ALGEBRAIC GROUPS UP TO ABSTRACT ISOMORPHISM

OLIVIER FRÉCON

Abstract. With any connected affine algebraic group G over an algebraically closed field K of characteristic zero, we associate another connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F)×K_{+}^{s}, where α is an abstract automorphism of D and s is an integer satisfying s = 0 when the derived subgroup of G contains the identity component of the center of G.

It follows from the latter that any two abstractly isomorphic connected algebraic groups over K have a common algebraic central extension.

The construction of D lies heavily on model theory and groups of finite Morley rank. In particular, it needs to prove that, for any two connected algebraic groups over K, the elementary equivalence of the pure groups implies they are abstractly isomorphic.

Contents

1. Introduction 1 2. Facts 9 3. The ACF -expansion of a pure group 12 4. Unipotent groups over \overline{ℚ} 20 5. Definably affine groups 30 6. More on pseudo-tori 36 7. Analysis of G/Z(G) and of G′ 42 8. Canonical development of G 45 9. The final argument 60 Acknowledgements 63 References 64

1. Introduction

1.1. Main theorem. The aim of this paper is to study the connections between the abstract isomorphy and the algebraic isomorphy in the context of connected affine algebraic groups over an algebraically closed field K of characteristic zero. More precisely, we want to classify all connected affine algebraic groups over K abstractly isomorphic to a fixed connected affine algebraic group over K. A feature of this work is that no structural assumption on the groups is required. Thus the

Date: May 26, 2010. 2000 Mathematics Subject Classification. 03C60, 14L17, 20G15. Key words and phrases. Affine algebraic groups, Abstract isomorphisms, Groups of finite Mor-

ley rank, Burdges’ unipotence.

algebraic groups are just connected and affine, and even just connected for some consequencies of the main result (see Corollary 1.2 and Theorem 1.5), but they are not necessarily simple or solvable for example.

Some situations prevent the equivalence of the abstract isomorphy and isomorphy as algebraic groups. For example, for all the positive integers m and n, the additive groups K_{+}^{m} and K_{+}^{n} are abstractly isomorphic, but they are isomorphic as algebraic groups only if m = n. Moreover, Remark 8.11 describes another problematic situation, where the group has finite center and is a central product of two infinite closed subgroups with finite intersection. The main theorem of this paper, Theorem 1.1 below, shows that these are the only pathological cases. More precisely, when we fix a connected affine algebraic group G, it determines all the other connected affine algebraic groups H, abstractly isomorphic with it, up to isomorphism of algebraic groups. In this paper, we denote by G^{∘} the identity component of any algebraic group G. Moreover, we denote by Z(G) the center of any group G, and by G′ its derived subgroup.

Theorem 1.1. – For any connected affine algebraic group G over an algebraically closed field K of characteristic zero, there is a connected affine algebraic group D over K and a finite central subgroup F of D such that, up to isomorphism of algebraic groups, affine algebraic groups over K abstractly isomorphic to G are precisely of the form D/α(F) × K_{+}^{s}, where α is a quasi-standard automorphism of D and s is an integer satisfying s = 0 when G′ contains Z(G)^{∘}.

An isomorphism α : G → H between two algebraic groups over algebraically closed fields K and L is said to be standard if it is the composition of a field isomorphism φ : K → L and an L-isogeny (i.e. L-rational homomorphism, surjective with finite kernel). We point out that, as throughout this paper, we identify the field isomorphism φ to the corresponding map φ^{∘} : G → ^{φ}G, where ^{φ}G denotes the algebraic group over L obtained by transfer of base field. Then an isomorphism α : G → H is said to be quasi-standard if G = G_{1} ×⋯×G_{n} and H = H_{1} ×⋯×H_{n} for G_{1}, …, G_{n}, H_{1}, …, H_{n} some algebraic subgroups such that, for each i, we have α(G_{i}) = H_{i}, and α_{|G_{i}} : G_{i} → H_{i} is a standard isomorphism.

We will mention several consequences of Theorem 1.1. In particular, they are worth mentioning Theorems 1.5 and 1.9, whose proofs are not direct, and the following two results, which reformulate Theorem 1.1 in some interesting special cases.

Corollary 1.2. – Any two abstractly isomorphic connected (nonnecessarily affine) algebraic groups G and H over an algebraically closed field K of characteristic zero have a common algebraic central extension.

Moreover, G/Z(G) and H/Z(H) are isomorphic as algebraic groups.

Proof – By Lemma 1.6 below, there is a connected affine algebraic group G_{A} (resp. H_{A}) over K such that G_{A} and G (resp. H_{A} and H) are abstractly isomorphic, and such that G_{A}/Z(G_{A}) and G/Z(G) (resp. H_{A}/Z(H_{A}) and H/Z(H)) are isomorphic as algebraic groups. Since G_{A} and H_{A} are abstractly isomorphic, Theorem 1.1 provides a connected algebraic group D, a finite central subgroup F of D, a quasi-standard automorphism α of D, and two integer r and s such that the algebraic groups G_{A} and D/F × K_{+}^{r} (resp. H_{A} and D/α(F) × K_{+}^{s}) are isomorphic. Since F and α(F) are finite, and since D is connected, we have Z(D/F × K_{+}^{r}) = Z(D)/F × K_{+}^{r} and Z(D/α(F) × K_{+}^{s}) = Z(D)/F × K_{+}^{s}, so G_{A}/Z(G_{A}) and H_{A}/Z(H_{A}) are isomorphic to D/Z(D) as algebraic groups. Consequently, G/Z(G) and H/Z(H) are isomorphic as algebraic groups.

Finally, the preimage in G × H of the graph of any isomorphism of algebraic groups from G/Z(G) to H/Z(H) is a common algebraic central extension for G and H. □

Corollary 1.3. – Let G and H be two abstractly isomorphic connected affine algebraic groups over an algebraically closed field K of characteristic zero. If G and H are not a central product of two infinite closed subgroups with a finite intersection, then G and H are isomorphic as algebraic groups.

Proof – Theorem 1.1 provides a connected algebraic group D, a finite central subgroup F of D, and a quasi-standard automorphism α of D, such that G and D/F (resp. H and D/α(F)) are isomorphic as algebraic groups. Our hypotheses over G and H imply that D has no decomposition as a direct product of two infinite closed subgroups, so α is standard. Now the map \overline{α} : D/F → D/α(F) defined by \overline{α}(xF) = α(x)α(F) for each x ∈ D is a standard isomorphism, hence the algebraic groups D/F and D/α(F) are isomorphic. □

Our study is closely linked with the very classical problem of the abstract isomorphisms of algebraic groups, which has been intensively studied since the twenties, and whose main paper is [3]. In the particular context of the algebraically closed fields [18], the theory essentially says that, if G and H are two simple algebraic groups over algebraically closed fields K and L respectively, then any abstract isomorphism μ : G → H is standard. The purpose of [3] was to extend the result by considering abstract homomorphisms, and especially to analyze the case where the ground fields K and L are not algebraically closed, but the nonsimple case was not considered. Indeed, in the context of the nonnecessarily simple connected affine algebraic groups, an abstract isomorphism α between two connected affine algebraic groups G and H, over algebraically closed fields K and L of characteristic zero, is not necessarily standard, even if G and H are solvable and centerless, or perfect and centerless for instance (see Example 3.1 (1) and (3)). Furthermore, in Example 3.1 (1) and (3), the isomorphism is not even continuous for the Zariski topology. Based on this observation, the relevance of the abstract isomorphisms in the general case seems rather different from the simple case, since such an isomorphism does not necessarily preserve the geometric structure of G.

In the present article, we consider the abstract isomorphisms from another point of view. Indeed, if α : G → H is an abstract isomorphism between two connected affine algebraic groups G and H, then prehaps that α is not standard, but that the existence of α implies that of a standard isomorphism from G to H. This situation is interesting since, in this case, the abstract isomorphy implies the isomorphy of the ground fields, and even the isomorphy of the algebraic groups when K = L. This is the subject of Corollary 1.8. However, to attack this problem, the algebraic and geometric methods do not seem to work outside, and the proof of Theorem 1.1 lies heavily on the model theory.

The methods and tools that will be used in this paper comes from the theory of groups of finite Morley rank. For more details about these groups, we may refer to [1, 4, 15, 16]. Morley rank is an ordinal-valued abstract dimension notion that arose in model theory. Groups of finite Morley rank are defined as being the goups having as Morley rank a natural number. Model theory studies structure using a certain formalism. A structure is an underlying set together with the graphs of some distinguished relations and functions. Thus an “abstract” group, or a pure group as model theorists would say, is a set G with the group law, inversion and the constant 1, as the distinguished binary, unary and 0-ary operations (one writes (G, · , ^{−1}, 1)). Any subset of G^{n} (n ∈ ℕ) described using first-order logic and symbols defining the three functions, the pure group language, is a definable set. More generally, one defines an interpretable set as a quotient of a definable set by a definable equivalence relation.

Using this line, a (pure) field is a structure of the form (K, +, −, · , 0, 1), where +, − and · are binary functions and 0 and 1 are constants. In this structure, the definable sets coincide with the constructible ones. This structural equivalence between definability and constructibility in the case of algebraically closed fields has a consequence at the level of model-theoretic and geometric dimensions:

Morley rank = Zariski dimension.

One question in the present paper is to compare the definable sets in an affine algebraic group when this group is seen as a pure group structure and when it is seen as a constructible set in an algebraically closed field. This is related also to the main conjecture in the analysis of groups of finite Morley rank: the Cherlin-Zil’ber Algebraicity Conjecture. This conjecture 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. Such a statement can be seen as finding an isomorphism between an abstract group with a dimension notion and an affine algebraic group. Since affine algebraic groups are examples of groups of finite Morley rank, it is natural that model theoretic techniques are relevant for questions of algebro-geometric nature such as the ones addressed in this paper.

1.2. Methods. Since the geometric and algebraic classical methods seem not appropriate for the study of the abstract structure of nonnecessarily simple groups, we are going to consider the model theory. Indeed, in the 80’s, B.I. Zil’ber [19], then B. Poizat [16], showed that model theory is potentially effective for our subject, since they dealt with a model-theoretic proof of the theorem of the abstract isomorphisms of simple algebraic groups in the algebraically closed context. However the bulk of the present work concerns nonnecessarily simple groups, and the last part of the proof of [16, Corollary 4.17] provides a rather general piece of information about the abstract isomorphisms. We gives it below. Moreover, we note that a slightly more elaborate version of this result will be fundamental for us (see Fact 3.6).

Fact 1.4. – [16, Proof of Corollaire 4.17] Let α be an abstract isomorphism between two algebraic groups G and H over algebraically closed fields. If, in the pure group G, there is an interpretable field K and a definable isomorphism from G to an algebraic group over K, then α is a standard isomorphism.

Actually the proof of [16, Corollaire 4.17] shows that α = β ∘ φ for a quasimorphism β and a field isomorphism φ. If the characteristic of the ground field of H is zero, then β is an isomorphism and α is standard. Otherwise there is a power Fr^{m}, for m ∈ ℕ, of the Frobenius automorphism Fr of the ground field of H, such that β ∘ Fr^{m} is an isogeny, and the map α = (β ∘ Fr^{m}) ∘ (Fr^{−m} ∘ φ) is standard.

Usually, the two most natural ways for a model theorist to view an algebraic group is as a pure group, or as a group, interpretable in a pure field. In this paper, we introduce an intermediate notion: the ACF -expansion of a pure group (Definition 3.2). Indeed, on the one hand, the pure group is not sufficiently rich for our subject (see Example 3.1 (2) and (3)), and on the other hand abstract isomorphisms do not preserve all the sets interpretable in the pure field. However, the ACF -expansion of a pure group is effective just if the ground field is isomorphic to the algebraic closure \overline{ℚ} of the field ℚ (see Lemma 3.10 for instance). Thus, most of our analysis is concerned with algebraic groups over \overline{ℚ}. In the final argument (§9), a classical model-theoretical argument will generalize the result over \overline{ℚ} to any algebraically closed field of characteristic zero.

As in [16], in order to analyze the abstract structure of algebraic groups over \overline{ℚ}, and more generally the one of algebraic groups over an algebraically closed field, we remark that their associated pure groups are of finite Morley rank. In fact, we introduce in §4 the notion of a ACF -group, including the one of a pure group and the one of the ACF -expansion of a pure group. Then a very substantial part of the article is to prove Theorem 7.7. This one says that, if G is a connected algebraic group over \overline{ℚ} then, in the ACF -expansion of pure group G, the quotient G/Z(G) is definably linear, in the sense of Definition 4.2. The core of the proof relies on the Burdges’ unipotence theory [5], its improvements in [9] and [10, §3.2], and the theory of solvable groups of finite Morley rank. Another key point is the notion of a pseudo-torus (Definition 2.4), a concept derived from the one of a decent torus of a group of finite Morley rank, and of a torus of an algebraic group. Indeed, they are the basis of the analysis of Ũ-groups, and thus they make possible the use of the Burdges’ unipotence for algebraic groups.

1.3. Abstract isomorphy versus elementary equivalence. It is noticeable that the final argument provides a slightly stronger result than Theorem 1.1. Indeed, Theorem 1.5 below is proven in §9 for affine groups, simultaneously with the main theorem. Thus, in this paper, if K denotes an algebraically closed field of characteristic zero, we describe the structure of all the affine algebraic groups over K elementarily equivalent to a fixed connected affine algebraic group G over K, and not only abstractly isomorphic to G.

Theorem 1.5. – Let G and H be two connected (nonnecessarily affine) algebraic groups over an algebraically closed field K of characteristic zero. Then G and H are abstractly isomorphic if and only if their pure groups are elementarily equivalent.

Furthermore, thanks to Lemma 1.6 below, we note that the groups are not necessarily affine in Theorem 1.5.

Lemma 1.6. – Let G be any connected algebraic group over an algebraically closed field K of characteristic zero. Then there are a connected affine algebraic group G_{A} over K such that G_{A} and G are abstractly isomorphic.

Moreover, we may choose G_{A} such that G_{A}/Z(G_{A}) and G/Z(G) are isomorphic as algebraic groups.

Proof – Let N be the smallest connected normal algebraic subgroup of G such that G/N is affine [17, §5, Corollary 3 p.431], and let A be the largest connected affine algebraic subgroup of G [17, §5, Theorem 16]. Then, by [17, §5, Corollary 1 p.433], we have N ≤ Z(G), and by [17, §5, Corollary 5 p.440], we have G = AN and N contains only a finite number of elements of any given finite order. So (A ∩ N)^{∘} and N are abelian and divisible, and (A ∩ N)^{∘} has a (nonnecessarily algebraic) divisible complement R in N.

By the choice of A, the group (A ∩ N)^{∘} is the largest connected affine algebraic subgroup of N, and N/(A ∩ N)^{∘} is an abelian variety by [17, §5, Theorem 16]. Consequently, for each integer n, the group R ≃ N/(A ∩ N)^{∘} has n^{2g} elements of order dividing n, where g denotes the dimension of N/(A ∩ N)^{∘}. This implies that R is abstractly isomorphic to the torus (K^{∗})^{2g}, so the affine algebraic group A × (K^{∗})^{2g} is abstractly isomorphic to A × R. But we have G = AN = AR and A ∩ R ≤ (A ∩ N) ∩ R is finite, so there is an abstract surjective homomorphism τ : A × (K^{∗})^{2g} → G such that τ(x, 1) = x for each x ∈ A, and with finite kernel E. Hence there is an abstract isomorphism σ from G to the connected affine algebraic group G_{A} := (A × (K^{∗})^{2g})/E, such that its restriction σ_{|A} : A → (A × {1})E/E is an isomorphism of algebraic groups. Thus, since we have N ≤ Z(G) and G = AN, we obtain G = AZ(G), and the map \overline{σ} : G/Z(G) → G_{A}/Z(G_{A}) induced by σ is an isomorphism of algebraic groups. □

1.4. What happens when the ground field is not fixed. If we consider two abstractly isomorphic connected affine algebraic groups G and H over distinct algebraically closed fields of characteristic zero, thanks to the following lemma and to Theorem 1.1, we can describe the structure of H from the one of G (Corollary 1.8). The proof of Lemma 1.7 is based on the fact that the pure group G has finite Morley rank.

Lemma 1.7. – Let G and H be two abstractly isomorphic connected (nonnecessarily affine) algebraic groups over algebraically closed fields K and L of characteristic zero. If G is either nonabelian or uncountable, there is a field isomorphism between K and L.

Otherwise, either G is trivial and there is no condition over L, or G is infinite and L is any countable algebraically closed field of characteristic zero.

Proof – We may assume that G is nontrivial. If G is nonabelian, its Borel subgroups, i.e. its maximal solvable connected closed subgroups, are nonabelian too. Let B be such a subgroup of G. Then B is a maximal solvable subgroup of G, so it is definable in the pure group G [4, Corollary 5.38]. Moreover B is connected in the pure field K, and since all the definable sets of the pure group G are definable in K too, B is connected in the pure group G. We consider the Fitting subgroup F (B) of B (see §2.1), which is a normal nilpotent definable subgroup of B. Then Fact 2.16 says that F (B)/Z(B) is a Ũ-group (see Definition 2.8). In particular, F (B)/Z(B) is connected in the pure group G. Since B is a solvable connected algebraic group, then B′ is unipotent, so F (B) contains B′, and B/F (B) is abelian. Thus, if F (B) = Z(B), then B′ is central in B and B is nilpotent, so we obtain B = F (B) = Z(B), contradicting that B is nonabelian. Consequently F (B)/Z(B) is infinite. But K has characteristic zero and B is algebraic over K, so B/Z(B) has no infinite subgroup of bounded exponent. Hence it follows from Definition 2.8 that there is a field K_{0} of characteristic zero, interpretable in the pure group G. By Fact 2.2, the fields K and K_{0} are definably isomorphic in the pure field K. Since G and H are abstractly isomorphic, there is a field L_{0} interpretable in the pure group H, with L_{0} isomorphic to K_{0}. In the same way, L_{0} and L are isomorphic, so there is a field isomorphism between K and L. Hence we may assume that G is abelian.

