2)}y^{2}v_{0}V ⊆ By^{2}v_{0}B

v_{0}w_{1}Bv_{0} = v_{0}w_{1}T Y_{1}V v_{0} = T Y_{2}^{(1)}v_{0}w_{1}v_{0} ⊆ Bv_{0}w_{1}v_{0}B

and thus

zBv_{0} = (zv_{0})v_{0}Bv_{0} ⊆ w_{0}Bv_{0}B ⊆ Bw_{0}v_{0}B = BzB
w_{1}v_{0}Bv_{0} ⊆ w_{1}Bv_{0}B ⊆ Bw_{1}v_{0}B
w_{2}Bv_{0} = (w_{2}v_{0})v_{0}Bv_{0} ⊆ v_{0}w_{1}Bv_{0}B ⊆ Bv_{0}w_{1}v_{0}B = Bw_{2}B.

Finally

yBv_{0} = yT Y_{1}V v_{0} = T Y_{1}^{(1)}yv_{0}V = T Y_{1}^{(1)}v_{1}V = T v_{1}Y_{1}^{(2)}V ⊆ Byv_{0}B

which gives us

v_{1}Bv_{0} = (v_{1}v_{0})v_{0}Bv_{0} ⊆ yBv_{0}B ⊆ Byv_{0}B = Bv_{1}B.

On the other hand

1Bw_{1} ⊆ B1w_{1}B
w_{1}Bw_{1} = w_{1}T X_{2}^{(1)}Mw_{1} ⊆ T X_{2}^{(1)}w_{1}X_{2}^{(1)}M ⊆ Bw_{1}B
v_{1}Bw_{1} = v_{1}T X_{2}^{(1)}Mw_{1} = T X_{2}^{(1)}v_{1}w_{1}M ⊆ Bv_{1}w_{1}B
w_{2}Bw_{1} = w_{2}T X_{2}^{(1)}Mw_{1} = T X_{1}w_{2}w_{1}M ⊆ Bw_{2}w_{1}B
v_{0}Bw_{1} = v_{0}T X_{2}^{(1)}Mw_{1} = T X_{2}^{(2)}v_{0}w_{1}M ⊆ Bv_{0}w_{1}B
yBw_{1} = yT X_{2}^{(1)}Mw_{1} = T X_{2}^{(2)}yw_{1}M ⊆ Byw_{1}B
w_{1}v_{0}Bw_{1} = w_{1}v_{0}T X_{2}^{(1)}Mw_{1} = T X_{1}w_{1}v_{0}w_{1} ⊆ Bw_{1}v_{0}w_{1}B

and thus

zBw_{1} = (zw_{1})w_{1}Bw_{1} ⊆ v_{1}Bw_{1}B ⊆ Bv_{1}w_{1}B = BzB
v_{0}w_{1}Bw_{1} ⊆ v_{0}Bw_{1}B ⊆ Bv_{0}w_{1}B
w_{2}Bw_{1} = (w_{2}w_{1})w_{1}Bw_{1} ⊆ yBw_{1}B ⊆ Byw_{1}B = Bw_{2}B.

Finally

y^{2}Bw_{1} = y^{2}T X_{2}^{(1)}Mw_{1} = T X_{2}y^{2}w_{1}M = T X_{2}w_{0}M = T w_{0}X_{1}M ⊆ By^{2}w_{1}B which gives us

v_{2}Bw_{1} = (v_{2}w_{1})w_{1}Bw_{1} ⊆ w_{1}v_{0}Bw_{1}B ⊆ Bw_{1}v_{0}w_{1}B = Bv_{2}B.

and BN3 is valid. By Proposition 4.72 again Q^{v_{0}} = V Y_{2} ≠ V Y_{1} = Q and Q^{w_{2}} = MX_{1}^{(1)} ≠ MX_{1}^{(2)} = Q which proves BN4 and gives us the theorem. □

Part II Black Box Groups Reflexive Tori

We are interested in the 2-local structure of finite groups of Lie type over fields of odd characteristic. In this case, many elementary abelian 2-subgroups are contained in maximal tori and normalizers of maximal tori are 2-local subgroups. In this chapter we investigate the possible orders of strongly real elements in maximal tori.

5.1 Basic Properties

As described in Section 1.4, a very good way to study finite groups of Lie type is to deduce their properties from the corresponding properties of algebraic groups by means of the Frobenius map. We are hence going to study strongly real elements of algebraic groups first.

Let G be a connected semisimple algebraic group over some algebraically closed field of odd characteristic. We call a torus of G reflexive, if it is inverted by an involution of G. Let i ∈ G be an involution. If T is a maximal torus of G, denote by T [i] the subgroup of all elements in T which are inverted by i. The Zariski connected component T [i]^{∘} is a reflexive torus by [42, 16.2]. Consider the set of tori which are inverted by a conjugate of i and let Y_{i} be the set of the maximal ones.

We are going to show that a reflexive torus S contained in a maximal torus T can be assumed to be inverted by an involution in N_{G}(T). Call an automorphism of an algebraic group H semisimple automorphism if it can be achieved by conjugation by a semisimple element of some algebraic group containing H.

Fact 5.1 ([53]) Every semisimple automorphism of an algebraic group G fixes a Borel subgroup and a maximal torus thereof.

Corollary 5.2 Let G be a connected algebraic group, T a maximal torus of G and S ≤ T a reflexive torus inverted by i for some i ∈ I(G). Then N_{G}(S) ⊆ N_{G}(T)C_{G}(S) and there exists x ∈ C_{G}(S) such that i^{x} ∈ N_{G}(T).

Proof. As C_{G}(S) is connected by [42, 22.3], all maximal tori of C_{G}(S) are conjugate [42, 21.3]. Let x ∈ N_{G}(S). Then T^{x} ≤ C_{G}(S) and as T is a maximal torus of C_{G}(S), T^{x} = T^{z} for some z ∈ C_{G}(S). Thus N_{G}(S) ≤ N_{G}(T)C_{G}(S).

Finally, conjugation by i is a semisimple automorphism of C_{G}(S) and i normalises a maximal torus of C_{G}(S) by Fact 5.1. Hence there exists x ∈ C_{G}(S) such that i^{x} ∈ N_{G}(T). □

Lemma 5.3 G has finitely many conjugacy classes Y_{1}, … Y_{s} on Y_{i}. Let F be a Frobenius map. Y_{m} contains a torus S_{m} fixed by F for all 1 ≤ m ≤ s. Let A_{m} := N_{G}(S_{m}). Then G^{F} has |H^{1}(F, A_{m})| = |H^{1}(F, N_{G}(S_{m})/C_{G}(S_{m}))| conjugacy classes on Y_{m}^{F}. If G is a Chevalley group, then |H^{1}(F, A_{m})| is the number of conjugacy classes in N_{G}(S_{m})/C_{G}(S_{m}).

Proof. Let T be a maximal torus in G. As maximal tori are conjugate, every S ∈ Y_{i} is conjugate to a subtorus of T. Assume that S ∈ Y_{i}, S ≤ T is inverted by i^{g}. Then S is inverted by i^{x} ∈ N_{G}(T) for some x ∈ C_{G}(S) by Corollary 5.2 and S = T [i^{x}]^{∘}. As the Weyl-group N_{G}(T)/T of G is finite, G has finitely many conjugacy classes on Y_{i}. The result follows now by Fact 1.12 as in the case of maximal tori. □

Let W_{m} := N_{G}(S_{m})/C_{G}(S_{m}) the Weyl-group of a maximal reflexive torus S_{m} fixed by F. As for maximal tori, we say that (S_{m}^{g^{−1}})^{F} is a reflexive torus of G^{F} twisted by w := g^{−1}F (g)C_{G}(S_{m}) ∈ W_{m}. To determine W_{m}, we will need some trivial facts about Coxeter and Weyl groups.

A Coxeter system is a pair (W, S), where W has generators s ∈ S and relations (ss′)^{m(s,s′)} where m(s, s) = 1 and m(s, s′) = m(s′, s) ≥ 2 for s ≠ s′. W is also called Coxeter group. Elements of W are called reflections, if they are conjugate in W to an element of S. The length l(w) of an element w ∈ W is the minimum length of an expression of the form w = s_{1} ⋯ s_{r} where s_{i} ∈ S. If l(w) = r and w = s_{1} ⋯ s_{r} where s_{i} ∈ S, then s_{1} ⋯ s_{r} is called a reduced expression for w.

The classical examples of a Coexter groups are Weyl groups of root systems. In this case the definitions of reflections coincide.

Lemma 5.4 Let w be an involution in a Coxeter group W. Then w is a product of commuting reflections.

Proof. By induction on the length of w. Assume that the claim is true for l(w) = n and let s_{1} … s_{n}s_{n+1} be a reduced expression for w. Since w^{2} = 1, l(ws_{1}) = n = l(s_{1}w). Consider now w_{1} = s_{1}ws_{1}. If l(w_{1}) = n − 1, then w_{1} is a product of commuting reflections by induction hypothesis. This implies that w = w_{1}^{s_{1}} is the product of commuting reflections as well. If on the other hand l(w_{1}) = n + 1 = l(w), then w = s_{1}ws_{1} by [43, 7.2] and s_{1} commutes with w_{2} = s_{2} … s_{n+1}. This implies that w_{2} is an involution and the claim follows by induction hypothesis. □.

Lemma 5.5 Let W be the Weyl-group of a root system Φ over the vector-space V and w ∈ W an involution. Then V = E_{1} ⊕ E_{−1}, where E_{k} is the eigenspace of w for the eigenvalue k, k = ±1. Furthermore E_{−1} is spanned by Φ ∩ E_{−1}.

Proof. Let A_{rs} := \frac{2(r,s)}{(r,r)} for all r, s ∈ Φ. As w is an involution, there exists a minimal n ∈ ℕ and pairwise commuting reflections w_{r_{i}} for r ∈ Φ (1 ≤ i ≤ n) such that w = w_{r_{1}} … w_{r_{n}} by Lemma 5.4. Since w_{r_{i}}w_{r_{j}}w_{r_{i}} = w_{w_{r_{i}}(r_{j})} = w_{r_{j}−A_{r_{i}r_{j}} r_{i}}, the r_{i} are pairwise orthogonal for 1 ≤ i ≤ n. This implies that

w(v) = v − ∑_{i=1}^{n} A_{r_{i}v}r_{i}

for all v ∈ V. Hence w(v) = −v if and only if v ∈ ⟨r_{1}, …, r_{n}⟩. Let E_{1} be the eigenspace of w for the eigenvalue 1, E_{−1} the eigenspace of w for the eigenvalue −1. Then V = E_{1} ⊕ E_{−1} and E_{−1} is spanned by Φ ∩ E_{−1}. □

We are assuming from now on, that G is a Chevalley group over an algebraically closed field K of odd characteristic. Most of the theory can be developed in a more general context, but for the present purpose this setting is sufficient.

Proposition 5.6 Let G be a Chevalley group over an algebraically closed field K, i ∈ G an involution and S ∈ Y_{i} a reflexive torus. Then, if T ≥ S is a maximal torus of G, S is inverted by j ∈ N_{G}(T). Let Φ be a root system for G, W the Weyl-group of Φ. Then N_{G}(S)/C_{G}(S) ≅ N_{W} (W_{J})/C_{W} (J) where J is a fundamental system of the set of roots inverted by the involution v ∈ W corresponding to j.

Proof. Let Δ a fundamental subsystem of Φ, and h and n be the homomorphisms corresponding to T as defined in Section 1.3. As C_{G}(T) = T, S in nontrivial and S = T [j]^{∘} for some involution j conjugate to i. By Lemma 5.2 we may assume that j ∈ N_{G}(T).

v := n(j) is an involution of the Weyl group W of Φ. Let χ, χ′ ∈ X such that χ′(r) = χ(v(r)) for all r ∈ Δ. Then h(χ) ∈ T [j] if and only if

χ(r)χ′(−r) = χ(r)χ(v(−r)) = χ(r + v(−r)) = 1

for all r ∈ Δ. As v is an involution of the Weyl group of Φ, V is the orthogonal sum of the eigenspaces E_{1} and E_{−1} as in Lemma 5.5. Let r ∈ Δ and r = r_{1} + r_{2} where r_{1} ∈ E_{1} and r_{2} ∈ E_{−1}. Then v(−r) = −r_{1} + r_{2} and r + v(−r) = 2r_{2}. Thus h(χ) ∈ T [i] if and only if χ(r_{2}) = ±1 for all r_{2} ∈ E_{−1} such that (E_{1} +r_{2})∩Δ ≠ ∅. This implies that S is generated by h(χ_{r,λ}) where r ∈ Ψ := Φ ∩ E_{−1} and λ ∈ K^{∗}.

Ψ is the set of roots inverted by v. Furthermore Ψ is a root system of E_{−1} of its own right as E_{−1} is spanned by Ψ by Lemma 5.4 and v(w_{r}(s)) = −s+ \frac{2(r,s)}{(r,r)}r = −w_{r}(s) for all r, s ∈ Ψ. Hence we can choose our fundamental system Δ such that J := Δ ∩ Ψ is a fundamental system for Ψ. Especially Ψ = Φ_{J} has the Weyl-group W_{J} := ⟨w_{r}|r ∈ J⟩ by [25, 2.5].

As N_{G}(S) ⊆ N_{G}(T)C_{G}(S), N_{G}(S) = (N_{G}(T) ∩ N_{G}(S))C_{G}(S) and

N_{G}(S)/C_{G}(S) ≅ (N_{G}(T) ∩ N_{G}(S))/((N_{G}(T) ∩ C_{G}(S)).