We note that G, K, H and L have the same cardinality. Moreover, K and L are elementarily equivalent by completeness of ACF_{0} [15, Corollary 3.2.3]. So, if G is uncountable, then K and L are two uncountable algebraically closed fields of characteristic zero with the same cardinality, hence they are isomorphic by the κ-categoricity of ACF_{0} [15, Proposition 2.2.5].

If G is abelian and countable, then K and L are necessarily countable. By Lemma 1.6, there is a connected affine algebraic group G_{A} over K such that G_{A} and G are abstractly isomorphic. Then we find two integers r and s with rs ≠ 0 such that the algebraic groups G_{A} and (K^{∗})^{r} × K_{+}^{s} are isomorphic. But, for any countable algebraically closed field F of characteristic zero, (K^{∗})^{r}×K_{+}^{s} is abstractly isomorphic to (F^{∗})^{r} × F_{+}^{s}, so we obtain the result. □

Then when the ground fields are not fixed, Theorem 1.1 becomes as follows.

Corollary 1.8. – Let G be a connected affine algebraic group over an algebraically closed field K of characteristic zero. Unless G is abelian and countable, there is a connected affine algebraic group D over K and a finite central subgroup F of D such that, for each connected affine algebraic group H over an algebraically closed field L of characteristic zero, if H is abstractly isomorphic to G, then there are a quasistandard automorphism α of D and an integer s such that H and α(D)/α(F) × L_{+}^{s} are isomorphic as algebraic groups.

Now, by Corollaries 1.2 and 1.3, we easily obtain some other general results concerning the abstract isomorphisms and standard isomorphisms.

Moreover, Theorem 1.1 and Lemma 1.7 allow to associate with any abstract group a very particular integer. Indeed, let G, D and F be as in Theorem 1.1. Since K has characteristic zero, the number of central subgroups of D isomorphic to F is finite, so there are finitely many algebraic groups of the form D/α(F) for a quasi-standard automorphism α of D. We denote by n(G) their number up to isomorphism of algebraic groups. Then Theorem 1.1 and Lemma 1.7 yield another important result to this paper.

Theorem 1.9. – For any abstract group G, there is an integer n(G) such that, for each algebraically closed field K of characteristic zero, and each integer d, up to isomorphism of algebraic groups, there are either zero, or n(G), connected affine algebraic groups H over K of dimension d and abstractly isomorphic to G.

Proof – We may assume that G is nontrivial, and that there is an algebraically closed field K of characteristic zero such that G is a connected affine algebraic group over K. Moreover, by Theorem 1.1 and Lemma 1.7, we may assume that G is abelian and countable. In this last case, there are two integer r and s with rs ≠ 0 such that the algebraic groups G and (K^{∗})^{r} ×K_{+}^{s} are isomorphic. Now let L be any algebraically closed field of characteristic zero such that there is a connected affine algebraic group over L, abstractly isomorphic to G. Then, up to isomorphism of algebraic groups, the connected affine algebraic groups over L abstractly isomorphic to G are of the form (L^{∗})^{r} × L_{+}^{t}, where t is any integer such that rt ≠ 0. This implies that, for each integer d, up to isomorphism of algebraic groups, there are at most one connected affine algebraic groups H over L of dimension d such that H is abstractly isomorphic to G. □

1.5. Plan of the article. After reminders of known facts in §2, we analyze the following very special group in §3:

G = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} .

For that, we introduce the notion of a ACF -expansion of a pure group (Definition 3.2), and we prove that the maximal tori of G are definable in the ACF -expansion of the pure group G (Proposition 3.11).

In §4, we study the unipotent groups over \overline{ℚ}, and we prove that, for each connected nilpotent algebraic group G over \overline{ℚ}, the quotient G/Z(G) is definably linear (in the sense of Definition 4.2) in the ACF -expansion of the pure group G (Theorem 4.3). In order to study the minimal configuration, we need Proposition 3.11. Moreover, a key point of this section is the Hoschild-Mostow Theorem (Fact 4.5), of which we provide a model-theoretic proof for the special case used here (Fact 4.6).

Section §5 is devoted to the study of the ACF -groups and ACF_{0}-groups (see the beginning of §4 for the definitions). These concepts encompass different ways of seeing an algebraic group for the model-theorist: as a pure group, as the ACF - expansion of a pure group, or as a group interpretable in a pure field for instance. Moreover, it appears that the notion of a definably linear group is not sufficient to analyze the ACF -groups (see Example 5.1). Therefore we introduce a slightly thinness concept: the definably affine groups (Definition 5.2). In particular, each ACF -group G has a largest connected definably affine subgroup (Corollary 5.10), and a smallest normal definable subgroup W (G) such that G/W (G) is definably affine (Corollary 5.17). It is noticeable that the proofs are somewhat more formal in this section, and most results are valid for the algebraic groups over any algebraically closed field, not just over \overline{ℚ}.

When the ground field of an algebraic group G is algebraically closed of characteristic zero, the definability of tori becomes a problem. We approach this problem using decent tori and pseudo-tori (Definition 2.4). However, even if G is a centerless connected algebraic group over \overline{ℚ}, the maximal tori of the pure group G may not be definable (Example 3.1 (2)). In §6, we prove that, when G is a connected solvable algebraic group over \overline{ℚ}, then T ∩ F (G) is central in G for each pseudo-torus T of the ACF -expansion of the pure group G (Theorem 6.3), where F (G) denotes the Fitting subgroup of G (see §2.1). Moreover, it is noticeable that, in order to study the minimal configuration, we need Proposition 3.11 again.

From this result, from the analysis of the unipotent groups in §4, and from the one of the ACF -groups in §5, we deduce Theorem 7.7, which is fundamental for us. Indeed, it says that when G is a connected algebraic group over \overline{ℚ}, then G/Z(G) is definably linear in the ACF -expansion of the pure group G. It is the base of the construction of the group D in the main theorem. Actually, this last group is built in §8. Its construction is rather natural, but above all, we have a serious problem with some central finite subgroups of it (see the proof of Theorem 8.14).

The transition from the algebraic groups over \overline{ℚ} to the algebraic groups over any algebraically closed field of characteristic zero is made in §9, essentially by a classical model-theoretic argument. Furthermore, it is noticeable that the end of the proof of the main result provides simultaneously the one of Theorem 1.5 in the affine case (the general case of Theorem 1.5 follows from the affine case and from Lemma 1.6).

2. Facts

The notations will be as in [4], which is also our main reference for groups of finite Morley rank. In this section we recall some definitions and known results.

Here is a list of the main notations and definitions used in this paper, with where they are defined.

d( ·) §2.1
·^{∘}  §2.1
F ( ·) §2.1
J( ·) §2.4
U_{K}( ·) Not. 2.7
Ũ( ·) Def. 2.8
V ( ·) Not. 4.10
A( ·) Cor. 5.10
W ( ·) Cor. 5.17
Φ( ·) §6
S( ·) §7
T ( ·) §7
Q( ·) §7
T ( ·) §8
D( ·) §8

Fitting subgroup  §2.1
S-minimal  §2.1
Decent torus  Def. 2.4
Pseudo-torus  Def. 2.4
Indecomposable group  §2.4
U_{K}-group  Def. 2.8
Ũ-group  Def. 2.8
𝒯 -expansion  Def. 3.2
ACF -group  §4
Definably linear group  Def. 4.2
Definably affine group  Def. 5.2
Carter subgroup  §6
Frattini subgroup  §6
Quasiunipotent radical  §7
Centrally indecomposable group Def. 8.3

2.1. Generalities. If X is a subset of a group G of finite Morley rank, then the definable hull of X, denoted by d(X), is the intersection of all the definable subgroups of G which contain X. By the descending chain condition on definable subgroups, this intersection is definable.

If H is a definable subgroup of a group G of finite Morley rank, its connected component is the smallest definable subgroup of finite index in H, and it is denoted by H^{∘}. The link between the connected component of a group of finite Morley rank and the identity component of an algebraic group is clarified by Lemma 4.1.

The Fitting subgroup F (G) of a group G is the subgroup generated by all the normal nilpotent subgroups of G. Nesin proved that the Fitting subgroup is definable and nilpotent in any group of finite Morley rank [4, Theorem 7.3].

A subgroup A of a group G of finite Morley rank is said to be S-minimal, where S is a subset of G, if A is infinite, definable, normalized by S and minimal for these conditions. It is noticeable that, when G is solvable and connected, then any G-minimal subgroup of G is abelian [4, Proposition 7.7].

2.2. Fields of finite Morley rank. The following result due to B. Zil’ber confers a central importance on fields of finite Morley rank. Moreover, we recall that, by a theorem of A. Macintyre [4, Theorem 8.1], an infinite field of finite Morley rank is always algebraically closed.

Fact 2.1. – [4, Theorem 9.1] Let G = A ⋊ H be a group of finite Morley rank where A and H are two infinite definable abelian subgroups, A is H-minimal and C_{H}(A) = 1. Then G interprets an algebraically closed field K such that A ≃ K_{+} definably, and H is definably isomorphic to a subgroup of K^{∗}.

Fact 2.2. – [16, Théorème 4.15] Let F be an algebraically closed field. Then, in the pure field F, every infinite definable field K is definably isomorphic to F.

Fact 2.3. – [16, Corollaire 3.3] Let K be a field of finite Morley rank of characteristic zero. Then K_{+} has no nontrivial proper definable subgroup.

2.3. Decent tori and pseudo-tori. In [7], G. Cherlin defines decent tori as an analogue to algebraic tori. In [12], we introduce pseudo-tori, as a more general notion, independant of torsion. Here we relate just the more general results used in this paper concerning pseudo-tori, but their proofs are very often similar to their analogues in [7] concerning decent tori. Moreover, pseudo-tori are the basis of Ũ-groups defined below.

Definition 2.4. – Let T be a radicable abelian group of finite Morley rank. • We say that T is a decent torus if T is the definable hull of its torsion.

• We say that T is a pseudo-torus if no definable quotient of T is definably isomorphic to the additive group K_{+} of an interpretable field K.

We note that any decent torus is a pseudo-torus, and any pseudo-torus is connected. In the following fact, we summarize the main properties of pseudo-tori.

Fact 2.5. – Let G be a group of finite Morley rank. Then, (i) [12, Theorem 1.7] the maximal pseudo-tori of G are conjugate; (ii) [12, Proposition 2.7] for any pseudo-torus T of G, N_{G}(T)^{∘} centralizes T; (iii) [12, Corollaries 2.8 and 2.9] F (G) has a unique maximal pseudo-torus, and

this one is central in G^{∘}; (iν) [12, Corollary 2.13] if N is a normal definable subgroup of G, the maximal

pseudo-tori of G/N are the images of the maximal pseudo-tori of G.

2.4. Unipotence. In [5], J. Burdges introduced some analogues of the algebraic unipotence for groups of finite Morley rank. In [9, 10], we continued the analysis of these concepts. Here are considered Ũ-groups introduced in [10]. This notion heavily depends on pseudo-tori [12]. We summarize the general results needed, and we refer to [5, 9, 10] for a more complete introduction to these unipotence notions.

To obtain a notion analogous to the unipotence in algebraic groups, Burdges [5] first introduces the notion of indecomposable group.

An abelian connected group A of finite Morley rank is indecomposable if it is not the sum of two proper definable subgroups. If A ≠ 1, then A has a unique maximal proper definable connected subgroup J(A), and if A = 1, let J(1) = 1.

The first result is nontrivial and is consequence of Fact 2.5 (iν) above.

Fact 2.6. – [12, Lemma 2.2] Let G be a group of finite Morley rank and H a definable normal subgroup of G. If \overline{B} is a radicable indecomposable definable abelian subgroup of G/H, then there is an indecomposable definable abelian subgroup A of G such that \overline{B} = AH/H.

In particular, Fact 2.6 says that any (nonnecessarily abelian) radicable group of finite Morley rank is generated by its indecomposable definable abelian subgroups.

We recall the definitions and properties of Ũ-groups.

Notation 2.7. – For any group G of finite Morley rank and any interpretable field K of characteristic zero, we denote by U_{K}(G) the subgroup of G generated by its indecomposable definable abelian subgroups A such that A/J(A) is definably isomorphic to K_{+}.

Definition 2.8. –

• A group G of finite Morley rank is said to be a U_{K}-group, where K is an interpretable field of characteristic zero, if G = U_{K}(G). We say that a U_{K}-group G is homogeneous if each definable connected subgroup of G is a U_{K}-subgroup.

• For every group G of finite Morley rank, we denote by Ũ(G) the subgroup of G generated by its normal homogeneous U_{K}-subgroups, for the interpretable fields K of characteristic zero, and by its normal definable connected subgroups of bounded exponent.

• A Ũ-group is a group G of finite Morley rank such that G = Ũ(G).

Remark 2.9. –

• By Fact 2.3, any radicable indecomposable definable abelian subgroup is either a pseudo-torus or a U_{K}-group for an interpretable field K of characteristic 0.

• By Fact 2.5 (iν) and Fact 2.12 below, in any Ũ-group, each pseudo-torus is trivial.

The below facts 2.10-2.16 comes from [10], but their proofs are not given in [10]. Indeed, they are mainly obtained by using [12] in place of [7], and their proofs are similar to the ones of [5, 6, 9].

Fact 2.10. – (see [10, Fact 3.10] and [6, Theorem 3.4]) Let G be a radicable nilpotent group of finite Morley rank, and let T be its maximal pseudo-torus. Then G interprets some fields K_{1}, ⋯, K_{n} of characteristic zero such that

G = T ∗ U_{K_{1}}(G) ∗ U_{K_{2}}(G) ∗ ⋯ ∗ U_{K_{n}}(G),

where ∗ denotes the central product.

Fact 2.11. – (see [10, Fact 3.11] and [9, Theorem 4.11]) Let G be a connected group of finite Morley rank, and K an interpretable field of characteristic zero. Assume that G acts definably by conjugation on H, a nilpotent U_{K}-group. Then [G, H] is a homogeneous U_{K}-subgroup.

Fact 2.12. – (see [10, Fact 3.13] and [9, Theorem 5.4]) Let G be a nilpotent Ũ-group. Then G interprets some algebraically closed fields K_{1}, ⋯, K_{n} of characteristic zero such that the following decomposition holds:

G = B × U_{K_{1}}(G) × U_{K_{2}}(G) × ⋯ × U_{K_{n}}(G)

where B is a definable connected characteristic subgroup of bounded exponent, and U_{K_{s}}(G) a homogeneous U_{K_{s}}-subgroup (for s ∈ {1, 2, …, n}).

Fact 2.13. – (see [10, Fact 3.15] and [9, Proposition 5.7]) Let G be a torsion-free group of finite Morley rank containing no nontrivial pseudo-torus. Then G is a Ũ-group if and only if, for each interpretable field K of characteristic zero, U_{K}(G) is a homogeneous U_{K}-subgroup.

Fact 2.14. – (see [10, Fact 3.16] and [9, Corollary 5.8])

• Every definable quotient of a Ũ-group is a Ũ-group.

• Every definable connected subgroup of a Ũ-group is a Ũ-group.

Fact 2.15. – (see [10, Fact 3.18] and [5, Lemma 2.11]) Let G be a group of finite Morley rank, U and V be two definable subgroups with V normal in G, and K be an interpretable field of characteristic zero. Then U_{K}(UV/V) = U_{K}(U)V/V.

Fact 2.16. – (see [10, Fact 3.25] and [9, Results 5.8, 6.12 and 6.20]) Let G be a solvable connected group of finite Morley rank. Then F (G)/Z(G) is a Ũ-group.

3. The ACF -expansion of a pure group

First we focus on a very particular algebraic group. We consider the algebraic closure \overline{ℚ} of ℚ and the following subgroup of GL(3, \overline{ℚ}):

\{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} .

This group will play a key role for us. However, the following examples show that the notion of a pure group is not sufficiently rich to analyze it. In order to remedy to this problem, we will define the ACF -expansion of a pure group (Definition 3.2).

Example 3.1. –

(1) Let ℚ̃ denote a proper elementary extension of the pure field \overline{ℚ}. We consider the following solvable centerless connected algebraic group

G̃ = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ ℚ̃^{∗}, (a, u, v) ∈ ℚ̃^{3}\} .

Then G̃ has an abstract automorphism α such that α(T) is not Zariski closed for any maximal torus T of G̃. Thus the maximal tori of G̃ are not definable in the pure group G̃.

Furthermore, the latter implies that the automorphism α is not continuous for the Zariski topology of G̃. In particular, α is not standard.

Indeed, since ℚ̃ is a proper extension of \overline{ℚ}, it has a nonzero derivation δ. Then we consider the map α : G̃ → G̃ defined by

α ( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) = ( \begin{matrix}t & a + δ(t) & u + δ(v) \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) ,

for each t ∈ ℚ̃^{∗} and each (a, u, v) ∈ ℚ̃^{3}. It is an abstract automorphism of G̃, so for each subset X of G̃, the set X is definable in the pure group G̃ if and only if α(X) is definable in the pure group G̃.

We verify α(T) is not Zariski closed for any maximal torus T of G̃. By conjugacy of maximal tori in G̃, we may assume that T is the diagonal subgroup of G̃. Then we have

T ∩ α(T) = {diag(t, t, 1) | t ∈ ℚ̃^{∗}, δ(t) = 0},

so neither T ∩ α(T), nor α(T) is Zariski closed.

(2) Now we consider the pure group

G = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} .

By Lemma 3.10 below, all of its abstract automorphisms are standard. Nevertheless, we cannot apply Fact 1.4 to G, since G does not verify its hypotheses.

Indeed, otherwise, in the pure group G, there would be an interpretable field K and a definable isomorphism ν from G to an algebraic group H over K. In particular, K and ν would be definable in the pure field \overline{ℚ}, so ν(T) would be a maximal torus of H for each maximal torus T of G. Then, by the definability of K in G, the subgroups ν(T) and T would be definable in G. But ℚ̃ is a proper elementary extension of the pure field \overline{ℚ}, so the pure group G̃ is an elementary extension of the pure group G, and the extension T̃ of T to G̃ is a maximal torus of G̃. Hence T̃ would be definable in the pure group G̃, contradicting the previous example.

We note that the argument above shows that, in the pure group G,

• there is no definable isomorphism from G to an algebraic group over an interpretable field.

• the maximal tori of G are not definable.