Consider N := n(N_{G}(T) ∩ N_{G}(S)) ≤ W. We claim that N = N_{W} (W_{J}) Let x ∈ N_{G}(S) ∩ N_{G}(T). xjx^{−1} inverts S. As above S is generated by h(χ_{r,λ}) where r ∈ Ψ′, λ ∈ K^{∗} and Ψ′ are the roots inverted by h(xjx^{−1}). This forces Ψ′ to equal Ψ. Set n := n(x) and let r ∈ J. Then nvn^{−1}(r) = −r and v(n^{−1}(r)) = −n^{−1}(r). Thus n^{−1}(r) ∈ Ψ. On the other hand w_{r}^{n} = w_{n^{−1}(r)} by [25, 2.4] and n ∈ N_{W} (W_{J}).

Set C := n(N_{G}(T) ∩ C_{G}(S)). We show that C = C_{W} (J). Let x ∈ N_{G}(T) ∩ C_{G}(S). As x ∈ C_{G}(S), χ_{r,λ}(s) = χ_{r,λ}(n(x)^{−1}(s)) for all r ∈ Ψ, λ ∈ K^{∗}, s ∈ Δ and χ_{r,λ}(s−n(x)^{−1}(s)) = 1. By definition of χ_{r,λ}, this implies that (r, s−n(x)^{−1}(s)) = 0 for all r ∈ Ψ, s ∈ Δ. However E_{−1} is spanned by Ψ and s−n(x)^{−1}(s) ∈ E_{1} for all s ∈ Δ. Furthermore v ∈ W_{J} by [25, 2.5.1], n(x) ∈ N_{W} (W_{J}) and n(x)vn(x)^{−1} ∈ W_{J}. Thus v(s − n(x)^{−1}(s)) = −s + n(x)^{−1}(s) = −(s − n(x)^{−1}(s)) for all s ∈ J. Then s − n(x)^{−1}(s) ∈ E_{−1} ∩ E_{1} = 0 for all s ∈ J and n(x)(s) = s for all s ∈ J. This yields the claim and proves the proposition. □

Let Y_{1}, ⋯ Y_{m} be the conjugacy classes of G on Y_{i}. Let F be a Frobenius map and T a maximal torus fixed by G. We may take T to be the diagonal of G. Let furthermore S_{k} ≤ T be a reflexive torus in Y_{k} for 1 ≤ k ≤ m.

Proposition 5.7 Let S ∈ Y_{i}. Assume that S^{F} ∈ Y_{k} is a reflexive torus twisted by w ∈ W_{k} := N_{G}(S_{k})/C_{G}(S_{k}) for some 1 ≤ k ≤ m. Then the order of S^{F} is |χ_{w}(q)| where χ_{w} is the characteristic polynomial of w in W_{k} and q the number of elements of the base field of G^{F}.

Proof. By Corollary 5.2 we may assume that w = xC_{G}(S_{k}), where x ∈ N_{G}(T). Then S ≤ T_{w_{0}}, where w_{0} := xT ∈ N_{G}(T)/T. To determine the order of S^{F}, note that the dual of S, X(S) is contained in the dual of T_{w} and that hence F acts on X(S) as qw, where q is he number of elements of the base field of G^{F}. The order of S^{F} is therefore |det(qw − 1)| computed in X(S). This, however, is nothing but |χ_{w}(q)| where χ_{w} is the characteristic polynomial of w in W_{k} as in [52]. □

5.2 Examples of reflexive tori

We are going to investigate reflexive tori in Chevalley groups of type B_{n} and C_{n} for n ≥ 2 over a finite field K. As the unlabeled Dynkin diagrams of type B_{n} and C_{n} are equal, the corresponding root systems Φ := Φ_{B_{n}} and Φ_{C_{n}} are isomorphic and hence W := W (Φ_{B_{n}}) ≅ W (Φ_{C_{n}}) for all n ≥ 2. Let Δ = {α_{1}, .....α_{n}} be a set of simple roots of Φ. We can represent Δ by α_{k} = e_{k} − e_{k+1} for 2 ≤ k ≤ n and α_{1} = 2e_{1} if Φ is of type C_{n} and α_{1} = e_{1} if Φ is of type B_{n} where {e_{1}, ....e_{n}} is an orthonormal basis of V. W is isomorphic to ℤ_{2}^{n} ⋊ S_{n} and each element determines a product of disjoint cycles (j_{1} ⋯ j_{m}) or (j_{1} ⋯ j_{m} −j_{1} ⋯−j_{m}) where j_{i} ∈ {1, ⋯, n, −1, ⋯ − n}. The first kind of cycle is said to be positive of length m, the second kind negative of length m. The signed cycle-type of w is the set of lengths of the disjoint cycle together with their signs. The action of W on Δ is as follows. The normal subgroup S_{n} acts as permutation group on the set of basis vectors of V, π ∈ S_{n} sending e_{i} to e_{π(i)} for all 1 ≤ i ≤ n. The complement ℤ_{2}^{n} acts by e_{i} ⟶ ρ_{i}(z)e_{i} for 1 ≤ i ≤ n where z ∈ ℤ_{2}^{n} and ρ_{i} is the i-th projection map.

Fact 5.8 ([25]) Two elements of W are conjugate if and only if they have the same cycle type. The characteristic polynomial χ_{w} of an element w ∈ W is the product of the following polynomials: (i) (q^{m}−1) for every positive cycle of length m and (ii) q^{m} + 1 for every negative cycle of length m.

Let G be a Chevalley group over the Lie algebra L(K), where L is of type B_{n} or C_{n} and K is any field of odd characteristic. Then G ≅ PSp_{2n}(K) or G ≅ P Ω_{2n+1}(K) by Fact 1.8. If K is finite, then K = GF (q), where q is an arbitrary power of char(K). For the following compare [56, 11.52,11.53] and [37, 4.3].

Let K be finite or algebraically closed. Then PSp_{2n}(K) has [\frac{n}{2}] + 1 conjugacy classes of involutions. Let s ∈ Sp(2n, K) represent an element of order 2 in PSp_{2n}(K). Then either s^{2} = −I or s^{2} = I. All involutions which are represented by an element t such that s^{2} = −I form one conjugacy class of involutions of type t_{n}. Furthermore for every even number 2 ≤ m ≤ n there is one conjugacy class of involutions of type t_{m/2}. If s is an involution of type t_{m/2}, then the eigenvalues of s on the underlying natural module are −1 with multiplicity m and 1 with multiplicity 2n − m.

Ω_{2n+1}(K) has n conjugacy classes of involutions. There is one conjugacy class of involutions of type t_{m/2} for each even number 2 ≤ m ≤ 2n. An involution s is of type t_{m} if the eigenvalues of s on the underlying natural module are −1 with multiplicity m and 1 with multiplicity 2n + 1 − m.

Lemma 5.9 Let G = P Sp_{2n}(K) for an algebraically closed field K of odd characteristic and n ≥ 2. Let T be a maximal torus in G and i ∈ N_{G}(T) be an involution.