(3) The algebraic groups G̃ and G above are solvable of class two, centerless and connected. However, we obtain the same conclusions by considering the algebraic groups H̃ and H below instead of G̃ and G respectively, whereas H̃ and H are perfect, centerless and connected.

Indeed, consider the following perfect connected affine algebraic group

R = \{( \begin{matrix}a & b & r & s \\ c & d & t & u \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{matrix} ) | (a, b, c, d, r, s, t, u) ∈ \overline{ℚ}^{8}, ad − bc = 1\} ,

and the group H := R/Z(R). Then H is a perfect centreless connected algebraic group over \overline{ℚ}.

Now let ℚ̃ be as above, and let R̃ and H̃ be the elementary extensions of R and H respectively to ℚ̃. Then H̃ is a perfect centreless connected algebraic group over ℚ̃.

If we consider a nonzero derivation δ of ℚ̃, and the abstract automorphism β of R̃ defined by

β ( \begin{matrix}a & b & r & s \\ c & d & t & u \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{matrix} ) = ( \begin{matrix}a & b & r + δ(a) & s + δ(b) \\ c & d & t + δ(c) & u + δ(d) \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{matrix} ) ,

for each (a, b, c, d, r, s, t, u) ∈ ℚ̃^{8} satisfying ad − bc = 1. Then β induces an automorphism \overline{β} of H̃. Arguing as for the automorphism α above, we show that \overline{β} is not continuous for the Zariski topology of G̃. In particular, \overline{β} is not standard.

Thus, as in (2), H does not verify the hypotheses of Fact 1.4. Even so, since H is not a direct product of two proper subgroups, it follows from Fact 3.6, Lemma 5.6 and Theorem 7.7 below, that all the abstract automorphisms of H are standard.

The three examples in 3.1 above show that the pure group structure of a connected affine algebraic group may not be as rich as its definable geometric structure, even when the group is centerless. This motivates the following definition. We note that, in this paper, the word definable signifies definable with parameters.

Definition 3.2. – Let 𝒯 be a theory in a language ℒ, and let ℳ = (M; ⋯) be a structure in a language ℒ′. We assume that ℳ is interpretable in a model of 𝒯 .

• Let n ∈ ℕ and A ⊆ M^{n}. Then A is said to be 𝒯 -definable if, for each ℒ′- isomorphism f from ℳ to another ℒ′-structure ℳ_{1}, and for each model 𝒩 of 𝒯 interpreting ℳ_{1}, the set f(A) is definable in 𝒩 .

• For each n ∈ ℕ, let 𝒜_{n} be the family of the 𝒯 -definable subsets of M^{n} and, for each A ∈ 𝒜_{n}, let ℛ_{A} be a symbol of an n-ary relation. We consider 𝒜 = ∪_{n∈ℕ} 𝒜_{n} and ℒ^{∗} = ℒ′ ∪ (⋃_{A∈𝒜} ℛ_{A}).

• The 𝒯 -expansion of ℳ = (M; ⋯) is the ℒ^{∗}-expansion ℳ^{∗} = (M; ⋯) of ℳ, where ℛ_{A}^{ℳ^{∗}} = A for each A ∈ 𝒜.

By the following results, if a structure ℳ is interpretable in a model of a theory 𝒯 , then for any set X and any automorphism α of the structure ℳ, if X is definable in the 𝒯 -expansion ℳ^{∗}, its image α(X) is definable in ℳ^{∗} too (Corollary 3.5). Thus, in our context where we consider group isomorphisms, and since ℳ^{∗} seems to be richer than ℳ, it appears more interesting to work in ℳ^{∗} than in ℳ.

Lemma 3.3. – Let 𝒯 be a theory in a language ℒ, and let ℳ = (M; ⋯) be a structure interpretable in a model of 𝒯 . Then, for each n ∈ ℕ and each A ⊆ M^{n}, the set A is definable in the 𝒯 -expansion ℳ^{∗} of ℳ if and only if it is 𝒯 -definable.

Proof – We may assume that A definable in ℳ^{∗}, and we have just to prove that A is 𝒯 -definable. We assume toward a contradiction that A is not 𝒯 -definable. In particular, it is not definable in ℳ, its complement is not 𝒯 -definable, and it is neither a union nor an intersection of two 𝒯 -definable sets. Moreover, it is not of the form M × X for a 𝒯 -definable set X, and there is no 𝒯 -definable subset X of M^{n+1} such that A = π(X) where π : M^{n+1} → M^{n} denotes the projection map defined for any (x_{1}, …, x_{n+1}) ∈ M^{n+1} by π(x_{1}, …, x_{n+1}) = (x_{1}, …, x_{n}). Hence [15, Proposition 1.3.4] says that we may assume that there exists m ∈ ℕ, a 𝒯 - definable subset X of M^{n+m} and b ∈ M^{m} such that A = {a ∈ M^{n} | (a, b) ∈ X}. But again in this last case, it follows from the definition that A is 𝒯 -definable, as claimed. □

Corollary 3.4. – Let 𝒯 be a theory in a language ℒ, and let ℳ = (M; ⋯) and ℳ′ = (M′; ⋯) be two ℒ′-structures interpretable in some models of 𝒯 , where ℒ′ is another language. Let f : ℳ → ℳ′ be an ℒ′-isomorphism, and let ℳ^{∗} and ℳ^{′∗} be the 𝒯 -expansions of ℳ and ℳ′ respectively. Then, for each n ∈ ℕ and each A ⊆ M^{n} definable in ℳ^{∗}, the set f(A) is definable in ℳ^{′∗}.

Proof – Let g be an ℒ′-isomorphism from ℳ′ to another ℒ′-structure ℳ_{1}. We assume that ℳ_{1} is interpretable in a model 𝒩 of 𝒯 . Then g ∘ f is an ℒ′- isomorphism from ℳ to ℳ_{1}. Now, since A is 𝒯 -definable by Lemma 3.3, the set g(f(A)) is definable in 𝒩 . This proves that f(A) is definable in ℳ^{′∗}. □

Corollary 3.5. – Let 𝒯 be a theory in a language ℒ, and let ℳ be a structure interpretable in a model of 𝒯 . If X is any definable set in the 𝒯 -expansion ℳ^{∗}, then its image by any automorphism of the structure ℳ is definable in ℳ^{∗} too.

It is noticeable that, by the previous result and the remark after Fact 1.4, the proof of [16, Corollaire 4.17] provides the very general following fact, which is fundamental for us.

Fact 3.6. – [16, Proof of Corollaire 4.17] Let α be an abstract isomorphism between two algebraic groups G and H over algebraically closed fields. If, in the ACF - expansion of the pure group G, there are an interpretable field K and a definable isomorphism from a definable section U/V to an algebraic group over K, then the isomorphism α_{|U/V} : U/V → α(U)/α(V) is standard.

Furthermore, we notice that the definition of the 𝒯 -expansion of a structure ℳ for a theory 𝒯 is rather robust.

Lemma 3.7. – Let 𝒯 be a theory in a language ℒ, and let ℳ be a structure in a language ℒ′. We assume that ℳ is interpretable in a model of 𝒯 . Let ℳ_{0} be another structure, interpretable in the 𝒯 -expansion of ℳ. If A is a set, definable in the 𝒯 -expansion of ℳ_{0}, then A is definable in the 𝒯 -expansion of ℳ too.

Proof – Let M (resp. M_{0}) be the domain of ℳ (resp. ℳ_{0}). Let A be a subset of M_{0}^{n} for n ∈ ℕ, with A definable in the 𝒯 -expansion of ℳ_{0}. Since ℳ_{0} is interpretable in the 𝒯 -expansion of ℳ, there are m ∈ ℕ, a subset B of M^{m} and an equivalence relation R over B, with B and R definable in the 𝒯 -expansion of ℳ, such that M_{0} = B/R. We denote by A_{M} the preimage of A in (M^{m})^{n}. Then we have just to prove that A_{M} is definable in the 𝒯 -expansion of ℳ.

Let f be an ℒ′-isomorphism from ℳ to another ℒ′-structure ℳ_{1}, and let 𝒩 be a model of 𝒯 interpreting ℳ_{1}. Then f induces an isomorphism \overline{f} from the structure ℳ_{0} to a structure ℳ_{B} with domain f(B)/f(R). Moreover, since the structure ℳ_{0} is interpretable in the 𝒯 -expansion of ℳ, then ℳ_{B} is interpretable in 𝒩 . Now, since \overline{f} is an isomorphism from ℳ_{0} to ℳ_{B}, then \overline{f}(A) is definable in 𝒩 . Thus f(A_{M}) is definable in 𝒩 , and A is definable in the 𝒯 -expansion of ℳ. □

Nevertheless, by the following remark, this notion is not preserved by elementary extensions.

Remark 3.8. – Let G̃ and G be as in Example 3.1 (1) and (2) respectively. Then the pure group G̃ is an elementary extension of G. Moreover, if T denotes any maximal torus of G, then its extension T̃ to G̃ is a maximal torus of G̃. But T is definable in the ACF -expansion of the pure group G by Lemma 3.11 below, while that T̃ is not definable in the ACF -expansion of the pure group G̃. Indeed, if α is as in Example 3.1 (1), then α(T̃) is not Zariski closed (see Example 3.1 (1)), so neither α(T̃), nor T̃ by Corollary 3.5, is definable in the ACF -expansion of the pure group G̃.

From now on, we study the abstract automorphisms of some algebraic groups.

Lemma 3.9. – Let K be an algebraically closed field of characteristic zero. Then each abstract automorphism α of the group

G = \{( \begin{matrix}t & a \\ 0 & 1 \end{matrix} ) | t ∈ K^{∗}, a ∈ K\}

has the form α = β ∘ μ for μ a field automorphism of K, and β an inner automorphism of G.

Proof – Let u ∈ K \ \{−1, 0, 1\}, x = ( \begin{matrix}1 & u \\ 0 & 1 \end{matrix} ) and y = ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ). Then we

have G = A ⋊ H for A = C_{G}(x) and H = C_{G}(y). In particular, A and H are definable in the pure group G, and Fact 2.1 says that G interprets an algebraically closed field L such that A is definably isomorphic to L_{+} and such that H is definably isomorphic to a subgroup of L^{∗}, acting by multiplication on A. But, in the pure field K, the fields K and L are definably isomorphic by Fact 2.2, so L^{∗} has no proper infinite definable subgroup, and we obtain H ≃ L^{∗} definably. Hence the

groups G = A ⋊ H and L_{+} ⋊ L^{∗} ≃ \{( \begin{matrix}t & a \\ 0 & 1 \end{matrix} ) | t ∈ L^{∗}, a ∈ L\} are isomorphic,

definably in the pure group G. Thus, by Fact 3.6 (or Fact 1.4), we have α = β ∘ μ for μ a field automorphism of K, and β an algebraic automorphism of G.

We show that β is an inner automorphism of G. Since H is abelian and selfnormalizing in G, it is a Cartan subgroup, and β(H) is a Cartan subgroup too. Hence H and β(H) are conjugate, and we may assume that β stabilizes H. The action of H on A \ {0} is transitive, so there exists h ∈ H such that β(x) = x^{h}, and we may assume that β fixes x. Therefore, since A ≃ K_{+} is torsion-free of dimension one, and since x is nontrivial, β centralizes A. Thus we have β(x^{y}) = x^{y}. But if β inverts H, then we have β(x^{y}) = x^{y^{−1}}, and y^{2} centralizes x, contradicting u ∉ {−1, 0, 1}. Hence β does not invert H. Since H is a torus of dimension one, its algebraic automorphisms are the identity map and the inversion map, so we obtain β(h) = h for each h ∈ H, and β centralizes G = AH, proving the result. □

Lemma 3.10. – Let K be an algebraically closed field of characteristic zero. Then each abstract automorphism α of the group

G = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ K^{∗}, (a, u, v) ∈ K^{3}\}

has the form α = β_{r,δ} ∘ γ ∘ μ for μ a field automorphism of K, γ an inner automorphism of G and β_{r,δ} an abstract automorphism of G such that, for each t ∈ K^{∗} and each (a, u, v) ∈ K^{3}, we have

β_{r,δ}(( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}t & ra + δ(t) & ru + δ(v) \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) ,

where δ is a derivation of K and r an element of K.

Proof – The unipotent part of G is its Fitting subgroup, so it is characteristic

and definable in the pure group G, and its center Z = \{( \begin{matrix}1 & 0 & u \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) | u ∈ K\}

is characteristic and definable too. Also the preimage in G of the center of G/Z

is U = \{( \begin{matrix}1 & a & u \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) | (a, u) ∈ K^{2}\} , so U is characteristic and definable in the

pure group G. Moreover, there is an isomorphism of algebraic groups between the

quotient group G/U and \{( \begin{matrix}t & a \\ 0 & 1 \end{matrix} ) | t ∈ K^{∗}, a ∈ K\}, and Lemma 3.9 shows that

the automorphism \overline{α} of G/U induced by α has the form \overline{α} = \overline{γ} ∘ μ for μ a field automorphism of K, and \overline{γ} an inner automorphism of G/U. Hence we may assume that α centralizes G/U, and we have just to prove that α has the form α = β_{r,δ} ∘ γ for γ an inner automorphism of G induced by an element of U.

We consider

C = \{( \begin{matrix}t & a & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) | t ∈ K^{∗}, a ∈ K\} .

This subgroup is abelian and self-normalizing in G, so it is definable in the pure group G, and it is a Cartan subgroup of G. In particular, C and α(C) are two Cartan subgroups of V = UC = Z ⋊ C, and they are conjugate by an element of Z ≤ U. Hence we may assume that α normalizes C and we have just to prove that α has the form α = β_{r,δ} ∘ γ for γ an inner automorphism of G induced by an element of C ∩ U.

The center of V is C∩U, and there is an isomorphism of algebraic groups between V/(C ∩ U) and

\{( \begin{matrix}t & a \\ 0 & 1 \end{matrix} ) | t ∈ K^{∗}, a ∈ K\} .

Then Lemma 3.9 shows that the automorphism \overline{α} of V/(C ∩ U) induced by α has the form \overline{α} = \overline{γ} ∘ μ for μ a field automorphism of K, and \overline{γ} an inner automorphism of V/(C ∩ U). Since U/(C ∩ U) is the unique nontrivial normal abelian subgroup of V/(C ∩ U), it is characteristic in V/(C ∩ U), and \overline{γ} and μ normalize it. But V/U is abelian, so \overline{γ} centralizes it, and since α centralizes G/U, the field automorphism μ centralizes V/U too. Furthermore, V/U is a torus of dimension one, hence μ is a field automorphism centralizing K^{∗} and, consequently, μ is the identity map. Now \overline{α} = \overline{γ} is an inner automorphism of V/(C ∩ U), and there exists r ∈ K such that,

for each t ∈ K^{∗} and each (a, u) ∈ K^{2}, there is b ∈ K such that α(( \begin{matrix}t & a & u \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} )) =

( \begin{matrix}t & b & ru \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ). Moreover, since α normalizes C, the element b depends just on t

and a, and there is a map ν_{0} : K^{∗} × K → K such that for each t ∈ K^{∗} and each a ∈ K, we have:

α(( \begin{matrix}t & a & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}t & ν_{0}(t, a) & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) .

In addition, we have G′ = \{( \begin{matrix}1 & 0 & u \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} ) | (u, v) ∈ K^{2}\}, and α centralizes G/U,

so there is a map μ_{0} : K × K → K such that, for each (u, v) ∈ K^{2}, we have

α(( \begin{matrix}1 & 0 & u \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & 0 & μ_{0}(u, v) \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} ) ,

and μ_{0} satisfies μ_{0}(u, 0) = ru for each u ∈ K. Thus, for each t ∈ K^{∗} and each (a, u, v) ∈ K^{3}, we have

\begin{aligned}α(( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} )) & = & α(( \begin{matrix}1 & 0 & u \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} )) · α(( \begin{matrix}t & a & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} )) \\ & = & ( \begin{matrix}t & ν_{0}(t, a) & μ_{0}(u, v) \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) . \end{aligned}

By considering t ∈ K^{∗}, (a, u, v) ∈ K^{3} and the equality α(xy) = α(x)α(y) for

x = ( \begin{matrix}t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) (resp. x = ( \begin{matrix}1 & 0 & u \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )) and y = ( \begin{matrix}1 & t^{−1}a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) (resp.

y = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} )), we obtain ν_{0}(t, a) = ν_{0}(t, 0) + tν_{0}(1, t^{−1}a) (resp. μ_{0}(u, v) =

ru+μ_{0}(0, v)). Moreover, by considering a ∈ K, and the equality α(xy) = α(x)α(y)

for x = ( \begin{matrix}1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and y = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ), we obtain μ_{0}(a, 1) = μ_{0}(0, 1) +

ν_{0}(1, a). Since we have μ_{0}(a, 1) = ra + μ_{0}(0, 1), this yields ν_{0}(1, a) = ra for each a ∈ K, and ν_{0}(t, a) = ν_{0}(t, 0) + ra for each (t, a) ∈ K^{∗} × K.

Now we consider the maps ν : K → K and μ : K → K defined by ν(0) = 0 and, for each t ∈ K^{∗} and each v ∈ K, ν(t) = ν_{0}(t, 0) and μ(v) = μ_{0}(0, v). Then, for each t ∈ K^{∗}, (a, u, v) ∈ K^{3}, we have

α(( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}t & ra + ν(t) & ru + μ(v) \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) .

We note that we have ν(1) = 0 and μ(0) = 0. Moreover, if γ denotes the conjugation

by ( \begin{matrix}1 & −r^{−1}μ(1) & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ∈ C ∩ U, we obtain

(α ∘ γ^{−1})(( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) ,

so we may assume μ(1) = 0, and we have just to prove that α = β_{r,δ} for a derivation δ of K.

We consider t ∈ K^{∗} and the equality α(xy) = α(x)α(y) for x = ( \begin{matrix}t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} )

and y = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ). Since μ(1) = 0, we obtain ν(t) = μ(t), and since ν(0) = 0 = μ(0), we find ν = μ. From now on, we have just to prove that ν is a derivation of K.

Firstly, for each (v, v′) ∈ K^{2}, by considering the equality α(xy) = α(x)α(y) for

x = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} ) and y = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & v′ \\ 0 & 0 & 1 \end{matrix} ), we find ν(v + v′) = ν(v) + ν(v′).