(i) If i is of type t_{m}, where 1 ≤ m ≤ [\frac{n}{2}], then iT can be written as the product of l positive 2-cycles for 0 ≤ l ≤ m.

(ii) If i is of type t_{n}, then iT can be written as the product of l_{1} positive 2-cycles and l_{2} negative 1-cycles for l_{1}, l_{2} ∈ ℕ such that 0 ≤ 2(l_{1}) + l_{2} ≤ n.

(iii) T ∈ Y_{i} if and only if i is of type t_{n}.

Especially iT contains involutions from [\frac{n}{2}] − 1 − k different conjugacy classes, if iT corresponds to the product of k + 1 positive 2-cycles for 0 ≤ k < [\frac{n}{2}].

Proof. iT is an involution in W and can thus be written as the product of commuting reflections by Lemma 5.4. The only reflections in W are positive 2cycles and negative 1-cycles. Assume that iT corresponds to a negative 1-cycle. As det(i) = 1, i is conjugate to an element in i_{0}T where

i_{0} := ( J_{k} ; \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ; J_{n−(k+1)} )

and J_{k} is a 2k × 2k matrix of the form diag(j, −j, ⋯ j, −j) for all 0 ≤ k ≤ n where j ∈ K is an element of order 4. Especially i is of type t_{n}. If on the other hand iT corresponds to a positive 2-cycle, then i is conjugate to an element in j_{o}T, where

j_{0} := ( \begin{matrix}0 & I_{2} \\ I_{2} & 0 \\ & & I_{n−4} \end{matrix} ) .

Thus i is of type t_{m} for 1 ≤ m ≤ [\frac{n}{2}]. This implies the claim. □

Corollary 5.10 Let G = P Sp_{2n}(K) for an algebraically closed field K of odd characteristic, n ≥ 2 and i ∈ G an involution. Then G acts transitively on Y_{i}.

Proof. Let T be a maximal torus. It is sufficient to show that G acts transitively on the reflexive subtori S ≤ T, where S ∈ Y_{i}. Furthermore we may assume that i ∈ N_{G}(T) by Lemma 5.2. If i is of type t_{n}, then T ∈ Y_{i} by Lemma 5.9 and we are done. Assume that i is of type t_{m} for 1 ≤ m < n. Then T [j]^{∘} ∈ Y_{i} for an involution j ∈ N_{G}(T) of type t_{m} if and only if i^{g} corresponds to the product of m positive 2-cycles by maximality of elements of Y_{i} and Lemma 5.9. However, all positive 2-cycles in the Weyl-group of G are conjugate by Fact 5.8. This yields the claim. □

Lemma 5.11 Let G = P Ω_{2n+1}(K) for an algebraically closed field K of odd characteristic and n ≥ 2. Let T be a maximal torus in G and i ∈ N_{G}(T) be an involution.

(i) If i is of type t_{m}, where 1 ≤ m ≤ [\frac{n}{2}], then iT an be written as the product of l_{1} negative 1-cycles and l_{2} positive 2-cycles, where 0 ≤ [\frac{l_{1}+1}{2}] + l_{2} ≤ m.

(i) If i is of type t_{m}, where [\frac{n}{2}] + 1 ≤ m ≤ n, then iT can be written as the product of l_{1} negative 1-cycles and l_{2} positive 2-cycles, where 0 ≤ [\frac{l_{1}}{2} ] + l_{2} ≤ n − m.

(iii) If n is even, then T ∈ Y_{i} if and only if i is of type t_{n/2}.

Especially iT contains involutions from n − 1 − 2k different conjugacy classes, if iT corresponds to the product of k + 1 positive 2-cycles for 0 ≤ k < [\frac{n}{2}].

Proof. iT is an involution in W and can thus be written as the product of commuting reflections by Lemma 5.4. The only reflections in W are positive 2-cycles and negative 1-cycles. Assume that iT corresponds to a positive 2-cycle. Then i is conjugate to an element in i_{0}T, where

i_{0} := ( I_{n−2}^{0 1 1 0}  0 ; I_{n−2}^{0 1 1 0} ; 0  1 )

Especially i is of type t_{k} for some 1 ≤ k ≤ n − 1. If iT corresponds to a negative 1-cycle, then is conjugate to an element in j_{o}T, where

j_{0} := ( \begin{matrix}I_{n−1}_{1  0}^{0  1} & 0 \\ 0 & I_{n−1}_{−1} \end{matrix} )

Hence i is of type t_{k} for some 1 ≤ k ≤ n. If furthermore n is even and i is of type t_{n/2}, then i is conjugate to an involution k in N_{G}(T) that corresponds to n negative 1-cycles. k inverts T and thus T ∈ Y_{i}. □

Corollary 5.12 Let G = P Ω(2n, K) for an algebraically closed field K of odd characteristic, n ≥ 2 and i ∈ G. Then G acts transitively on Y_{i}.

Proof. Let T be a maximal torus. It is sufficient to show that G acts transitively on the reflexive subtori S ≤ T, where S ∈ Y_{i}. If i is of type t_{n}, then iT correspond to a single negative 1-cycle and there is nothing to prove. Let i be of type t_{m} for 1 ≤ m ≤ [\frac{n}{2}]. Then T [j]^{∘} ∈ Y_{i} for some j ∈ N_{G}(T) if and only if j corresponds to the product of 2l_{1} negative 1-cycles and l_{2} positive 2-cycles for l_{1} + l_{2} = m by maximality of elements of Y_{i} and Lemma 5.9. Furthermore T [j_{0}]^{∘} ≥ T [j_{1}]^{∘} if j_{0}, j_{1} ∈ N_{G}(T) such that j_{0} corresponds to (1 − 1)(2 − 2) and j_{1} to (1 2). Hence l_{2} = 0 by maximality of elements of Y_{i}. This implies the claim for m ≤ [\frac{n}{2}] as elements of the Weyl-group with the same cycle-type are conjugate by Fact 5.8.

Let i be of type t_{m} for [\frac{n}{2}] + 1 ≤ m < n. Then T [j]^{∘} ∈ Y_{i} for some j ∈ N_{G}(T) if and only if j corresponds to the product of 2(n − m) + 1 negative 1-cycles by maximality of elements of Y_{i} and Lemma 5.9 as in the previous paragraph. This implies that all reflexive subtori S ∈ Y_{i} are conjugate and that S[j]^{∘} ∈ Y_{i} if and only if j corresponds to the product of 2(n − m) + 1 negative 1-cycles. □

It is not known, however, if a semisimple algebraic group acts in general transitively on Y_{i} for any involution i.

Proposition 5.13 Let G be a Chevalley group of type C_{n} over an algebraically closed field of of odd characteristic. Let i ∈ G be an involution of type t_{m} and F a standard Frobenius map. If q is the number of elements of the base field of G^{F}, then the order of S^{F} for any S ∈ Y_{i} is |χ(q)|, where χ is the characteristic polynomial of an element of W (Φ_{C_{m}}).