Secondly, for each (t, t′) ∈ (K^{∗})^{2}, by considering the equality α(xy) = α(x)α(y)

for x = ( \begin{matrix}t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) and y = ( \begin{matrix}t′ & 0 & 0 \\ 0 & t′ & 0 \\ 0 & 0 & 1 \end{matrix} ), we find ν(tt′) = tν(t′) + t′ν(t).

Since we have ν(tt′) = 0 = tν(t′) + t′ν(t) as soon as t = 0 or t′ = 0, the map ν is indeed a derivation, and this finishes the proof. □

Proposition 3.11. – We consider the group

G = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} .

Then its maximal tori are definable in the ACF -expansion of the pure group G.

Proof – We consider another algebraically closed field L, a pure group G_{L} interpretable in L, and an abstract group isomorphism μ : G → G_{L}. We remark that G has no proper subgroup of finite index, so the groups G and G_{L} are connected as pure groups as well as algebraic groups. We consider the subgroup H of G formed

by the matrices ( \begin{matrix}1 & 0 & u \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) for u ∈ \overline{ℚ}. Then H is a normal definable subgroup

of the pure group G, since it is the centralizer in G of the subgroup M formed by the

matrices ( \begin{matrix}1 & a & u \\ 0 & 1 & v \\ 0 & 0 & 1 \end{matrix} ) for (a, u, v) ∈ \overline{ℚ}^{3}. We may consider the groups H ⋊ G/M

and G′/H ⋊ G/M where G/M acts by conjugation on H and G′/H: they are definable in the pure group G because M = C_{G}(H) is definable. Moreover, the actions are transitive on H \ {0} and on G′/H \ {0} respectively, and faithfull, so in the pure group G, there are two interpretable fields K_{1} and K_{2} such that H (resp. G′/H) is definably isomorphic to (K_{1})_{+} (resp. (K_{2})_{+}) (Fact 2.1). We note also that, in the pure field \overline{ℚ}, the fields K_{1} and K_{2} are definably isomorphic to \overline{ℚ} (Fact 2.2).

Now there are two fields L_{1} and L_{2}, isomorphic to K_{1} and K_{2} respectively, and interpretable in the pure group G_{L}, such that μ(H) (resp. μ(G′)/μ(H)) is definably isomorphic to (L_{1})_{+} (resp. (L_{2})_{+}). So Fact 2.2 says that, in the pure field L, the fields L_{1} and L_{2} are definably isomorphic to L. In particular, this proves that the fields \overline{ℚ} and L are isomorphic. Moreover, μ(G′) is an abelian group and, in the pure field L, the groups μ(H) and μ(G′)/μ(H) are definably isomorphic to L_{+}, so μ(G′) is isomorphic to L_{+} × L_{+}, definably in the pure field L.

We consider a maximal torus T of G_{L} and its centralizer C. Then we have G_{L} = CG′_{L}, so Z(C) ∩ G′_{L} is central in G_{L}. Since G_{L} is centerless, Z(C) ∩ G′_{L} is trivial, and since C is nilpotent, we obtain G_{L} = G′_{L} ⋊ C. Since G′_{L} is a maximal abelian subgroup of G_{L}, the group C ≃ G_{L}/G′_{L} acts faithfully by conjugation on G′_{L}, and C is isomorphic to an abelian definable subgroup of GL(2, L), definably in the pure field L. Consequently, in the pure field L, there is a definable isomorphism f from G_{L} to a subgroup of

\{( \begin{matrix}t_{1} & a & u \\ b & t_{2} & v \\ 0 & 0 & 1 \end{matrix} ) | (t_{1}, t_{2}, a, b, u, v) ∈ L^{6}, t_{1}t_{2} − ab ≠ 0\} ,

such that • f(μ(H)) is formed by the matrices where t_{1} = t_{2} = 1 and a = b = v = 0;

• f(G′_{L}) = f(μ(G′)) by the matrices where t_{1} = t_{2} = 1 and a = b = 0;

• f(C) by some matrices where u = v = 0. In addition, since μ(H) is normal in G_{L}, we obtain b = 0 for each element of f(G_{L}). Also, μ(M)/G′_{L} is an infinite abelian torsion-free subgroup of G_{L}/G′_{L}, definable in the pure group G_{L}, so C ≃ G_{L}/G′_{L} contains a closed infinite torsion-free subgroup, that is a nontrivial unipotent subgroup. Since the unipotent part of

\{( \begin{matrix}t_{1} & a & 0 \\ 0 & t_{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) | (t_{1}, t_{2}) ∈ (L^{∗})^{2}, a ∈ L\}

has dimension one, it is contained in f(C), and we obtain

\{( \begin{matrix}1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) | a ∈ L\} ⊆ f(C) ⊆ \{( \begin{matrix}t_{1} & a & 0 \\ 0 & t_{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) | (t_{1}, t_{2}) ∈ (L^{∗})^{2}, a ∈ L\} .

Since f(C) is abelian, has torsion, and is connected because it is isomorphic to G/G′ which is radicable, this forces f(C) to be equal to

\{( \begin{matrix}t & a & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) | t ∈ L^{∗}, a ∈ L\} ,

and f(G_{L}) is equal to

\{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ L^{∗}, (a, u, v) ∈ L^{3}\} .

Since the fields \overline{ℚ} and L are isomorphic, we may consider a field isomorphism δ : L → \overline{ℚ}. Then δ ∘ f ∘ μ is an abstract automorphism of G, and Lemma 3.10 shows that δ ∘ f ∘ μ is a standard automorphism. So the image of any maximal torus of G by δ ∘ f ∘ μ is a maximal torus of G. But δ is a field isomorphism, and the isomorphism f is definable in the pure field L, so the preimage by δ ∘ f of any maximal torus of G is a maximal torus of G_{L}. Hence the image of any maximal torus of G by μ is a maximal torus of G_{L}. In particular, the image of any maximal torus of G is definable in L. Thus, the maximal tori of G are definable in the ACF -expansion of the pure group G. □

4. Unipotent groups over \overline{ℚ}

In this paper, we study the structures M = (G, · ,^{−1} , 1, ⋯) interpretable in a pure algebraically closed field K, where (G, · ,^{−1} , 1) is a group. Such a structure is said to be an ACF -group and, if p denotes the characteristic of K, M is said to be an ACF_{p}-group.

In particular, any ACF -group is a group of finite Morley rank, and each interpretable group in an ACF -group is an ACF -group too. Also, in the pure field K, any ACF -group G is constructible [16, §4.a], and there is an isomorphism definable in the pure field K, between G and an algebraic group over K [16, Théorème 4.13].

The connected component G^{∘} of an ACF -group G as a group of finite Morley rank is not necessarily equal to the one G^{∘∘} of G as a constructible group. Indeed, if p is a prime integer, the pure group K_{+} ⊕ 𝔽_{p} is connected as a group of finite Morley rank, and it is not connected as a constructible group. However, by Lemma 4.1, the situation is different for ACF_{0}-groups. Then, in the rest of this paper, for any ACF_{0}-group G, we will denote by G^{∘} its connected component as a group of finite Morley rank, as well as a constructible group.

Lemma 4.1. – For any ACF_{0}-group G, we have G^{∘} = G^{∘∘}.

Proof – Since G^{∘} contains G^{∘∘}, we may assume G = G^{∘}. We proceed by induction on the Morley rank of G. Let A be a G-normal definable connected subgroup of G^{∘∘}. If A ≠ 1, the induction hypothesis applied with G/A provides the result. Hence we may assume that G^{∘∘} contains no nontrivial G-normal definable connected subgroup. Then the Zil’ber’s Indecomposability Theorem [4, Results 5.26 and 5.29] yields [G^{∘∘}, G] = 1. Now Z(G) contains G^{∘∘} and it is a definable subgroup of finite index in the connected group G, so G = Z(G) is abelian.

Let n be the index of G^{∘∘} in G. Then {g^{n} | g ∈ G} is a connected definable subgroup of G, and it is contained in G^{∘∘}. so it is trivial and G has bounded exponent m. Since G is an ACF_{0}-group, it is constructible, and since it has exponent m, it is finite. Finally, G is trivial since it is a finite connected group of finite Morley rank. □

The following notion is central for our paper.

Definition 4.2. – A group of finite Morley rank is said to be definably linear (over finitely many interpretable fields K_{1}, …, K_{n}), if G has an interpretable faithful linear representation over the ring K_{1} ⊕ ⋯ ⊕ K_{n}.

In other words, G definably embeds in H_{1} × ⋯ × H_{n}, where H_{i} is an affine algebraic group over K_{i} for each i = 1, …, n.

In this section, we prove our first theorem. Its proof is based on the previous section and on a Hochschild-Mostow’s theorem (Facts 4.5 and 4.6).

Theorem 4.3. – Let G be a connected nilpotent algebraic group over \overline{ℚ}. Then, in the ACF -expansion of the pure group G, the quotient G/Z(G) is definably linear.

Remark 4.4. – This result fails when G is a nilpotent algebraic group over an algebraically closed field K of characteristic zero not isomorphic to \overline{ℚ}.

Indeed, consider a nonzero derivation δ of K, the group

G = \{( \begin{matrix}1 & a & b & x \\ 0 & 1 & a & y \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} ) | (a, b, x, y, z) ∈ K^{5}\} ,

and the group automorphism α : G → G defined for each (a, b, x, y, z) ∈ K^{5} by:

α(( \begin{matrix}1 & a & b & x \\ 0 & 1 & a & y \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b + δ(a) & x + δ(y) \\ 0 & 1 & a & y + δ(z) \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} ) .

Then suppose toward a contradiction that G/Z(G) is definably linear in the ACF - expansion of the pure group G. That is G/Z(G) is definably isomorphic to H_{1} × ⋯ H_{n}, where H_{i} is an affine algebraic group over an interpretable field K_{i} for each i (Lemma 5.6). Moreover, we may assume that, for each i ≠ j, the fields K_{i} and K_{j} are not definably isomorphic, and that H_{i} is not trivial. Since H_{i} is a U_{K_{i}}-group for each i, and since G′ contains Z(G), Facts 2.10 and 2.15 provide G = U_{K_{1}}(G) ∗ ⋯ ∗ U_{K_{n}}(G), where ∗ denotes the central product, and Fact 2.11 gives G′ = U_{K_{1}}(G)′ × ⋯ × U_{K_{n}}(G)′. If, for some i, we have U_{K_{i}}(G)′ = 1, then U_{K_{i}}(G) is central in G, contradicting that H_{i} is nontrivial. Since G is nilpotent, this implies that Z(G) ∩ U_{K_{i}}(G)′ is infinite for each i. Since Z(G) is torsion-free and of dimension one over K, its Morley rank is one in the ACF -expansion of the pure group G too, and we obtain Z(G) ≤ U_{K_{i}}(G)′ for each i. Thus we find n = 1, and G/Z(G) is definably linear over K_{1}.

From now on, since K_{1} and K are definably isomorphic in the pure field K (Fact 2.2), the definable subsets of G/Z(G) are precisely its constructible subsets. In particular, for each constructible subset X/Z(G) of G/Z(G), the image of X by any group automorphism of G is constructible too. But, if we set

X = \{( \begin{matrix}1 & 0 & 0 & x \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & z \\ 0 & 0 & 0 & 1 \end{matrix} ) | (x, z) ∈ K^{2}\} ,

then X is constructible and X∩α(X) is not constructible, contradicting that G/Z(G) is definably linear.

Our study starts with the analysis of some automorphisms of algebraic groups. The following theorem by Hochschild-Mostow, together with Lemma 4.7, has several consequences very useful for us.

An affine algebraic group is said to be conservative if the action of its algebraic automorphism group on its algebra of polynomial functions is locally finite. Actually, by [14], a group G is conservative if and only if the holomorph of G inherit an affine algebraic group structure with which it is the semidirect product, in the sense of affine algebraic groups, of G and its algebraic automorphism group.

Fact 4.5. – [14, Theorem 3.2] Let G be a connected affine algebraic group over the algebraic closed field F of characteristic zero. Then G is conservative if and only if one of the following two conditions is satisfied:

(1) the center of G is finite over its unipotent part;

(2) the center of a maximal reductive subgroup of G has dimension at most 1.

Actually, only a special case of this result will be used: when the group is generated by its unipotent elements. Fortunately, for this particular case, we can provide below a model theoretic proof for Fact 4.5.

Fact 4.6. – (Special case of Fact 4.5) Let K be an algebraically closed field of characteristic zero, and let G be an affine algebraic group over K. If G is generated by its unipotent elements, then there is an affine algebraic group of the form H = G ⋊ A, where A is an algebraic subgroup acting faithfully on G by conjugation and such that, for each algebraic group automorphism φ of G, there exists a ∈ A satisfying φ(g) = g^{a} for each g ∈ G.

Proof – We may assume G ≠ 1. We consider the pure language of fields. Then G is an interpretable group and it has a finite Morley rank. Moreover, G × G definably embeds in L = GL_{m}(K) for a positive integer m. Since there is finitely many Jordan decompositions for unipotent elements in L, there is a finite subset X of nontrivial unipotent elements of G × G such that each element of the set 𝒰 of the nontrivial unipotent elements of G × G is conjugate in L with a unique element of X. For each x ∈ X, let V_{x} = d(x). Since each x ∈ X is nontrivial and unipotent, V_{x} is definably isomorphic to K_{+}. Thus ℱ = {V_{x}^{g} | (x, g) ∈ X × L, x^{g} ∈ G × G} is a uniformly definable family of subgroups of G × G such that V_{1} ∩ V_{2} = 1 for each distinct elements V_{1} and V_{2} of ℱ. Moreover, for each x ∈ X, the subgroup V_{x} is unipotent, so all the elements of ∪ℱ are unipotent and, by choice of X, we have ∪ℱ = 𝒰 ∪ {1}.

We consider n nontrivial unipotent elements u_{1}, ⋯, u_{n} of G, such that u_{i+1} ∉ d(u_{1}, ⋯, u_{i}) for each i < n, and such that G = d(u_{1}, ⋯, u_{n}). The existence of n is ensured by the Zil’ber Indecomposability Theorem ([16, Theorem 2.9] or [4, Theorem 5.26]), and because, for each nontrivial unipotent element u of G, the subgroup d(u) is definably isomorphic to K_{+}, so it is connected. In particular, we notice that G is connected.

From now on, by the Zil’ber Indecomposability Theorem, there is an integer k such that, for any elements F_{1}, ⋯, F_{n} of ℱ, the product (F_{1} ⋯ F_{n})^{k} is a subgroup of G × G. In particular, the family ℋ_{0} of these products is a uniformly definable family of subgroups of G × G. Let

ℋ = {F ∈ ℋ_{0} | rk(F) = rk(G), F ∩ (G × {1}) = 1, F ∩ ({1} × G) = 1}.

Then ℋ is a uniformly definable family of subgroups of G × G, and since rk(F) = rk(G) for each F ∈ ℋ , the connectedness of G implies that each element of ℋ is the graph of an algebraic group automorphism of G. Conversely, if α is an algebraic group automorphism of G, then for each i, the element (u_{i}, α(u_{i})) is nontrivial and unipotent, so d((u_{i}, α(u_{i}))) ∈ ℱ. Moreover, the graph Δ of α contains the subgroup H ∈ ℋ_{0} generated by the subgroups d((u_{i}, α(u_{i}))) for i = 1, ⋯, n. But we have G = d(u_{1}, ⋯, u_{n}), so there is a definable surjection from H to G, and rk(H) ≥ rk(G). Since Δ contains H, we obtain rk(H) = rk(Δ) = rk(G) and H ∈ ℋ . Now, since Δ is the graph of an automorphism of G, it is connected, so Δ = H belongs to ℱ. This proves that ℋ is the set of graphs of the algebraic group automorphisms of G. In particular the group Aut_{alg}(G) of all the algebraic group automorphisms of G is interpretable in K.

Furthermore, our argument proves that each φ ∈ Aut_{alg}(G) is characterized by φ(u_{1}), ⋯, φ(u_{n}), so the multiplication in Aut_{alg}(G) is interpretable too, as well as its action over G. Thus, there is an algebraic group of the form G ⋊ A, where A is an algebraic subgroup acting faithfully on G by conjugation and such that, for each φ ∈ Aut_{alg}(G), there exists a ∈ A satisfying φ(g) = g^{a} for each g ∈ G.

From now on, we have to show that the algebraic group G ⋊ A is affine. Since G^{∗} = G ⋊ A^{∘} is connected and since A^{∘} ∩ Z(G^{∗}) = 1, the group A^{∘} is affine by the Rosenlicht’s Theorem [17, §5, Theorem 13]. Then G^{∗} is affine by [17, §5, Theorem 16 p.439], and G ⋊ A is affine too by [17, §5, Corollary 1 p.430], as desired. □

Lemma 4.7. – Let K be a field of finite Morley rank of characteristic zero, and let G be an affine algebraic group over K. If G is generated by its unipotent elements, then any definable automorphism of G is an algebraic automorphism.

Proof – We may assume G ≠ 1. For any nontrivial unipotent element u of G, the subgroup d(u) is contained in the Zariski closure \overline{⟨u⟩} of ⟨u⟩, which is definably isomorphic to K_{+}. By Fact 2.3, we obtain d(u) = \overline{⟨u⟩}, and d(u) is closed and connected (as an algebraic group, as well as a group of finite Morley rank). Let u_{1}, ⋯, u_{n} be n nontrivial unipotent elements of G such that u_{i+1} ∉ d(u_{1}, ⋯, u_{i}) for each i < n, and such that G = d(u_{1}, ⋯, u_{n}). The existence of n is ensured since d(u) is connected for each unipotent element u, and by the finiteness of the Morley rank of G.

Let α be a definable automorphism of G. We show that α is an algebraic group automorphism of G. Let Δ_{α} be the graph of α. Then Δ_{α} is a definable subgroup of G × G, and it is definably isomorphic to G. In particular Δ_{α} is connected (as a group of finite Morley rank). But G = d(u_{1}, ⋯, u_{n}), so we have Δ_{α} = d(⟨(u_{i}, α(u_{i})) | i = 1, ⋯, n⟩). For each i = 1, …, n, the subgroup d(u_{i}, α(u_{i})) is contained in \overline{⟨(u_{i}, α(u_{i}))⟩}, which is definably isomorphic to K_{+}. So we obtain d(u_{i}, α(u_{i})) = \overline{⟨(u_{i}, α(u_{i}))⟩} (Fact 2.3), and Δ_{α} contains Δ_{0} = ⟨\overline{⟨(u_{i}, α(u_{i}))⟩} | i = 1, ⋯, n⟩. Since \overline{⟨(u_{i}, α(u_{i}))⟩} is a connected closed subgroup for each i, the group Δ_{0} is connected and closed too. In particular, it is definable. But it is contained in Δ_{α}, and it contains (u_{i}, α(u_{i})) for each i, hence it is equal to Δ_{α}, and Δ_{α} is closed. Thus, the automorphism α is interpretable in the pure field K, and α is algebraic. □

Corollary 4.8. – Let G be a group of finite Morley rank interpreting a field K of characteristic zero. We assume that G acts faithfully and definably on an affine algebraic group U over K. If U is generated by its unipotent elements, then U ⋊ G is definably isomorphic to a definable subgroup of an affine algebraic group over K.

Proof – By Fact 4.5 or 4.6, there is an affine algebraic group of the form H = U ⋊A for a closed subgroup A, such that H is isomorphic to the holomorph of U. By Lemma 4.7, each element of G acts on U by algebraic group automorphism, so there is a definable isomorphism from G to a definable subgroup of A. □

Proposition 4.9. – Let G be a nilpotent group of finite Morley rank interpreting a field K of characteristic zero. Then G has a largest definable subgroup definably isomorphic to a unipotent algebraic group over K.

Proof – We may assume that G is connected. We proceed by induction on the Morley rank of G. Let N be a definable subgroup, maximal among the ones definably isomorphic to a unipotent algebraic group over K. We may assume N < G and we consider a maximal proper definable subgroup M of G containing N. By induction hypothesis, N is definably characteristic in M, so N is normal in G.

Let U be a definable subgroup of G definably isomorphic to a unipotent algebraic group over K. We may assume that N does not contain U and that U is minimal for these conditions. Then, by induction hypothesis, we have G = NU. Also, U has a normal definable subgroup V such that V is definably isomorphic to a unipotent algebraic group over K, and such that U = V ⋊ A for a definable subgroup A definably isomorphic to K_{+}. By minimality of U, N contains V and, finally, we have U = A ≃ K_{+}. Hence, by Fact 2.3, we have G = N ⋊U, and either C_{U}(N) = 1 or C_{U}(N) = U.

In the first case, Corollary 4.8 gives the result. In the second case, G is definably isomorphic to N × K_{+}, so we have the result too. □

In the pure language of fields, if A is an indecomposable unipotent algebraic group over a field K of characteristic zero, then J(A) = 1. This remark induces the following notion.

Notation 4.10. – In each group G of finite Morley rank, we consider

V (G) = ⟨J(A) | A is a radicable indecomposable definable abelian subgroup of G⟩.

Remark 4.11. –

• By the Zil’ber Indecomposability Theorem, for each group G of finite Morley rank, V (G) is definable and connected.

• Moreover, by Fact 2.6, V (G/H) = V (G)H/H for every normal definable subgroup H of G.

Lemma 4.12. – Let G be a nilpotent group of finite Morley rank, and let K be an interpretable field of characteristic zero such that G is a U_{K}-group. Then G is definably isomorphic to a unipotent algebraic group over K if and only if V (G) = 1.

Moreover, in this case, G is a homogeneous U_{K}-group.

Proof – First we assume that G is definably isomorphic to a unipotent algebraic group G̃ over K. Then each nontrivial indecomposable definable abelian subgroup A of G̃ is contained in a closed subgroup B definably isomorphic to K_{+}. So Fact 2.3 yields A = B and J(A) = 1, and we have V (G̃) = 1 and V (G) = 1. Now each indecomposable subgroup of G is a U_{K}-group, so G is a homogeneous U_{K}-group.

If V (G) = 1, we proceed by induction on the rank of G. We may assume G ≠ 1. Let M be a maximal proper connected definable subgroup of G. Then we have V (M) ≤ V (G) = 1 and, by induction hypothesis, M is definably isomorphic to a unipotent algebraic group over K. Since G is a U_{K}-group, there is an indecomposable definable abelian subgroup A of G not contained in M such that A/J(A) is definably isomorphic to K_{+}. In particular, by the maximality of M, we have G = MA. On the other hand, since J(A) ≤ V (G) is trivial, A is definably isomorphic to K_{+}. Now Proposition 4.9 says that G = MA is definably isomorphic to a unipotent algebraic group over K. □

Corollary 4.13. – Let G be a nilpotent U_{K}-group of finite Morley rank for an interpretable field K of characteristic zero. If there are finitely many algebraically closed fields K_{1}, …, K_{n} such that G is definably isomorphic to a direct product of unipotent groups U_{1}, …, U_{n} over K_{1}, …, K_{n} respectively, then V (G) = 1.

Proof – In this case, each nontrivial indecomposable definable abelian subgroup A of G is contained in a definable subgroup B which is definably isomorphic to (K_{1})_{+} × ⋯ × (K_{n})_{+}. So Fact 2.3 says that A is definably isomorphic to (K_{i})_{+} for some i and that J(A) = 1. In particular, this proves that V (G) = 1. □

Now, by a relatively technical argument, we can prove Theorem 4.3, as a consequence of the previous study.

Proof of Theorem 4.3 – We proceed by induction on the nilpotence class of G and on the Morley rank of G. By Fact 2.10, if T_{0} is the maximal pseudo-torus of G, then G interprets some fields K_{1}, ⋯, K_{n} of characteristic zero such that G = T_{0} ∗ U_{K_{1}}(G) ∗ ⋯ ∗ U_{K_{n}}(G), where ∗ denotes the central product, so G/Z(G) is definably isomorphic to U_{K_{1}}(G)/Z(U_{K_{1}}(G)) × ⋯ × U_{K_{n}}(G)/Z(U_{K_{n}}(G)). But, if G is not a U_{K_{i}}-group for some i = 1, …, n, then the induction hypothesis shows that U_{K_{i}}(G)/Z(U_{K_{i}}(G)) satisfies the result for each i, so G satisfies the result. Hence we may assume that G is a U_{L}-group for an interpretable field L. In particular, by Lemma 4.12, we have just to prove that V (G) is central in G, and we may assume V (G) ≰ Z(G). We note that, by Facts 2.13 and 2.16, G/Z(G) is a homogeneous U_{L}-group.

If G has two distinct G-minimal subgroups A_{1} and A_{2}, we consider Z_{i}/A_{i} = Z(G/A_{i}) for i = 1, 2. By induction hypothesis and Corollary 4.13, we have V (G) ≤ Z_{1} ∩ Z_{2}. Now V (G) is central in G, contradicting V (G) ≰ Z(G). Hence G has a unique G-minimal subgroup A.

We denote by ℳ the set of the maximal proper connected definable subgroups of G. If there are two distinct elements M_{1} and M_{2} of ℳ such that V (M_{1}) and V (M_{2}) are contained in Z(G), then Proposition 4.9 and Lemma 4.12 show that G/Z(G) = M_{1}M_{2}/Z(G) satisfies the theorem. Hence there is at most one element M_{V} of ℳ such that Z(G) contains V (M_{V}). We denote by ℳ^{∗} the set of the elements M of ℳ such that Z(G) does not contain V (M).

For each M ∈ ℳ, by induction hypothesis, Z(M) contains V (M). If Z(M) does not contain Z(G), then M does not contain Z(G). Moreover, since G is nilpotent and connected, its torsion part is contained in its maximal torus, which is central in G, so Z(G) is connected, and we have G = MZ(G) by maximality of M. But this implies that Z(M) = Z(G) ∩ M, so G/Z(G) is definably isomorphic to M/Z(M), and we obtain V (G) ≤ Z(G), contradicting V (G) ≰ Z(G). Hence Z(M) contains Z(G) for each M ∈ ℳ.

Let M_{1} ∈ ℳ. Since G is a U_{L}-group, G/M_{1} is definably isomorphic to L_{+} (Facts 2.3 and 2.15). By Fact 2.6, G has an indecomposable subgroup I_{0} covering G/M_{1}. Since G is nonabelian, we have G ≠ I_{0}, and there exists M_{2} ∈ ℳ containing I_{0}. Therefore G/M_{2} is definably isomorphic to L_{+}, and G/(M_{1} ∩ M_{2}) is definably isomorphic to L_{+} × L_{+}. Let N = M_{1} ∩ M_{2}. Since Z(M_{1}) ∩ Z(M_{2}) centralizes M_{1}M_{2} = G, the previous paragraph yields Z(G) = Z(M_{1}) ∩ Z(M_{2}). In particular, N contains Z(G) and we have V (N) ≤ V (M_{1})∩V (M_{2}) ≤ Z(M_{1})∩Z(M_{2}) = Z(G), so N/Z(G) ≤ G/Z(G) is definably isomorphic to a unipotent group over L (Lemma 4.12). Moreover, since G/N ≃ L_{+} × L_{+}, we have V (G) ≤ N.

If there exists M_{3} ∈ ℳ does not containing N, then G/(N ∩ M_{3}) is definably isomorphic to L_{+} × L_{+} × L_{+}. Now we find N_{1}, N_{2} and N_{3} such that, for i = 1, 2, 3, we have N_{i} = M_{1}^{i} ∩ M_{2}^{i} for some maximal definable subgroups M_{1}^{i} and M_{2}^{i}, and such that G = ⟨N_{1}, N_{2}, N_{3}⟩. By the previous paragraph, N_{i} contains Z(G) for i = 1, 2, 3, the quotient N_{i}/Z(G) is definably isomorphic to a unipotent group over L, and Proposition 4.9 shows that G/Z(G) is definably isomorphic to a unipotent group over L too. Now Lemma 4.12 contradicts V (G) ≰ Z(G). Hence each element of ℳ contains N. Moreover, we have G′V (G) ≤ N and G/N ≃ L_{+} × L_{+}, and G/G′V (G) is definably isomorphic to a direct product of copies of L_{+} by Lemma 4.12, so the intersection of the elements of ℳ is contained in G′V (G). Hence we obtain N = G′V (G).

We consider the subgroup Z defined by Z/A = Z(G/A). By induction hypothesis, Z contains V (G), and G/Z is definably isomorphic to a unipotent group over L (Lemma 4.12). For each z ∈ Z \ Z(G), the groups G/C_{G}(z) and A are definably isomorphic, so we have C_{G}(z) ∈ ℳ, and N ≤ C_{G}(z). Thus N centralizes Z. Moreover, C_{G}(z) ∈ ℳ implies that A ≃ G/C_{G}(z) is definably isomorphic to L_{+}. We show that N contains Z. If G = Z, then we have N ≤ C_{G}(Z) = Z(G), contradicting V (G) ≤ N and V (G) ≰ Z(G). Therefore Z is contained in an element M_{Z} of ℳ. Since M_{Z} ∈ ℳ contains N, and since G/N is definably isomorphic to L_{+} × L_{+}, either N contains Z or NZ = M_{Z}. In the second case, let I_{1} be an indecomposable subgroup of G such that J(I_{1}) is not contained in Z(G). Let x ∈ J(I_{1}) \ Z(G). Then x belongs to V (G) ≤ Z, so C_{G}(x) belongs to ℳ and C_{G}(x) contains N. Moreover, we have x ∈ V (G) ≤ N, so x centralizes Z, and we obtain C_{G}(x) = NZ. But x centralizes I_{1}, hence I_{1} is contained in NZ. This shows that each indecomposable subgroup I_{1} satisfying J(I_{1}) ≰ Z(G) is contained in NZ. Since G/N is definably isomorphic to L_{+} × L_{+}, there exists M^{∗} ∈ ℳ^{∗} \ {NZ}, and we find an indecomposable subgroup I_{2} in M^{∗} such that J(I_{2}) ≰ Z(G). Consequently, we have I_{2} ≤ NZ ∩ M^{∗} = N, contradicting V (N) ≤ Z(G). This proves that N contains Z. In particular, since Z contains V (G), we obtain N = G′Z.

We show that Z/Z(G) is definably isomorphic L_{+} × L_{+}. Since we have Z(G) < Z ≤ Z(N), we have either C_{G}(Z) = N, or C_{G}(Z) ∈ ℳ. We consider two distinct elements H_{1} and H_{2} of ℳ \ {C_{G}(Z)}. For i = 1, 2, we fix h_{i} ∈ H_{i} \ C_{G}(Z), and let γ_{i} : Z → A be the homomorphism defined by γ_{i}(z) = [z, h_{i}]. Then Z/C_{Z}(h_{i}) is definably isomorphic to A ≃ L_{+}. Thus, since we have G = Nd(h_{1}, h_{2}), we obtain C_{Z}(h_{1}, h_{2}) = Z(G), and Z/Z(G) is either definably isomorphic to L_{+}, or definably isomorphic to L_{+} × L_{+}. In the first case, for i = 1, 2, we have C_{Z}(h_{i}) = Z(G), so V (H_{i}) is contained in Z ∩ Z(H_{i}) = Z(G), and H_{i} belongs to ℳ \ ℳ^{∗}. This contradicts that ℳ \ ℳ^{∗} has at most one element. Hence Z/Z(G) is definably isomorphic L_{+} × L_{+}. Furthermore, if C_{G}(Z) belongs to ℳ, we consider g ∈ G \ C_{G}(Z). Then C_{Z}(g) centralizes d(C_{G}(Z), g) = G, and we have Z(G) = C_{Z}(g). But the homorphism γ_{g} : Z → A defined by γ_{g}(z) = [g, z] is surjective by Gminimality of A, hence Z/C_{Z}(g) is definably isomorphic to A ≃ L_{+}, contradicting that Z/Z(G) is definably isomorphic L_{+} ×L_{+}. Thus C_{G}(Z) does not belong to ℳ, and we have N = C_{G}(Z).

By Fact 2.2, the fields \overline{ℚ} and L are isomorphic, definably in the pure field \overline{ℚ}. Hence there is an isomorphism f, definable in the pure field \overline{ℚ}, from G to a connected nilpotent algebraic group G_{L} over L. We consider the induced isomorphism \overline{f} : G/Z → G_{L}/f(Z), and its graph \overline{Δ}. In the ACF -expansion of the pure group G, there is a definable isomorphism i between G/Z × G_{L}/f(Z) and a unipotent group U_{1} over L. Since, in the pure field \overline{ℚ}, the fields \overline{ℚ} and L are definably isomorphic, there is an isomorphism j : U_{1} → U_{2}, definable in \overline{ℚ}, from U_{1} to a unipotent group U_{2} over \overline{ℚ}, such that the preimage of each closed subgroup of U_{2} is a subgroup of U_{1}, definable in the ACF -expansion of the pure group G. But (j ∘ i)(\overline{Δ}) is a subgroup of U_{2}, definable in \overline{ℚ}, so it is a closed subgroup of U_{2}, and \overline{Δ} is a subgroup of G/Z × G_{L}/f(Z), definable in the ACF -expansion of the pure group G.

We consider the preimage Δ of \overline{Δ} in G×G_{L}. It is a definable subgroup of G×G_{L}, and if Δ^{∗} is the graph of f, then we have Δ = (Z × {1})Δ^{∗}. Since G centralizes Z/A and does not centralize Z, we have Δ′ = (A × {1})(Δ^{∗})′. In the same way, since G centralizes A, if Δ^{2} denotes [Δ, Δ′], we have Δ^{2} = (Δ^{∗})^{2}. In particular, Δ^{2} ∩ (G × {1}) is trivial. We consider Δ/Δ^{2}, and let Z_{Δ}/Δ^{2} be its center. Then [Δ, Z_{Δ}∩(G×{1})] is contained in Δ^{2}∩(G×{1}) = 1, so Z_{Δ}∩(G×{1}) ≤ Z(G)×{1}. If the nilpotence class of G is not 2, then the induction hypothesis applied with Δ/Δ^{2} shows that V (Δ) is contained in Z_{Δ}. But G_{L} is a unipotent group over L, so we have V (G_{L}) = 1 and V (G × G_{L}) is contained in G × {1}. Hence V (Δ) is contained in Z_{Δ} ∩ (G × {1}) ≤ Z(G) × {1}, and V (G) is contained in Z(G). This contradiction implies that the nilpotence class of G is 2. In particular, G′ is contained in Z(G) ≤ Z and we have N = G′Z = Z.

We show that ℳ = ℳ^{∗}. Indeed, if M_{V} exists, we consider M ∈ ℳ^{∗}. Then Proposition 4.9 and Lemma 4.12 show that V (G) = V (MM_{V}) is contained in V (M)Z(G). But we have Z = N = G′V (G) = Z(G)V (G), so we obtain N = Z(G)V (M) ≤ Z(M), and M ≤ C_{G}(N) = Z, contradicting our choice of M. Hence we have V (M) ≰ Z(G) for each M ∈ ℳ.

Now, for each g ∈ G \ N, we consider the map γ_{g} : G → Z(G) defined by γ_{g}(x) = [g, x]. Since g ∉ Z does not centralize N = Z and centralizes Z/A, we have γ_{g}(N) = A ≃ L_{+} and, since Z/Z(G) is definably isomorphic to L_{+} × L_{+}, the group C_{Z}(g)/Z(G) is definably isomorphic to L_{+}. Moreover, since g ∉ Z(G), we have C_{G}(g) < G and, since N ≤ M for each M ∈ ℳ, we have C_{G}(g)N < G. Since g ∉ N and since G/N is definably isomorphic to L_{+} × L_{+}, this implies that C_{G}(g)N/N is definably isomorphic to L_{+}. Now, if we consider h ∈ G \ C_{G}(g)N, then we have C_{G}(h)N/N ≃ L_{+} too, and since h ∈ C_{G}(h) \ C_{G}(g)N, Fact 2.3 gives G = d(g)d(h)N. Thus, for each z ∈ C_{G}(g, h), we have G = C_{G}(z)N, Therefore, since N is contained in each element of ℳ, we have G = C_{G}(z) and z ∈ Z(G), so C_{G}(g)/Z(G) ∩ C_{G}(h)/Z(G) is trivial. On the other hand, again since N is contained in each element of ℳ, we obtain G = C_{G}(g)C_{G}(h), and G/Z(G) is the direct product of C_{G}(g)/Z(G) and C_{G}(h)/Z(G).