Proof. Let T be a maximal torus of G. G acts transitively on Y_{i} by Corollary 5.10. We may assume that S = T [i]^{∘} ∈ Y_{i} for some i ∈ N_{G}(T) by Lemma 5.2. If i is of type t_{n}, then S = T and the claim follows by Fact 1.14. If i is of type t_{m} for 1 ≤ m ≤ [\frac{n}{2}], then we may assume that iT corresponds to (1 2)(3 4) … (2m − 1 2m) by Lemma 5.9 and Corollary 5.10. Thus N_{G}(S)/C_{G}(S) ≅ N_{W} (W_{J})/C_{W} (J) where J is a fundamental system of the set of roots inverted by the involution v ∈ W corresponding to i by Proposition 5.6. J = {α_{1}, α_{3}, ⋯, α_{m}} consists of m components of type A_{1} and hence N_{W} (W_{J}) ≅ W_{J} ⋊ W (Φ_{C_{m}}) where W_{J} ≅ ℤ_{2}^{m} by [40]. Furthermore C_{W} (J) consists of the reflections corresponding to the long roots of the component C_{m} by [40] again. Thus

N_{W} (W_{J})/C_{W} (J) ≅ W_{J} ⋊ S_{m} ≅ ℤ_{2}^{m} ⋊ S_{m},

the Weyl-group of C_{m}. By Proposition 5.7 the order of S is thus a characteristic polynomial of an element w ∈ W (Φ_{C_{m}}), evaluated at q. □

Proposition 5.14 Let G be a Chevalley group of type B_{n} over an algebraically closed field of of characteristic p. Let i ∈ G be an involution of type t_{n/2} and F a standard Frobenius map. If q is the number of elements of the base field of G^{F}, then the order of S^{F} for any S ∈ Y_{i} is |χ(q)|, where χ is the characteristic polyomial of an element of

• W (Φ_{B_{2m}}) if 1 ≤ m ≤ [\frac{n}{2}] and of

• W (Φ_{B_{k}}), where k := 2(n − m) + 1 for [\frac{n}{2}] + 1 ≤ m ≤ n.

Proof. Let T be a maximal torus of G. G acts transitively on Y_{i} by Corollary 5.12. We may assume that S = T [i]^{∘} ∈ Y_{i} for some i ∈ N_{G}(T) by Lemma 5.2. If i is of type t_{m} for 1 ≤ m ≤ [\frac{n}{2}] then we may assume that iT corresponds to (1 −1) ⋯ (2m −2m) by Corollary 5.12. Thus N_{G}(S)/C_{G}(S) ≅ N_{W} (W_{J})/C_{W} (J) where J is a fundamental system of the set of roots inverted by the involution v ∈ W corresponding to i by Proposition 5.6. J = {α_{1}, α_{2}, ⋯, α_{2m}} consists of one component of type B_{2m} and hence N_{W} (W_{J}) ≅ W_{J} ⋊ W (Φ_{B_{2m}}) where W_{J} ≅ W (Φ_{B_{2m}}) by [40]. Furthermore C_{W} (J) consists of all reflections corresponding to the roots of the component B_{2m} by [40] again. Thus

N_{W} (W_{J})/C_{W} (J) ≅ W_{J} ≅ W (Φ_{B_{2m}}),

the Weyl-group of B_{2m}. By Proposition 5.7 the order of S is thus a characteristic polynomial of an element w ∈ W (Φ_{B_{2m}}),evaluated at q.

If i is of type t_{m} for [\frac{n}{2}] + 1 ≤ m then to we may assume that iT corresponds to we may assume that iT corresponds to (1 − 1) ⋯ (k − k) by Corollary 5.12, where k = 2(n − m) + 1. The above paragraph implies that the order of S is thus a characteristic polynomial of an element w ∈ W (Φ_{B_{k}}), evaluated at q. □

Since the characteristic polynomials of elements in the Weyl-group of Chevalley groups are given in [25], we have thus a complete description of the possible orders of strongly real elements in PSp_{2n}(q) and Ω_{2n+1}(q) for q odd and n ≥ 2.

Distinguishing B_{n} and C_{n}

The aim of this section is to propose a fast Monte-Carlo algorithm to distinguish PSp_{2n}(q) from Ω_{2n+1}(q) where n and q are given. Namely we prove the following theorem

Theorem 6.1 There is a Monte-Carlo algorithm which, when given a black box group G isomorphic to a simple classical group of type B_{n} or C_{n} defined over some field of given size q, can decide if G is of type C_{n} in time O(μn^{3} log q + ξn). The algorithm is one-sided if n is odd.

In the classical setting it is assumed the black box operations can be performed at unit cost. We will modify these assumptions slightly, in accordance with [46], and set μ to be an upper bound on the time for each multiplication or inversion within the group G. If thus ξ is an upper bound on the time requirement per element for the construction of independent, (nearly) uniformly distributed random elements of G, then ξ can be expressed as a polynomial function of N and μ by Fact 1.16.

In the following we assume for the sake of argument that the Black Box gives us independent uniformly distributed elements of G. For a discussion of this assumption see [11].

As PSp_{4}(q) ≅ P Ω_{5}(q) for all q and PSp_{2n}(2^{m}) ≅ P Ω_{2n+1}(2^{m}) for all n, m ∈ ℕ, it is enough to give an algorithm for n ≥ 3 and q odd.

The structure of the algorithm is very straightforward:

108 CHAPTER 6. DISTINGUISHING B_{N} AND C_{N}

(1) Seek an element z of good order. (This would guarantee that z is contained in some specified maximal torus of even order.) If it cannot be found after examination of at least 20n randomly selected elements of G then return no. Otherwise go to step (2). The algorithm errs with probability less than 0.01. This step takes O(μn^{3} log q + ξn) time.

(2) Compute the involution x in the cyclic group generated by z in O(μn log q) time.

(3) Compute y := x^{g}x^{h} for random elements g, h ∈ G. If o(y) divides q(q + 1) or q(q − 1) then go to (4). Otherwise return yes. The algorithm errs with probability 0 if n is odd and with probability \frac{2}{2+n} if n is even. This step takes O(ξ + μ log q) time.

(4) Repeat step (3) 60n times. If y always divides q(q + 1) or q(q − 1) then return no. The error probability is ≤ 1/2 in this case.

The number of necessary repetions in (4) is a very crude estimate, which was sufficient for our theoretical purposes. In practice, one does get a strongly real element which does not divide q(q + 1) or q(q − 1) at the very first step, if G is of type C_{n} and q > 3.

I have written a short GAP-programme, which uses this algorithm. Tests have shown that it is very fast and reliable.