We fix g ∈ G \ N and h ∈ G \ C_{G}(g)N. The previous paragraph shows that, for any definable isomorphism α : C_{G}(g)/Z(G) → C_{G}(h)/Z(G), there is a definable subgroup U/Z(G) of G/Z(G) representing the graph of α. We remark that, since G/N is definably isomorphic to L_{+} × L_{+}, we have either U ≤ N, or UN/N ≃ L_{+} definably, or UN/N = G/N. Since G = C_{G}(g)U and since G ≠ C_{G}(g)N, we have not U ≤ N. Moreover, since N is contained in each element of ℳ and since U < G, we do not have UN/N = G/N, and we obtain UN/N ≃ L_{+} and UN ∈ ℳ. In particular, we have V (UN) ≰ Z(G) and V (UN) ≤ Z(UN). We consider u ∈ U \N. Since UN/N ≃ L_{+}, the subgroup C_{G}(u)N contains UN, and since UN ∈ ℳ, we have either G = C_{G}(u)N or C_{G}(u) ≤ UN. In the first case, since each element of ℳ contains N, the group C_{G}(u) is contained in no element of ℳ, and we obtain G = C_{G}(u) and u ∈ Z(G), contradicting Z(G) ≤ N. Hence UN contains C_{G}(u). Then C_{G}(u) centralizes V (UN), and since the previous paragraph gives C_{Z}(u)/Z(G) ≃ L_{+}, we obtain C_{Z}(u) = V (UN)Z(G) = V (U)V (N)Z(G). Also, since Z(G) contains V (N), we have C_{Z}(u) = V (U)Z(G) ≤ U. But we have Z/Z(G) ≃ L_{+} × L_{+}, so either U∩Z = C_{Z}(u) or U contains Z. Since we have U/Z(G)∩C_{G}(g)/Z(G) = 1 and C_{Z}(g)/Z(G) ≃ L_{+}, U does not contain Z, and we obtain U ∩Z = C_{Z}(u). Moreover, since we have u ∈ U \N, we have C_{U}(u)N/N ≠ 1, and since the previous paragraph gives C_{G}(u)N/N ≃ L_{+} ≃ UN/N, we obtain C_{U}(u)N/N = C_{G}(u)N/N = UN/N. In particular, we have C_{G}(u) = C_{U}(u)N ∩ C_{G}(u) = C_{U}(u)C_{N}(u), and since U contains C_{N}(u), we obtain C_{G}(u) ≤ U. Furthermore, since U ∩ Z = C_{Z}(u), we obtain U = C_{G}(u)(N ∩ U) ≤ C_{G}(u) and U = C_{G}(u).

We note that, if u ∈ C_{G}(g)N, then we have UN = C_{G}(g)N because UN/N ≃ L_{+} ≃ C_{G}(g)N/N, contradicting G = C_{G}(g)U and C_{(}g)N < G. Thus we have u ∉ C_{G}(g)N and, in the same way, u ∉ C_{G}(h)N. In particular, the following uniformly definable family ℱ of subgroups of G/Z(G) contains the graph of each definable isomorphism from C_{G}(g)/Z(G) to C_{G}(h)/Z(G):

ℱ = {C_{G}(u)/Z(G) | u ∈ G \ (C_{G}(g)N ∪ C_{G}(h)N)}.

Conversely, for each u ∈ G \ (C_{G}(g)N ∪ C_{G}(h)N), the quotient group G/Z(G) is the direct product of C_{G}(g)/Z(G) and C_{G}(u)/Z(G), and also of C_{G}(u)/Z(G) and C_{G}(h)/Z(G), so C_{G}(u)/Z(G) is the graph of an isomorphism from C_{G}(g)/Z(G) to C_{G}(h)/Z(G). Now the set ℐ of the definable isomorphisms from C_{G}(g)/Z(G) to C_{G}(h)/Z(G) is a (nonempty) uniformly definable family. But, if we fix r ∈ ℐ, then the set of the definable automorphisms of C_{G}(g)/Z(G) is 𝒜 = {r^{−1} ∘ s | s ∈ ℐ }, and it is a uniformly definable family.

We show that the Morley rank of 𝒜 is 2. We note that, in the pure field \overline{ℚ}, we have rk(L_{+}) = 1 (Fact 2.2), so we have rk(L_{+}) = 1 in the ACF -expansion of the pure group G too. Since there are some definable bijections between ℱ, ℐ and 𝒜, we have to prove that ℱ has Morley rank 2. First we show that, if C_{G}(u)/Z(G) and C_{G}(v)/Z(G) are two distinct elements of ℱ, then C_{G}(u) ∩ C_{G}(v) is contained in N, that is the elements of ℱ are generically disjoint. The previous analysis shows that, if we have v ∉ C_{G}(u)N, then C_{G}(u) ∩ C_{G}(v) = Z(G), so we may assume v ∈ C_{G}(u)N. Therefore C_{G}(u)N/N ∩ C_{G}(v)N/N is not trivial and, since C_{G}(u)N/N and C_{G}(v)N/N are definably isomorphic to L_{+}, we obtain C_{G}(u)N = C_{G}(v)N. Also, the previous study gives C_{Z}(u) = V (C_{G}(u)N)Z(G) = V (C_{G}(v)N)Z(G) = C_{Z}(v). Since C_{G}(u)/C_{Z}(u) and C_{G}(v)/C_{Z}(v) are definably isomorphic to C_{G}(u)Z/Z = C_{G}(v)Z/Z ≃ L_{+}, we have either C_{G}(u) = C_{G}(v) or C_{G}(u) ∩ C_{G}(v) = C_{Z}(u) = C_{Z}(v) ≤ N. Hence the elements of ℱ are generically disjoint. Moreover, each element C_{G}(u)/Z(G) of ℱ has Morley rank

rk(C_{G}(u)/C_{Z}(u)) + rk(C_{Z}(u)/Z(G)) = 2rk(L_{+}) = 2,

and ∪ℱ contains G/Z(G) \ (C_{G}(g)N/Z(G) ∪ C_{G}(h)N/Z(G)) which is a generic subset of G/Z(G). So the Morley rank of ∪ℱ is rk(G/Z(G)) = rk(G/N) + rk(N/Z(G)) = 4rk(L_{+}) = 4, and the one of 𝒜 is rk(∪ℱ) − rk(C_{G}(u)/Z(G)) = 2, as claimed.

The same reasoning applied in the pure field \overline{ℚ} shows that the group 𝒜 has dimension 2 over \overline{ℚ} too.

In the pure field \overline{ℚ}, the group C_{G}(g)/Z(G) is definably isomorphic to \overline{ℚ}_{+} × \overline{ℚ}_{+}, since C_{G}(g)/C_{Z}(g) and C_{Z}(g)/Z(G) are definably isomorphic to L_{+} ≃ \overline{ℚ}_{+}. Hence there is a definable embedding μ from the canonical semidirect product C_{G}(g)/Z(G) ⋊ 𝒜 to (\overline{ℚ}_{+} × \overline{ℚ}_{+}) ⋊ GL(2, \overline{ℚ}), where GL(2, \overline{ℚ}) acts by multiplication on \overline{ℚ}_{+} × \overline{ℚ}_{+}, and such that μ(C_{G}(g)/Z(G)) = \overline{ℚ}_{+} × \overline{ℚ}_{+} and μ(𝒜) is a closed subgroup of GL(2, \overline{ℚ}). By Lemma 4.1, the group μ(𝒜^{∘}) is connected, and since 𝒜 has dimension 2 over \overline{ℚ}, the group μ(𝒜^{∘}) has dimension two. Let T be a maximal torus of μ(𝒜^{∘}). It is nontrivial since μ(𝒜^{∘}) is a closed subgroup of dimension two of GL(2, \overline{ℚ}). First we assume that μ^{−1}(T) is definable in the ACF -expansion of the pure group G. Since T is nontrivial, there exists e_{1} ∈ \overline{ℚ}_{+} × \overline{ℚ}_{+} such that T ·e_{1} = \overline{ℚ}^{∗} ·e_{1}, and the subset V_{1} := (T ·e_{1})∪\{(0, 0)\} = \overline{ℚ}·e_{1} is an infinite subgroup of \overline{ℚ}_{+} × \overline{ℚ}_{+}. Now either T centralizes (\overline{ℚ}_{+} × \overline{ℚ}_{+})/V_{1} and V_{2} := C_{\overline{ℚ} _{+}×\overline{ℚ} _{+}}(T) is a complement of V_{1} in \overline{ℚ}_{+} × \overline{ℚ}_{+}, or T does not centralize (\overline{ℚ}_{+} × \overline{ℚ}_{+})/V_{1} and we find e_{2} ∈ (\overline{ℚ}_{+} × \overline{ℚ}_{+}) \ V_{1} such that T · e_{2} = \overline{ℚ}^{∗} · e_{2}. In this last case, the subset V_{2} := (T · e_{2}) ∪ \{(0, 0)\} = \overline{ℚ} · e_{2} is an infinite complement of V_{1} in \overline{ℚ}_{+} × \overline{ℚ}_{+}. Thus, for i = 1, 2, the preimage μ^{−1}(V_{i}) = (μ^{−1}(T) · μ^{−1}(e_{i})) ∪ {μ^{−1}(0, 0)} of V_{i} is definable in the ACF -expansion of the pure group G. Since V_{1} and V_{2} are infinite and satisfy \overline{ℚ}_{+} ×\overline{ℚ}_{+} = V_{1} ⊕V_{2}, and since C_{G}(g)/Z(G) has Morley rank two then, for i = 1, 2, the subgroup μ^{−1}(V_{i}) is torsion-free of Morley rank one and we have C_{G}(g)/Z(G) = μ^{−1}(V_{1})⊕μ^{−1}(V_{2}). Since C_{G}(g)/Z(G) is an homogeneous U_{L}- group, this implies that μ^{−1}(V_{i}) is definably isomorphic to L_{+} for i = 1, 2, and that C_{G}(g)/Z(G) is definably isomorphic to L_{+} × L_{+} in the ACF -expansion of the pure group G. But C_{G}(h)/Z(G) is definably isomorphic to C_{G}(g)/Z(G), and G/Z(G) is the direct product of C_{G}(g)/Z(G) and C_{G}(h)/Z(G), so G/Z(G) is isomorphic to (L_{+})^{4}, definably in the ACF -expansion of the pure group G. This contradicts V (G) ≰ Z(G). Hence μ^{−1}(T) is not definable in the ACF -expansion of the pure group G.

Since μ^{−1}(T) is not definable in G, the group μ(𝒜^{∘}) is not a torus, and since it has dimension two while the maximal unipotent subgroups of GL(2, \overline{ℚ}) have dimension one, we have μ(𝒜^{∘}) = P ⋊ T for P a maximal unipotent subgroup of GL(2, \overline{ℚ}). Moreover, since μ^{−1}(T) is not definable in G, we have μ^{−1}(T) < C_{𝒜^{∘}} (μ^{−1}(T)) and T < C_{μ(𝒜^{∘})} (T), and since P is a torsion-free group of dimension one, we obtain μ(𝒜^{∘}) = C_{μ(𝒜^{∘})} (T) and μ(𝒜^{∘}) is abelian. Now T centralizes a nontrivial unipotent subgroup of GL(2, \overline{ℚ}), hence T is central in GL(2, \overline{ℚ}) and it is conjugate with the following subgroup of GL(2, \overline{ℚ}):

R = \{( \begin{matrix}t & a \\ 0 & t \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, a ∈ \overline{ℚ}\} .

Thus there is an isomorphism ν, definable in \overline{ℚ}, from (\overline{ℚ}_{+} × \overline{ℚ}_{+}) ⋊ μ(𝒜^{∘}) to H_{1} = (\overline{ℚ}_{+} × \overline{ℚ}_{+}) ⋊ R, where R acts on \overline{ℚ}_{+} × \overline{ℚ}_{+} by multiplication, and such that ν(T) is the maximal torus of R. But there is an isomorphism γ, definable in \overline{ℚ}, from H_{1} to the following group

H_{2} = \{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} ,

and satisfying (γ ∘ ν)(T) = \{( \begin{matrix}t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}\}. Since μ^{−1}(T) is not de-

finable in G, this implies that, in the ACF -expansion of the pure group H_{2}, the maximal torus (γ ∘ ν)(T) is not definable. This contradicts Lemma 3.7 and Proposition 3.11, and finishes the proof. □

5. Definably affine groups

Generally a quotient of a definably linear group of finite Morley rank by a normal definable subgroup, even finite, is not definably linear (Example 5.1). In order to overcome this obstacle, we introduce definably affine groups (Definition 5.2). Then we obtain a structural result about the definably affine ACF -groups (Proposition 5.11). Moreover, we introduce two new subgroups A(G) and W (G) for any ACF - group G (Corollaries 5.10 and 5.17). We notice that the subsections 5.1 and 5.2 concern all the ACF -groups, not just the ACF_{0}-groups, since we do not need Lemma 4.1 in them.

Example 5.1. – We consider the pure group G = H_{1} × H_{2}, where H_{1} and H_{2} are two copies of SL_{2}(K) for an algebraically closed field K of characteristic zero. On the one hand, for i = 1, 2, the quotient H_{i}/Z(H_{i}) is definably linear [16, Corollaire 4.16], and H_{i} is definably linear too (Proposition 5.14). Hence G is definably linear. On the other hand, if i and j denote the involutions of Z(H_{1}) and Z(H_{2}) respectively, then \overline{G} = G/⟨(i, j)⟩ is not definably linear.

Indeed, let μ be a nontrivial field automorphism of K, and let μ^{∗} be the automorphism of H_{1} induced by μ. We consider the automorphism α of \overline{G} defined for each \overline{(h_{1}, h_{2})} ∈ \overline{G} by α(\overline{(h_{1}, h_{2})}) = \overline{(μ^{∗}(h_{1}), h_{2})}. Then, if Δ is the graph in G of an algebraic group isomorphism from H_{1} to H_{2}, and if \overline{Δ} := Δ/⟨(i, j)⟩ denotes its image in \overline{G}, the intersection of \overline{Δ} and of α(\overline{Δ}) is not definable. So α(\overline{Δ}) is not definable and α is not a standard isomorphism. Consequently, by Fact 3.6 (or Fact 1.4), the pure group \overline{G} is not definably linear over one interpretable field. But if \overline{G} is definably linear over several interpretable fields, then Lemma 5.6 shows that \overline{G} is a direct product of two proper definable subgroups. This contradicts that each nontrivial normal subgroup of \overline{G} contains the central involution of \overline{G}. Hence \overline{G} is not definably linear.

5.1. Definition and generalities.

Definition 5.2. – A group G of finite Morley rank is said to be definably affine (over finitely many fields K_{1}, …, K_{n}, interpretable in G) if G is definably isomorphic to a definable section of H_{1} × ⋯ × H_{n}, where H_{i} is an affine algebraic group over K_{i} for each i = 1, …, n.

Remark 5.3. –

(1) If G is a definably affine ACF -group, interpretable in the pure algebraically closed field K, then G is affine by [16, §4.e (2)] and Fact 2.2.

(2) Every definable quotient and every definable subgroup of a definably affine group is definably affine too.

(3) By [16, §4.e (2)] and Fact 2.2, if an ACF -group G is definably affine over an algebraically closed field L, there is a definable isomorphism ρ : G → H for an affine algebraic group H over L, and any subgroup of H is definable in G if and only if it is closed.

Our first result is a remark concerning some definable fields isomorphisms. For the fields of finite Morley rank and of characteristic zero, a result of the same vein is known [10, Corollary 2.8].

Lemma 5.4. – Let K and L be two fields, definable in an ACF -group G. If one of the following two conditions is satisfied, then K and L are definably isomorphic:

• K_{+} and L_{+} are definably isomorphic;

• K^{∗} and L^{∗} are definably isomorphic.

Proof – Let F be an algebraically closed field such that G is interpretable in the pure field F. We may assume that K and L are infinite. Thus there is a field isomorphism α : K → L, definable in the pure field F (Fact 2.2). If δ : K_{+} → L_{+} is a group isomorphism, definable in G, then μ = δ^{−1} ∘ α is a group automorphism of K_{+}, definable in F. By Fact 2.2, the map μ is definable in the pure field K, so α = δ ∘ μ is definable in G.

If γ : K^{∗} → L^{∗} is a group isomorphism, definable in G, we consider the map β : K → L defined by β(x) = γ(x) for each x ∈ K^{∗} and by β(0) = 0. Then ν = β^{−1} ∘ α is a permutation of K definable in F, and ν is definable in the pure field K (Fact 2.2). Thus α = β ∘ ν is definable in G. □

Corollary 5.5. – Let G be an infinite ACF -group. Suppose that K and L are two interpretable fields such that G is definably affine over K (resp. L). Then K and L are definably isomorphic.

Proof – We may assume that G has no proper infinite definable subgroup. Let H_{K} (resp. H_{L}) be an affine algebraic group over K (resp. L) such that G is definably isomorphic to H_{K} (resp. H_{L}). By the minimality of G, the group H_{K} (resp. H_{L}) is definably isomorphic to either K^{∗} (resp. L^{∗}), or K_{+} (resp. L_{+}). Now Lemma 5.4 provides the result. □

Lemma 5.6. – Let G be an ACF -group, and let K_{1}, …, K_{n} be n infinite definable fields, such that K_{i} is not definably isomorphic to K_{j} for i ≠ j. For i = 1, …, n, we consider an affine algebraic group H_{i}. Then, for each connected definable subgroup U of H = H_{1} × ⋯ × H_{n}, we have U = (U ∩ H_{1}) × ⋯ × (U ∩ H_{n}).

Proof – We assume toward a contradiction that U is a counterexample of minimal Morley rank. Then, for each proper connected definable subgroup M_{0} of U, we have M_{0} = (M_{0} ∩ H_{1}) × ⋯ × (M_{0} ∩ H_{n}), so U has a unique maximal proper connected definable subgroup M. For each i, we denote by ρ_{i} : U → H_{i} the projection map from U to H_{i}. We may assume ρ_{i}(U) = H_{i} ≠ 1 for each i. From now on, M = (M ∩ H_{1}) × ⋯ × (M ∩ H_{n}) is normal in H. Thus, if M ≠ 1, we have H_{i} ∩ M ≠ 1 for some i, and H/H_{i} ∩ M is equal to H_{1}(H_{i} ∩ M)/(H_{i} ∩ M) × ⋯ × H_{n}(H_{i} ∩ M)/(H_{i} ∩ M). Then, in H/(H_{i} ∩ M), the minimality of the Morley rank of U gives

U/(H_{i} ∩ M) = (U ∩ H_{1})(H_{i} ∩ M)/(H_{i} ∩ M) × ⋯ × (U ∩ H_{n})(H_{i} ∩ M)/(H_{i} ∩ M),

and U = (U ∩ H_{1}) × ⋯ × (U ∩ H_{n}), contradicting the choice of U, so M = 1.

For each i, since ρ_{i} is definable, the triviality of M implies that each proper closed subgroup of H_{i} is finite, so H_{i} is definably isomorphic either to (K_{i})^{∗}, or to (K_{i})_{+}. Moreover, for each i, since ρ_{i}(U) = H_{i} ≠ 1, the kernel R_{i} of ρ_{i} is finite, and U/R_{i} is definably isomorphic either to (K_{i})^{∗}, or to (K_{i})_{+}. Now R = ⟨R_{i} | i = 1, …, n⟩ is finite and, for each i, U/R is definably isomorphic either to (K_{i})^{∗}, or to (K_{i})_{+}. Hence Lemma 5.4 yields a contradiction. □

Corollary 5.7. – Let G be an ACF -group, definably affine over interpretable fields K_{1}, …, K_{n}. Then, for each i = 1, …, n, there is an affine algebraic group H_{i} over K_{i}, such that G is definably isomorphic to U/F, for a definable subgroup U of H_{1} × ⋯ × H_{n}, and a finite normal subgroup F of U.

Proof – We may assume that K_{i} is not definably isomorphic to K_{j} for i ≠ j. Since G is definably affine, it is definably isomorphic to a definable section U/F of H_{1} × ⋯ × H_{n}, where H_{i} is an affine algebraic group over K_{i} for each i = 1, …, n. We may assume F ∩ H_{i} = 1 for each i. But Lemma 5.6 gives F^{∘} = (F^{∘} ∩ H_{1}) × ⋯ × (F^{∘} ∩ H_{n}) = 1, so F is finite. □

From now on, we connect the notions of definably linear groups and of definably affine groups.

At first we note that, if G is both an ACF -group and an affine algebraic group over an algebraically closed field K, then since any quotient of an affine algebraic group by a normal closed subgroup is affine, it follows that the ACF -group G is definably affine over one interpretable field if and only if it is definably linear over one interpretable field.

Corollary 5.8. – Let G be an ACF -group, definably affine over interpretable fields K_{1}, …, K_{n}. Then G has a finite normal subgroup E such that G/E is definably linear over K_{1}, …, K_{n}.

Proof – Let U/F and H_{1}, …, H_{n} be as in the previous result. We may assume G = U/F. For each i, let F_{i} be the projection of F on H_{i}. Then E = (F_{1} × ⋯ × F_{n})/F is convenient. □

5.2. The subgroup A(G).

Theorem 5.9. – Let G be an ACF -group. If G is generated by its connected definably affine subgroups over interpretable fields K_{1}, …, K_{n}, then G is definably affine over K_{1}, …, K_{n}.

Proof – Let L be an algebraically closed field such that G is interpretable in the pure field L. First we assume that G is generated by its connected definably affine subgroups over an interpretable field K. We may assume that K is infinite, therefore K is isomorphic to L, definably in the pure field L (Fact 2.2). Let φ be an isomorphism from G to an algebraic group G_{K} over K, definable in the pure field L, and let Δ be its graph. By the Zilber’s Indecomposability Theorem, G is connected and there exist finitely many connected definably affine subgroups R_{1}, …, R_{m} of G over K such that G = ⟨R_{i} | i = 1, …, m⟩. Since K and L are definably isomorphic in L, for each i, the subgroup φ(R_{i}) of G_{K} is affine, and any subgroup of R_{i} × φ(R_{i}) is definable in G if and only if it is definable in L. In particular, the subgroup S_{i} = {(x, φ(x)) | x ∈ R_{i}} of Δ is affine and definable in G for each i. Moreover, for each i, since R_{i} is connected (in G), this implies that S_{i} and φ(R_{i}) are connected in G. Consequently, the Zilber’s Indecomposability Theorem says that Δ = ⟨S_{i} | i = 1, …, n⟩ is definable in G, so φ is definable in G. Moreover, for each i, since φ(R_{i}) is connected in G, it is connected in K too, and G_{K} = ⟨φ(R_{i}) | i = 1, …, n⟩ is an affine algebraic group. Hence G is definably linear over K.

For the proof of the result, we may assume that K_{i} is not definably isomorphic to K_{j} for each i ≠ j. For each i, we denote by U_{i} the (normal) subgroup of G generated by the connected definable subgroups of G, definably affine over K_{i}. By the Zilber’s Indecomposability Theorem, U_{i} is definable and connected for each i, and the previous paragraph says that U_{i} is definably linear over K_{i} for each i. Now, for each i ≠ j, by the Zilber’s Indecomposability Theorem, [U_{i}, U_{j}] is a connected definable subgroup of U_{i} ∩ U_{j}, and it is definably affine over K_{i} (resp. K_{j}). Hence Corollary 5.5 yields [U_{i}, U_{j}] = 1.

We show that G is the central product of U_{1}, …, U_{n}. By the Zilber’s Indecomposability Theorem, G is connected and there exist finitely many connected definably affine subgroups R_{1}, …, R_{m} of G such that G = ⟨R_{i} | i = 1, …, m⟩. By Lemma 5.6 and Corollary 5.7, for each i = 1, …, n and each j = 1, …, m, there is an affine algebraic group H_{ij} over K_{i}, such that R_{j} is definably isomorphic to V_{j}/F_{j}, for a definable subgroup V_{j} of H_{1j} ×⋯×H_{nj}, and a finite normal subgroup F_{j} of V_{j}. Since R_{j} is connected for each j, we may assume that V_{j} is connected for each j. Now, for each j = 1, …, m, Lemma 5.6 shows that R_{j} has some connected definable subgroups S_{1j}, …, S_{nj} such that R_{j} = ⟨S_{ij} | i = 1, …, n⟩ and such that S_{ij} is definably affine over K_{i} for each i. Since for each i, j, the subgroup S_{ij} is contained in U_{i}, this shows that G is generated by U_{1}, …, U_{n}. Since [U_{i}, U_{j}] = 1 for each i ≠ j, the group G is the central product of U_{1}, …, U_{n}. Hence we find a definable epimorphism from U_{1} × ⋯ × U_{n} to G. Since U_{i} is definably linear over K_{i} for each i, this finishes the proof. □

Corollary 5.10. – Any ACF -group G has a largest connected definably affine subgroup denoted A(G).

Proposition 5.11. – Let G be a connected ACF -group. Suppose that G is definably affine over the fields K_{1}, …, K_{n}, and that K_{i} is not definably isomorphic to K_{j} for each i ≠ j. For each i, let G_{i} be the largest connected subgroup of G definably linear over K_{i}. Then the following conditions hold:

• G is the central product of G_{1}, …, G_{n};

• G has a nontrivial finite normal subgroup E such that G/E is the direct product of G_{1}E/E, …, G_{n}E/E; in particular G_{i} ∩ G_{j} is finite for each i ≠ j;

• {G_{1}, ⋯, G_{n}} is stable by each automorphism of the ACF -group G.

Proof – By Corollary 5.7, for each i, there is an affine algebraic group H_{i} over K_{i} such that G is definably isomorphic to U/F, for U a definable subgroup of H_{1} × ⋯ × H_{n} and a finite normal subgroup F of U. We may assume G = U/F. Since G is connected, we may assume that U is connected, and Lemma 5.6 gives U = (U ∩ H_{1}) × ⋯ × (U ∩ H_{n}). In particular, U ∩ H_{i} is connected for each i, and we may assume H_{i} ≤ U for each i. Now G is the central product of H_{1}F/F, …, H_{n}F/F, and H_{i}F/F ∩ H_{j}F/F is finite for each i ≠ j.

For each i, we consider V_{i} such that G_{i} = V_{i}/F. By Lemma 5.6, we have V_{i}^{∘} = (V_{i}^{∘} ∩ H_{1}) × ⋯ × (V_{i}^{∘} ∩ H_{n}) for each i. In particular, V_{i}^{∘} ∩ H_{j} is connected for each i, j. Thus, for each i, j, the subgroup (V_{i}^{∘} ∩ H_{j})F/F ≤ G_{i} ∩ H_{j}F/F is connected and definably affine over K_{i} (resp. K_{j}). Hence Corollary 5.5 gives V_{i}^{∘} ∩ H_{j} = 1 for each i ≠ j, so V_{i}^{∘} is contained in H_{i} for each i. Since G_{i} contains H_{i}F/F for each i, we obtain G_{i} = H_{i}F/F for each i. This proves that G is the central product of G_{1}, …, G_{n}, and that G_{i} ∩ G_{j} is finite for each i ≠ j.

Moreover, Corollary 5.8 provides a finite normal subgroup E such that G/E is definably linear over K_{1}, …, K_{n}, and Lemma 5.6 says that G/E is definably isomorphic to A_{1} × ⋯ × A_{n} for some affine algebraic groups A_{1}, …, A_{n} over K_{1}, …, K_{n} respectively. Then Corollary 5.5 and Lemma 5.6 imply that G_{i}E/E is definably isomorphic to A_{i} for each i, and that G/E is the direct product of G_{1}E/E, …, G_{n}E/E.

Let φ be an automorphism of the ACF -group G. We fix i ∈ {1, …, n}, and we show that φ(G_{i}) ∈ {G_{1}, …, G_{n}}. Then, as for G_{i} in G, there is an infinite interpretable field L such that φ(G_{i}) is the largest connected definable subgroup of G, definably linear over L. If L is not definably isomorphic to K_{j} for each j, then G is definably affine over K_{1}, …, K_{n}, L, and the previous paragraphs applied with this situation give dim(G) = dim(G_{1}) + ⋯ + dim(G_{n}) + dim(φ(G_{i})). On the other hand, by the previous paragraph, we have dim(G) = dim(G_{1}) + ⋯ + dim(G_{n}), so φ(G_{i}) is finite. Since φ(G_{i}) is connected, we obtain φ(G_{i}) = 1, so G_{i} = 1 = φ(G_{i}). Hence we may assume that L is definably isomorphic to K_{j} for some j. Since φ(G_{i}) is the largest connected definable subgroup of G, definably linear over L, we obtain φ(G_{i}) = G_{j}. This finishes the proof. □

5.3. The subgroup W (G). We provide some crucial criterions for the ACF - groups to be definably affine (Theorems 5.15 and 5.9). Moreover, similarly to the existence of the subgroup A(G) in any ACF -group G (Corollary 5.10), we show that any such a group G has a smallest normal definable subgroup W (G) such that G/W (G) is definably affine, and that W (G) is connected when G is an ACF_{0}-group (Corollary 5.17).

In the proof of Corollary 5.13, we use the following result, due to A.V. Borovik and G. Cherlin.

Fact 5.12. – [11, Proof of proposition 4.3] Let H be a normal definable subgroup of finite index of a group G of finite Morley rank. Then G definably embeds in the wreath product of H by G/H.

Corollary 5.13. – Let G be a group of finite Morley rank. If G^{∘} is definably affine over interpretable fields K_{1}, …, K_{n}, then G is definably affine over K_{1}, …, K_{n} too.

Proof – For each i = 1, …, n, let H_{i} be an affine algebraic group over K_{i} such that G^{∘} is definably isomorphic to the definable section U/F of H = H_{1} ×⋯×H_{n}. Then the wreath product W of G^{∘} ≃ U/F by G/G^{∘} is definably isomorphic to a definable section of the wreath product of H by G/G^{∘}, which is definably linear over K_{1}, …, K_{n}. Since Fact 5.12 says that G definably embeds in W, we obtain the result. □

Proposition 5.14. – Let G be an ACF_{0}-group, and let E be a finite normal subgroup of G. If G/E is definably linear over an interpretable field L, then G is definably linear over L too.

Proof – Let K be an algebraically closed field of characteristic zero such that the ACF_{0}-group G is interpretable in the pure field K. By Corollary 5.13, we may assume that G is connected and infinite. Moreover, G/E is definably linear over L, so G/E and G are affine over K by Fact 2.2, and we may assume that G is an infinite subgroup of GL_{n}(K) for an integer n. Since G/E is definably linear over L, we find an algebraic group A_{0} over L definably isomorphic to G/E. Since G is infinite, L is infinite too and L is isomorphic to K, definably in K by Fact 2.2. Let α be a field automorphism from K to L, definable in the pure field K. We consider A = α^{∗}(G) and B = α^{∗}(E), where α^{∗} is the isomorphism from GL_{n}(K) to GL_{n}(L) induced by α. Then A/B is isomorphic to A_{0}, definably in the pure field K. In the pure field K, for each i, the definable subsets of GL_{i}(L) are the images of the constructible subsets of GL_{i}(K) by the isomorphisms α_{i}^{∗} induced by α. In particular A is definable in L and, for any isomorphism from A/B to A_{0}, definable in K, the graph Δ_{0} is definable in L too.

Let Z = B×E, and let Δ be the graph of the isomorphism u from A to G induced by α^{∗}. Let v be an isomorphism from G/E to A_{0} definable in G. Then u induces an isomorphism \overline{u} from A/B to G/E, definable in K, and v ∘ \overline{u} is an isomorphism from A/B to A_{0} definable in K. By the previous paragraph, v ∘ \overline{u} is definable in L too. Thus, since v is definable in G, \overline{u} is definable in G. This shows that ΔZ is a subgroup of A × G, definable in G. Moreover, since Z is abelian and finite, ΔZ/Δ is abelian and finite. Since, by Lemma 4.1, (ΔZ)^{∘}/((ΔZ)^{∘})′ is a connected abelian algebraic group over an algebraically closed field of characteristic zero, it is divisible and it has no proper subgroup of finite index. In particular, (ΔZ)^{∘} is contained in Δ, and since it has finite index in ΔZ, the group Δ is definable in G. This proves that u is definable in G, so G is definably linear over L. □

Theorem 5.15. – Let G be an ACF_{0}-group. Suppose that G has a finite normal subgroup F such that G/F is definably affine over interpretable fields K_{1}, …, K_{n}. Then G is definably affine over K_{1}, …, K_{n}

Proof – By Corollary 5.13, we may assume that G is connected. By Corollary 5.8, G has a finite normal subgroup E containing F such that there exists a definable isomorphism ρ from G/E to a definable subgroup U of H = H_{1} × ⋯ × H_{n}, where H_{i} is an affine algebraic group over K_{i} for each i. By Lemma 5.6, we have U = (U ∩ H_{1}) × ⋯ × (U ∩ H_{n}).

For each i, we consider V_{i}/E = ρ^{−1}(U ∩ H_{i}). Then we have G/E = V_{1}/E × ⋯ × V_{n}/E and, for each i ≠ j, we have [V_{i}, V_{j}] ≤ E. Since V_{i}/E is connected for each i and since E is finite, the Zil’ber’s Indecomposability Theorem [4, Corollary 5.29] shows that [V_{i}, V_{j}] = 1 for each i ≠ j. Thus G is the central product of V_{1}, …, V_{n}, and there is a definable epimorphism f from V_{1} × ⋯ × V_{n} to G with a finite kernel. Hence, by Corollary 5.13, we have just to prove that V_{i}^{∘} is definably affine over K_{i} for each i. But, for each i, the quotient V_{i}^{∘}/(V_{i}^{∘} ∩ E) is definably linear over K_{i}, hence Proposition 5.14 provides the result. □

Lemma 5.16. – Let G be a group of finite Morley rank. If G is residually definably affine over interpretable fields K_{1}, …, K_{n}, then G is definably affine over K_{1}, …, K_{n}.

Proof – We find finitely many normal definable subgroups S_{1}, …, S_{m} of G such that ∩_{i=1}^{m}S_{i} = 1 and such that G/S_{i} is definably affine over K_{1}, …, K_{n} for each i. Then G definably embeds in (G/S_{1}) × ⋯ × (G/S_{m}), and the result follows. □

Corollary 5.17. – Any ACF -group G has a smallest normal definable subgroup W (G) such that G/W (G) is definably affine.

Furthermore, if G is an ACF_{0}-group, then W (G) is connected.

Proof – The existence of W (G) follows from Lemma 5.16. Since G/G^{∘} is definably linear over any finite field, we have W (G) ≤ G^{∘}. In particular, W (G)/W (G)^{∘} is a central subgroup of G^{∘}/W (G)^{∘}, and it is abelian. Now, by Corollary 5.13 and Theorem 5.15, we obtain W (G) = W (G)^{∘}, and this equality finishes the proof. □

6. More on pseudo-tori

In this section, we consider an algebraic group G over \overline{ℚ}, and its pseudo-tori in the ACF -expansion of the pure group G. Ideally, these subgroups are the tori of G. However, even if G is centerless and connected, the maximal tori of the pure group G may be not definable (Example 3.1 (2)).

Our first result concerns the ACF -expansion of the pure group G, when G is a connected centerless algebraic group over \overline{ℚ}, with G solvable of class two (Proposition 6.2). We should note that, by the main theorem of [8], such a group interprets finitely many connected, solvable of class two and centerless algebraic groups G_{1}, …, G_{n} over algebraically closed fields K_{1}, …, K_{n} respectively, in such a way that G imbeds in G_{1} × ⋯ × G_{n}. However, contrary to what is announced, the embedding in [8] is not necessarily definable. Indeed, otherwise the group G in Example 3.1 (2) would be definably linear, and since it is not a direct product of two proper subgroups, it would be definably linear over one interpretable field (Lemma 5.6), contradicting Example 3.1 (2).