6.1 Preliminaries

We fix the following notations. Let K be an algebraically closed field of odd characteristic p and let H be either PSp_{2n}(K) or Ω_{2n+1}(K). Thus H is an adjoint Chevalley group of type C_{n} or B_{n} over the field K. Let F be a standard Frobenius map such that number of elements of the base field is q. Set G := [H^{F} , H^{F}]. Then H^{F} /[H^{F} , H^{F}] ≅ ℤ_{2} and G is a Chevalley group of type C_{n} or B_{n} over GF (q). Let e ∈ ℕ such that q = p^{e}.

Let T be the diagonal in H. Denote by Φ the set of all roots of H, by Δ a fundamental root system and by W the Weyl-group of H. Let furthermore T_{w} be a maximal torus of H^{F} which is twisted by w ∈ W. The order of T_{w} is the product of numbers of the form q^{k} ± 1 for k ≤ n by Fact 1.14 and Section 5.2. As |H^{F} /G| = 2, a maximal torus T_{0} of G has order |T_{w}|/2 for some w ∈ W.

The algorithm works with involutions in the most frequent tori. These are the tori twisted by an element of cycle-type −n or n by Fact 1.14. We have to distinguish the following cases:

(i) q^{n} ≅ −1 mod n, i.e. n odd and q ≅ −1 mod 4.

(ii) q^{n} ≅ 1 mod n, i.e. n even or q ≅ 1 mod 4

In the first case the maximal tori of order \frac{q^{n}+1}{2} contain an involution, in the second case the maximal tori of order \frac{q^{n}−1}{2} contain an involution. Note that if n is even, the tori of order \frac{q^{n}−1}{2} form at least two conjugacy classes of involutions. One arising from twisting with elements of cycle-type n and one from twisting with an elements of cycle-type (\frac{n}{2}, −\frac{n}{2}). We call tori in these conjugacy classes maximal non-split tori of even order. However if n ≅ 0 mod 4, there is even another conjugacy class of tori of order \frac{q^{n}−1}{2}, namely the ones twisted by elements of cycle-type (\frac{n}{2}, \frac{n}{4}, −\frac{n}{4}). To distinguish these tori from the maximal non-split tori of even order, we introduce the concept of 2-height. Let G be a group and g ∈ G such that o(g) = 2^{k}m where m is odd. Then k is called the 2-height of g. We can show, that the 2-height of elements of even order in a torus of order \frac{q^{n}−1}{2} is maximal if and only if the torus is a maximal non-split torus of even order.

Lemma 6.2 Let w = w_{1}w_{2} ∈ W, where w_{1} and w_{2} operate on disjoint sets and set H_{m} := C_{T} (w_{m}) and C_{m} := C_{G}(H_{m})′ for m = 1, 2. Then (C_{1}C_{2})^{F} contains a torus T_{w} twisted by w. Furthermore T_{w} can be chosen to be T_{1}T_{2}S where T_{m} is a torus in C_{m}^{F} twisted by w_{m} for m = 1, 2 and S = T ∩ H_{1} ∩ H_{2}.

Proof. C_{1} and C_{2} are connected semisimple groups which are generated by the root-subgroups {X_{α}| α ∈ S_{1}} and {X_{β}| β ∈ S_{2}} respectively where S_{m} := {π ∈ Π : ∃ρ ∈ Π : π = ρ − w_{m}^{−1}(ρ)} by [52, II,4.1]. It is easy to see, that α + β ∉ Π for all α ∈ S_{1} and β ∈ S_{2}. Hence C_{1} and C_{2} commute by [26, 4.1]. Since furthermore S_{1} ∩ S_{2} = ∅, C_{1} ∩ C_{2} is finite. As C_{m}^{F} contains a torus T_{m} twisted by w_{m} the claim follows. □

Corollary 6.3 Let m ∈ ℕ be even. The 2-height of elements of H^{F} in tori twisted by an element of cycle type m is greater than the 2-height of elements in tori twisted by elements of cycle type (\frac{m}{2}, −\frac{m}{2}).

Proof. Set l := \frac{m}{2} and let w be an element of cycle type (l, −l). Then T_{w} = T_{1}T_{2} by Lemma 6.2, where T_{1} is a cyclic group of order q^{l} − 1 and T_{2} is a cyclic group of order q^{l} + 1. Since the order of T_{w} is q^{m} − 1, T_{w} contains an elementary abelian subgroup of order 4 and the maximal 2-height is smaller than the 2-height of of a torus twisted by an element of cycle type m. □

Notice that if n is even the maximal 2-height in two non-conjugate maximal non-split tori of even order is the same, since G has index 2 in H^{F}.

In the following we are looking for elements of maximal 2-height in a maximal non-split torus of even order. To find elements in a maximal non-split torus the notion of primitive prime divisor elements of G plays a crucial role as in [48] and [46]. Let p be a prime and m ≥ 2. A prime dividing p^{m} − 1 but not p^{i} − 1 for 1 ≤ i < m is called a primitive prime divisor of p^{m} − 1. In our case primitive prime divisors always exist by [58]. Furthermore an integer j is called a ppd(p, m) if j|p^{m} − 1 and j is divisible by a primitive prime divisor of p^{m} − 1. An element of G of order ppd(p, en) is called primitive prime divisor element.

Let P be a maximal non-split torus of even order and let |P | = 2^{k}m where k, m ∈ ℕ and m is odd. We say that an element g ∈ P is of maximal 2-height, if the 2-height of g is k. Since P is cyclic, elements of maximal 2-height exists. Furthermore \frac{1}{2} of the elements of P have maximal 2-height as the cyclic group generated by a 2-element x of maximal 2-height contains 2^{k−1} elements of maximal 2-height, namely x^{s} for s ∈ ℕ and s odd.

We call g good if it satisfies the following conditions:

• g has maximal 2-height.

• g has ppd(p, 2en) order in case (i) and ppd(p, en) order in case (ii).

• o(g) ∤ q^{n/2} + 1 if n is even and q ≅ 1 mod 4.

An element of good order lies in a maximal non-split tori of even order by Corollary 6.3.

Proposition 6.4 The share of good elements of maximal non-split tori of even order in G is at least \frac{|G|}{4n}. If n is even the share of good elements which lay in a torus twisted by an element of cycle-type (\frac{n}{2}, −\frac{n}{2}) in the set of good elements is \frac{2}{2+n}.

Proof. Since good elements are regular the result can be proven as the corresponding estimate in [48].

The number of good elements in G is the number of good elements in a maximal non-split torus of even order times the number of maximal non-split tori of even order. We first compute the number of good elements in H^{F}. By Fact 1.14 there are \frac{|H^{F} |}{|C_{W} (w)|} maximal non-split tori if n is odd where w ∈ W is of cycle-type n or −n. If n is even we get \frac{|H^{F} |}{|C_{W} (w_{1})|} + \frac{|H^{F} |}{|C_{W} (w_{2})|} maximal nonsplit tori where w_{1} ∈ W has cycle-type n and w_{2} ∈ W has cycle-type (\frac{n}{2}, −\frac{n}{2}). Since |H^{F} | = 2|G| and |C_{W} (w)| = \frac{|W |}{|w^{W} |} = \frac{2^{n}n!}{|w^{W} |} for any w ∈ W we get that \frac{|H^{F} |}{|C_{W} (w)|} = \frac{|G||w^{W} |}{2^{n−1}n!}. Furthermore |w^{W} | is the number of elements in W with the same cycle-type as w and hence |w^{W} | = 2^{n−1}(n − 1)! if w has cycle-type n or −n.

If w ∈ W has cycle-type (\frac{n}{2}, −\frac{n}{2}) then |w^{W} | = 2^{n} \frac{(n−1)!}{n}. Thus if n is odd we have \frac{|G|}{n} maximal non-split tori of even order, if n is even \frac{|G|}{n} + \frac{2|G|}{n^{2}}. If n is even the share of good elements which lay in a torus twisted by an element of cycle-type (\frac{n}{2}, −\frac{n}{2}) in the set of good elements is thus \frac{2}{n^{2}} /(\frac{1}{n} + \frac{2}{n^{2}} ) = \frac{2}{2+n}.

Let S be a maximal non-split tours of even order. Let us consider case (ii) first. S is a cyclic group, generated by t. Let Φ(n, e) the product of all the primitive prime divisors of q^{n} − 1, counting multiplicities. Let S_{0} be the set of good elements in S. Then S_{0} = S\⟨t^{Φ(n,e)}⟩ ∪ ⟨t^{2}⟩ ∪ ⟨t^{r}⟩, where r = \frac{q^{n/2}−1}{2} if n is even and q ≅ 1 mod n and r = q^{n} − 1 otherwise. Thus if n = 1 mod 2, then t^{r} = 1, if n = 0 mod 4, then r is even and t^{r} ∈ ⟨t^{2}⟩. In all other case, r ≥ 2n. Since furthermore Φ(n, e) ≥ n + 1,

(∗) |S_{0}| ≥ |S| − (\frac{1}{n + 1} − \frac{1}{2} − \frac{1}{2n})|S| = \frac{n − 2}{2(n + 1)}|S|.

If n ≤ 5 the last term of the sum in (∗) can be omitted and |S_{0}| ≥ |S|\frac{n−1}{2(n+1)} ≥ \frac{|S|}{4}. If n ≥ 6 then \frac{n−2}{n+1} ≥ \frac{1}{2} and |S_{0}| ≥ \frac{|S|}{4} again. The share of good elements is thus at least \frac{|G|}{4n} in this case.

In the second case S is cyclic as well, generated by x. Since every primitive prime divisor of q^{2n} − 1 divides q^{n} + 1, again S_{0} = S\⟨x^{Φ_{2n}(q)}⟩ ∪ ⟨x^{2}⟩ and hence the share of good elements in S is at least \frac{2n−1}{2(2n+1)} ≥ \frac{1}{4}. Hence more than \frac{|G|}{4n} elements of G have good order. □

6.2 Proof of the theorem

The algorithm spends most of its timing testing if elements are good. The following lemma corresponds to [46, 2.7.].

Lemma 6.5 Let m_{1} := ∏_{i=2}^{2n−1} q^{i} − 1 and m_{2} := ∏_{i=2}^{n−1} q^{i} − 1. An element g ∈ G is good

- in case (i), if o(g)|q^{2n} − 1 but o(g) ∤ m_{1} and o(g) ∤ \frac{q^{n}+1}{4},
- in case (ii) for n odd, if o(g)|q^{n} − 1 but o(g) ∤ m_{2} and o(g) ∤ \frac{q^{n}−1}{4},

- in case (ii) for n even, if o(g)|q^{n} − 1 but o(g) ∤ m_{2} and o(g) ∤ \frac{q^{n}−1}{4} and

o(g) ∤ q^{n/2} + 1.

If an element is good can thus be tested using O(n^{2} log q) multiplications.

Proof. An element is good if it satisfies the conditions in the lemma by definition of good. Furthermore, we can decide if o(g) divides k for any k ∈ ℕ by computing g^{k} and comparing g^{k} with 1. This requires O(log k) multiplications if we use the squaring method. □

Hence if 0 < ϵ < 1, we can find a good element x in time O(μn^{3} log q + ξn) with probability 1 − ϵ by Proposition 6.4. Especially, a sample of 20n random elements of G contains a good element with probability (1−(1− \frac{1}{4n})^{20n}) > 1−e^{−5}.

Since x is an element of maximal 2-height, i := x^{\frac{q^{n}+1}{4}} is an involution in case (i) and i := x^{\frac{q^{n}−1}{4}} is an involution in case (ii). Hence i can be computed in time O(μn log q) time.

Proposition 6.6 Let i be a involution in a maximal non-split torus T_{w} ∩ G of even order.

(i) If w is of cycle type −n or n, then i is of type t_{n}.

(ii) If w is of cycle type (\frac{n}{2}, −\frac{n}{2}), then i is of type t\frac{n}{2} .

Proof. The easiest way to prove the claim is to look which centralizers of involutions of type t_{k} can contain maximal non-split tori of even order. Conjugacy classes of maximal tori and their orders are given by the tables in [37, 4.5] together with Fact 1.14 and [52]. Thus a maximal non-split torus of order \frac{q+1}{2} can be only contained in centralizers of involutions of type t_{n} for both groups. A maximal non-split torus of order \frac{q−1}{2} can arise in different ways. It can be contained in centralizers of involutions of type t_{n} or t\frac{n}{2} if n is even. Assume that n is even and that i is of type t\frac{n}{2} . Then a torus in C_{G}(i) of order \frac{q−1}{2} arises as central product of a torus of order q^{n/2} − 1 and q^{n/2} + 1. That means that it is a torus twisted by an element of cycle type (\frac{n}{2}, −\frac{n}{2}). (Compare Lemma 6.2). This gives us the claim. □

The crucial point of our algorithm is that we can choose elements of good order uniformly in both groups and that we can thus generate involutions of type t_{n}. However involutions of type t_{n} behave very differently in the two groups. In Section 5.2. we have seen, that involutions of type t_{n} invert maximal tori if G is of type C_{n} and while involutions of type t_{n} only invert one-dimensional tori if G is of type B_{n}. We can make this more precise.

Lemma 6.7 Let G := Ω_{2n+1}(q) and i ∈ G an involution of type t_{n}. Then o(i^{g}i^{h})|q(q + 1) or o(i^{g}i^{h})|q(q − 1) for all g, h ∈ G.

Proof. Since i is of type t_{n} the dimension of V_{−} is 2n. Thus dim W_{g} ≥ 2n − 1 for all g ∈ G where W_{g} := {v ∈ V : ii^{g}(v) = v}. Therefore i^{g}i^{h} acts as an element of Gl(2, V) for all g, h ∈ G and the claim follows □

Lemma 6.8 Let G := PSp_{2n}(q) and i ∈ G an involution of type t_{n}. Then i^{g}i^{h} is good with probability p > \frac{1}{30n} for random g, h ∈ G.

Proof. Consider the map φ : i^{G} × i^{G} ⟶ G. We are interested in the fibers of this map. Let x ∈ G be a good element. Then x is contained in φ(i^{G} × i^{G}) by Lemma 5.9. Furthermore |φ^{−1}(x)| = |I(C_{G}^{∗}(x))| = |T |, where T is the maximal non-split torus in which x is contained. Let us first assume q^{n} ≅ −1 mod n. At least \frac{|G|}{4n} elements of G are good by Proposition 6.4. Let X be the collection of all good elements, then |φ(X)^{−1}| = |G|\frac{q^{n}+1}{4n}. On the other hand |i^{G} × i^{G}| = |G|^{2}|C_{G}(i)|^{−2} and the share of pairs of involutions which are sent to a good element is \frac{|C_{G}(i)|^{2}(q^{n}+1)}{|G|4n}. Since we are in the first case, the centralizer of i is a twisted group of Lie type extended by a torus of order q + 1 and an involution. Thus

|C_{G}(i)| = (q + 1)q^{\frac{n(n−1)}{2}} (q^{2} − 1)(q^{3} + 1) … (q^{n} + 1)

by [26]. Hence

\frac{|C_{G}(i)|^{2}(q^{n} + 1)}{|G|} = \frac{q^{n} + 1}{q^{n}} \frac{(q + 1)(q^{2} − 1) … (q^{n} + 1)}{(q − 1)(q^{2} + 1) … (q^{n} − 1)}

If we set x(n, q) := ∏_{i=1}^{n} \frac{q^{i}−(−1)^{i}}{q^{i}+(−1)^{i}} then x(n, q) = e^{∑_{i=0}^{n−1}(−1)^{i}f_{i}(q)} where f_{i}(q) := ln(q^{i+1} + 1) − ln(q^{i+1} − 1). Since f_{i}(q) is a strictly decreasing sequence for all q ∈ ℕ, q > 1, the sum is convergent and lim_{n→∞} x(n, q) exists. Since however by elementary analysis, f_{i}(q) > 2\frac{1}{q^{i+1}} , x(n, q) > e^{2 ∑_{i=0}^{n−1}(−1)^{i} \frac{1}{q^{i+1}}} . The right side converges to e^{2 \frac{1}{1+q}} and is always greater than 1. Hence the share of pairs of involutions which are sent to good elements is greater than \frac{1}{4n}. If q^{n} ≅ 1 mod n, similar arguments apply. Here the share of pairs of involutions which are sent good elements is greater than \frac{1}{4n}e^{−2}. As e(2) ∗ 4 < 30, this implies the claim. □

Hence an involution i is contained in a maximal non-split torus is of type t_{n} with probability 1 if n is odd, and probability at least \frac{2}{2+n} if n is even. If, however, i is of type t_{n}, the type of the group can be recognized by the orders of strongly real elements by Lemma 6.7 and Lemma 6.8. This proves Theorem 6.1.

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Index

algorithm Las-Vegas, 17 Monte-Carlo, 17

one-sided, 17 randomized, 16

automorphism semisimple, 97

B-conjecture, 9 BN-pair, 52

split, 52 Brauer-Fowler theorem, 7

C_{G}^{∗}(x), 7 component, 9

intrinsic, 10 connected component, 6 core, 38

trivial, 38, 45 Coxeter system, 98 cycle

negative, 101 positive, 101

cycle-type signed, 101

D^{∗}, 33 d(X), 7 definable closure, 7 Δ, 12 Dynkin diagram, 12

element Coxeter length of, 98 reduced expression for, 98

good, 110 invisible, 67 primitive prime divisor, 110 strongly real, 7 visible, 67

F (G), 9 F-conjugate, 15 field

bad, 2, 8 of finite Morley rank, 1 pure, 1

Frobenius map, 15–16

standard, 15 fundamental system, 12 F^{∗}(G), 9

G_{2}(K), 77, 79, 88–93 Γ_{S,2}(G), 9 graph automorphism, 14 group

algebraic, 10–12 bad, 1, 8 black box, 17–19, 97–113 Chevalley, 12–14 connected, 6 Coxeter, 98 grey box, 18 K^{∗}-, 2, 9 K-, 2, 9 of Lie type, 15 of finite Morley rank, 6–10,

35–93 of Lie type, 16 Prüfer p-, 9 quasi-simple, 9 semi-simple, 9 tame, 2, 8–10 Weyl

of a BN-pair, 52 of a Chevalley group, 14 of a reflexive torus, 98, 99 of a root system, 12

Zariski connected, 11

H^{1}(F, A), 15

I(G), 7 Inv(G), 67 involution, 7

classical, 10 conjugacy class

in Ω(2n, q), 101 in PSp(2n, q), 101

L(G), 9 Landau symbol, 17 layer, 9

m(H), 9

n(H), 9

O(G), 38 O(f), 17 Ω_{2n+1}(q), 101–106, 113 oracle, 18

P (G), 28 Φ, 12 ppd(p, m), 110 primality test, 16 primitive prime divisor, 110 pr(H), 9 PSL_{3}(K), 55–77 PSp_{4}(K), 77, 79, 85–88 PSp_{2n}(q), 101–106, 113

Q(G), 28

radical pseudounipotent, 28 quasiunipotent, 28 solvable, 11

rank 2-, 9 normal 2-, 9 of a root system, 12 Prüfer, 9 Prüfer 2, 9

R(T), 59 reflection

in Coxeter groups, 98 regular point, 59 root, 12

simple, 12 root lattice, 14 root system, 12

irreducible, 13

set definable, 6 Zariski closed, 11

σ(G), 11 signalizer functor

nilpotent, 38 signalizer functor, 38

complete, 38 σ-setup, 15 Sr(G), 7 subgroup

Borel, 11 Fitting, 9

generalized, 9 four-, 33 invisible, 67 local, 35 parabolic, 52 p-local, 35 pseudounipotent, 23–31

maximal, 28 quasiunipotent, 23–34

n-dimensional, 56

maximal, 28

normalizer of, 31–34 recognizable, 17 root, 14 strongly embedded, 35–45 Sylow 2-, 7 visible, 67

subset generic, 66

T [i], 97 Thompson rank formula, 7, 45–51 Tits rank of a BN-pair, 52 torus

maximal non-split of even order, 109 conjugacy classes, 15–16 non-degenerate, 16 order of, 16 twisted by w, 16

p-, 23 reflexive, 97–106

in PSp_{2n}(q), 101–106

in P Ω_{2n+1}(q), 101–106

order of, 104, 105 Trichotomy theorem, 10 T [w_{k}], 59 2-height, 109

maximal, 110 2-generated core, 9, 37, 44 type

even, 3, 7 odd, 3, 7

uniform ϵ, 18 near, 18

W (Φ), 12
w_{r}, 12