For the proof of Proposition 6.2 and of the main result of this section, that is Theorem 6.3, we need Carter subgroups. These subgroups are defined in any group of finite Morley rank as being definable, connected, nilpotent, and of finite index in their normalizers. They have turned out to be increasingly useful in the analysis of groups of finite Morley rank. In the algebraic groups over an algebraically closed field, and when the language is the one of the pure fields, these subgroups are precisely Cartan subgroups, namely the connected component of centralizers of the maximal tori. The following fact is a summarize, in the solvable context, of their properties useful for us. We refer to [13] for more details on Carter subgroups.

Fact 6.1. – Let G be a connected solvable group of finite Morley rank, and let N be a normal definable subgroup of G. Then the following conditions are satisfied:

(i) [12, Corollary 2.10] any pseudo-torus of G lies in a Carter subgroup of G; (ii) [13, Theorem 3.11] its Carter subgroups are self-normalizing; (iii) [13, Corollary 3.3] the Carter subgroups of G/N are exactly of the form

QN/N, with Q a Carter subgroup of G; (iν) [13, Proposition 3.19] if G is 2-solvable, then G has a definable connected

characteristic abelian subgroup A of G such that G = A⋊C for every Carter subgroup C of G.

Proposition 6.2. – Let G be an algebraic group over \overline{ℚ}. If G is connected, centerless, and solvable of class two, then the ACF -expansion of the pure group G is definably linear.

Proof – We notice that, since G is connected and centerless, F (G) is torsionfree. We suppose toward a contradiction that there exists a counterexample G of minimal Morley rank. For i = 1, 2, let A_{i} be a G-minimal subgroup, let Z_{i}/A_{i} be the center of G/A_{i}, and let W_{i} be the hypercenter of G/A_{i}. Since G is connected, W_{i} is definable for i = 1, 2 [16, Corollaire 3.15]. Moreover, since F (G) is torsionfree, A_{1} and A_{2} are torsion-free too. Thus, if we have A_{1} ≠ A_{2}, then A_{1} ∩ A_{2} is trivial, and Z_{1} ∩ Z_{2} ≤ Z(G) is trivial too, so W_{1} ∩ W_{2} is trivial. In this case, since G/W_{1} and G/W_{2} are definably affine by induction hypothesis, G is definably affine (Lemma 5.16). Hence G has a unique G-minimal subgroup A.

Let C be a Carter subgroup of G. By Fact 6.1 (iii), we have G = G′C. Then Z(C)∩C_{C}(G′) is central in G, so C_{C}(G′) is trivial. Moreover, since G is 2-solvable, C′ centralizes G′, so C is abelian. Hence C ∩ G′ is central in G, and we obtain G = G′ ⋊ C.

We show that, if A(G′) denotes the largest connected definably affine subgroup of G′ (Corollary 5.10), we have A ≤ A(G′) < G′, and that C/C_{C}(A(G′)) is definably linear over one interpretable field. By Facts 2.1, we have A ≤ A(G′). Moreover, since G′ ≤ F (G) is torsion-free, A(G′) is definably linear by Corollary 5.8. So the uniqueness of A and Lemma 5.6 show that A(G′) is definably linear over one interpretable field K. Moreover, since G′ is torsion-free, K is of characteristic zero and A(G′) is definably isomorphic to a unipotent algebraic group over K. Consequently, Corollary 4.8 says that C/C_{C}(A(G′)) is definaly linear over K. Therefore, if A(G′) = G′, since C_{C}(G′) is trivial, then C is definably linear over K. But, since G is solvable, we have G′H < G for each normal proper subgroup H of G, so the conjugates of C generate G. Hence the condition A(G′) = G′ implies that G is definably affine over K by Theorem 5.9. Thus we obtain A(G′) < G′.

We claim that G has a unique maximal proper normal connected definable subgroup. Indeed, suppose that N_{1} and N_{2} are two distinct maximal proper normal connected definable subgroups of G. In particular, N_{1} and N_{2} contain G′, and G is generated by N_{1} and N_{2}. We may assume that A is noncentral in N_{2}. Since Z(N_{2}) ≤ F (G) is torsion-free, and since it is normal in G and does not contains A, it is trivial. Hence N_{2} is definably linear by induction hypothesis. Moreover, by the maximality of N_{2} and since G is solvable, N_{2} contains G′, therefore G′ is definably linear. This contradicts the previous paragraph, so G has a unique maximal proper normal connected definable subgroup N.

We show that N = F (G). Indeed, we have F (G) ≤ N by the uniqueness of N. Let \overline{B} be a G-minimal section of G′. If C centralizes \overline{B}, then C covers \overline{B} by Fact 6.1 (iii), contradicting G′ ∩ C = 1. So Fact 2.1 says that G/C_{G}(\overline{B}) ≃ C/C_{C}(\overline{B}) is definably isomorphic to a subgroup of L^{∗} for an interpretable algebraically closed field L. But L and \overline{ℚ} are definably isomorphic in the pure field \overline{ℚ} (Fact 2.2), hence L^{∗} has no proper connected definable subgroup, and C_{G}(\overline{B})^{∘} is a maximal proper connected definable subgroup of G. Then, by the uniqueness of N, we obtain N = C_{G}(\overline{B})^{∘}, and N centralizes each G-minimal section of G′. This implies that N is nilpotent, so N = F (G).

We show that A = A(G′). We note that, since G is centerless and since A(G′) contains A, we have C_{G}(A(G′)) < G. So, by the uniqueness of N, the quotient NC_{G}(A(G′))/C_{G}(A(G′)) is the unique maximal proper normal connected definable subgroup of G/C_{G}(A(G′)). But G/C_{G}(A(G′)) is abelian since G′ is abelian, and G/C_{G}(A(G′)) ≃ C/C_{C}(A(G′)) is definably linear over K. Hence, since K and \overline{ℚ} are definably isomorphic in the pure field \overline{ℚ} (Fact 2.2), the quotient G/C_{G}(A(G′)) has dimension one over \overline{ℚ}, and the uniqueness of N gives N = C_{G}(A(G′))^{∘}. In particular, since the unipotent part of G is contained in F (G) = N, it centralizes A(G′), and G/C_{G}(A(G′)) is a torus. Moreover, A(G′) is a closed torsion-free abelian subgroup of G, so it is a \overline{ℚ}-vector space, and there is a base (a_{1}, …, a_{n}) of A(G′) such that G normalizes \overline{ℚ} · a_{i} for each i. Then, since A(G′) is definably linear over K and since K and \overline{ℚ} are definably isomorphic in the pure field \overline{ℚ}, the subgroups \overline{ℚ}·a_{i} are definable and normal in G for each i. So we obtain n = 1 by the uniqueness of A, and A(G′) is definably isomorphic to K_{+}. Now Fact 2.3 gives A(G′) = A, as desired.

We show that AC is the unique maximal proper connected definable subgroup of G containing C. Indeed, if G = AC, then we have G′ = A = A(G′), contradicting A(G′) < G′. Let M be a maximal proper connected definable subgroup M of G containing C. Since AC is proper in G, we have C < M. Then M ∩ G′ is a normal infinite definable subgroup of G, and we have A ≤ M by the uniqueness of A. Let Z be the hypercenter of M. It is definable by [16, Corollaire 3.15], and it is contained in C by Fact 6.1 (ii). In particular, Z ∩ G′ is trivial. Moreover, by induction hypothesis, M/Z is definably linear, so M ∩ G′ is definably linear, and the previous paragraph gives M ∩ G′ ≤ A(G′) = A. Now we have M = AC, as claimed.

From now on, we can prove that the maximal tori of G are definable. Indeed, by the maximality of AC and since A < G′, the quotient G′/A is C-minimal. Since G = G′ ⋊ C, Fact 6.1 (iii) says that C does not centralize G′/A, and Fact 2.1 provides an infinite interpretable field K_{1} such that G′/A is definably isomorphic to (K_{1})_{+}. Then Fact 2.2 shows that G′/A is isomorphic to \overline{ℚ}_{+}, definably in \overline{ℚ}. Since A ≃ \overline{ℚ} · a_{1} is isomorphic to \overline{ℚ}_{+}, definably in \overline{ℚ} too, G′ is isomorphic to a \overline{ℚ}-vector space of dimension two, definably in \overline{ℚ}. Thus, since C_{C}(G′) is trivial, C is isomorphic, definably in \overline{ℚ}, to an abelian connected closed subgroup of

\{( \begin{matrix}s & a \\ 0 & t \end{matrix} ) | (s, t) ∈ (\overline{ℚ}^{∗})^{2}, a ∈ \overline{ℚ}\} .

In particular, C has dimension at most two over \overline{ℚ}. Since C/C_{C}(A) is definably linear over K, Proposition 5.14 says that C/C_{C}(A)^{∘} is definably linear over K too. Thus, since the conjugates of C generate G, and since G is not definably affine over K, Theorem 5.9 shows that C_{C}(A)^{∘} is nontrivial, so C has dimension exactly two over \overline{ℚ}. Moreover, C_{C}(A)^{∘} ≤ N = F (G) is torsion-free, so it is a unipotent subgroup, and C is not a torus. This implies that C is isomorphic, definably in \overline{ℚ}, to

\{( \begin{matrix}t & a \\ 0 & t \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, a ∈ \overline{ℚ}\} .

Consequently, G = G′ ⋊ C is isomorphic, definably in \overline{ℚ}, to

\{( \begin{matrix}t & a & u \\ 0 & t & v \\ 0 & 0 & 1 \end{matrix} ) | t ∈ \overline{ℚ}^{∗}, (a, u, v) ∈ \overline{ℚ}^{3}\} ,

and Lemma 3.7 and Proposition 3.11 say that the maximal tori of G are definable.

Let T be the maximal torus of C. Since C ∩ N ≤ F (G) is torsion-free, C ∩ N is a unipotent subgroup. On the other hand, the unipotent part of C is contained in the one of G, which is contained in F (G) = N, so we obtain C = (C ∩ N) × T. Actually, since N = F (G) is torsion-free, it is the unipotent part of G, and since G = G′C = NC = N ⋊ T, the torus T is maximal in G. Consequently, by the previous paragraph, it is definable, and G′T is definable too. But, since G = G′ ⋊ C and since C ∩ N = C_{C}(A)^{∘} is nontrivial, G′T is a proper normal connected definable subgroup of G. Hence G′T is contained in N by the uniqueness of N. This contradicts G = N ⋊ T, and finishes the proof. □

From now on, we are going to prove the following fundamental result.

Theorem 6.3. – Let G be a connected solvable algebraic group over \overline{ℚ}. Then T ∩F (G) is central in G for each pseudo-torus T of the ACF -expansion of the pure group G.

First, we have to prove Lemma 6.5. Its proof uses the Frattini subgroup, which is defined for any group G of finite Morley rank, as being the intersection of all the maximal proper definable connected subgroups of G. It is denoted by Φ(G).

Fact 6.4. – Let G be a connected group of finite Morley rank. Then,

(i) [13, Lemma 2.14.b] if H is a definable subgroup of G such that G = Φ(G)H, we have G = H;

(ii) [13, Lemma 2.14.a] Φ(G/Φ(G)) is trivial; (iii) [13, Proposition 3.18] if G is solvable, Φ(G) is nilpotent and F (G)/Φ(G) =

F (G/Φ(G)) is abelian. In particular, G/Φ(G) is 2-solvable, and G is nilpotent if and only if G/Φ(G) is abelian;

(iν) [13, Proposition 6.5] if G is solvable, F (G) ∩ Φ(C) ≤ Φ(G) for each Carter

subgroup C of G; (ν) [11, Lemma 5.4] if G is nilpotent and if Φ(G) is finite, Φ(G/A) is finite for

each normal definable subgroup A of G.

Lemma 6.5. – Let C be a nilpotent group of finite Morley rank, and T a pseudotorus of C. Then, for each connected definable subgroup A of T, AΦ(C)/Φ(C) is a pseudo-torus.

Proof – By Facts 2.5 (iν) and 6.4 (ii), we may assume Φ(C) = 1. In particular C is abelian (Fact 6.4 (iii)). We assume toward a contradiction that A is not a pseudotorus. Since T is abelian and radicable, then A is abelian and radicable too, and it has a definable normal subgroup B such that A/B is definably isomorphic to K_{+} for an interpretable field K. Moreover, since A is radicable, the characteristic of K is zero, and A/B has no nontrivial proper definable subgroup (Fact 2.3). Since Φ(C/B) is finite (Fact 6.4 (ν)), there is a maximal proper connected definable subgroup M/B of C/B does not containing A/B. In particular, since A/B has no nontrivial proper definable subgroup, we have A ∩ M = B. Moreover, by maximality of M/B, we have C = MA = MT, and K_{+} ≃ A/B = A/(A ∩ M) is definably isomorphic to T/(T ∩ M), contradicting that T is a pseudo-torus. This finishes the proof. □

From now on, we can prove Theorem 6.3.

Proof – [Proof of Theorem 6.3] We assume toward a contradiction that G is a counterexample of minimal Morley rank and with Z(G) of minimal Morley degree. We may assume that T is a maximal pseudo-torus of G. We show that no proper normal definable subgroup of G contains T. Indeed, suppose toward a contradiction that M is a proper normal definable subgroup of G containing T. Then M^{∘} contains T since T is connected, and T ∩F (M^{∘}) is central in M^{∘} by the minimality of rk(G). But we have G = MN_{G}(T) by Fact 2.5 (i) and a Frattini Argument, therefore by Fact 2.5 (ii) and since G is connected, we find G = M^{∘}C_{G}(T). Hence we obtain T ∩ F (M^{∘}) ≤ Z(G). Since M is normal in G, we have F (M^{∘}) = M^{∘} ∩ F (G), so T ∩ F (G) is central in G, contradicting the choice of G. This proves that no proper normal definable subgroup of G contains T. In particular, we have G = G′T.

We show that Z(G) = 1. By Fact 2.5 (iν), T Z(G)/Z(G) is a pseudo-torus of G/Z(G). Consequently, if Z(G) is infinite, the minimality of rk(G) implies that (T ∩ F (G))Z(G)/Z(G) is central in G/Z(G), so C_{G}(t) is a normal subgroup of G for each t ∈ T ∩ F (G). Now the previous paragraph gives G = C_{G}(t) for each t ∈ T ∩ F (G), that is T ∩ F (G) ≤ Z(G), which contradicts our hypothesis. Thus Z(G) is finite and, since G is connected, G/Z(G) is centerless. If Z(G) ≠ 1, the minimality of the Morley degree of Z(G) yields (T ∩ F (G))Z(G)/Z(G) = 1 and T ∩ F (G) ≤ Z(G), contradicting our hypothesis on G, so G is centerless.

We show that F (G) is torsion-free. Indeed, for each prime p, since \overline{ℚ} has characteristic zero and since F (G) is nilpotent, the subset F_{p} of the elements of order p in F (G) is finite. Then, since G is connected and normalizes F_{p}, it centralizes F_{p}, and we obtain F_{p} = ∅ because Z(G) = 1. This proves that F (G) is torsion-free. Furthermore, this implies that F (G) is a unipotent subgroup of G.

By Fact 6.1 (i), T is contained in a Carter subgroup C of G. We show that Φ(C) contains T ∩ F (G). Since F (G) is torsion-free, T ∩ F (G) is connected. By Lemma 6.5, the group (T ∩ F (G))Φ(C)/Φ(C) is a pseudo-torus, and Fact 2.5 (iν) yields a pseudo-torus T_{0} of T ∩ F (G) such that T_{0}Φ(C) = (T ∩ F (G))Φ(C). By Fact 2.5 (iii), we have T_{0} ≤ Z(G) = 1, so (T ∩ F (G))Φ(C)/Φ(C) = 1 and T ∩ F (G) is contained in Φ(C). In particular, T ∩ F (G) is contained in Φ(G) (Fact 6.4 (iν)).

We show that T is contained in a unique maximal proper definable connected subgroup M of G. Indeed, since T < G, the subgroup M exists. Moreover, we have T ∩ F (M) ≤ Z(M) by the minimality of rk(G). Since M contains Φ(G) by the definition of Φ(G), and since Φ(G) is nilpotent (Fact 6.4 (iii)), we have Φ(G) ≤ F (M) and the previous paragraph yields T ∩F (G) ≤ T ∩F (M) ≤ Z(M). Thus, if T is contained in two distinct maximal proper definable connected subgroups M_{1} and M_{2} of G, then T ∩F (G) centralizes ⟨M_{1}, M_{2}⟩ = G, contradicting T ∩F (G) ≰ Z(G). Hence we obtain the uniqueness of M.

We consider Z/G′′ = Z(G/G′′), \overline{G} = G/Z and \overline{T} = T Z/Z. We show that \overline{G} = \overline{G}′ ⋊ \overline{T} and that \overline{G} is centerless. If G′′ = 1 then, since G is centerless, we have Z = 1 and Z(\overline{G}) = 1. Moreover G′ and T are abelian and, since the first paragraph gives G = G′T, we obtain G′ ∩ T ≤ Z(G) = 1, so G = G′ ⋊ T and \overline{G} = \overline{G}′ ⋊ \overline{T}. Hence we may assume G′′ ≠ 1. In particular, the minimality of rk(G) yields T G′′/G′′ ∩ F (G/G′′) ≤ Z/G′′. But Fact 6.4 (iii) gives G′′ ≤ Φ(G) ≤ F (G) and F (G/Φ(G)) = F (G)/Φ(G), hence we have F (G/G′′) = F (G)/G′′ and we obtain T ∩ F (G) ≤ Z. Also, the equalities Z/G′′ = Z(G/G′′) and F (G/G′′) = F (G)/G′′ imply Z ≤ F (G) and F (\overline{G}) = F (G)/Z, so we have \overline{T} ∩ F (\overline{G}) = 1. In particular, since F (\overline{G}) contains the abelian subgroup \overline{G}′, we have \overline{G}′ ∩ \overline{T} = 1, and since the first paragraph gives G = G′T, we obtain \overline{G} = \overline{G}′ ⋊ \overline{T}. Now, since \overline{G}′ ≤ F (\overline{G}) and since \overline{T} ∩ F (\overline{G}) = 1, we find \overline{G}′ = F (\overline{G}) = F (G)/Z. Moreover, \overline{C} = CZ/Z is a Carter subgroup of \overline{G} (Fact 6.1 (iii)), and Fact 6.1 (iν) provides a definable connected characteristic abelian subgroup \overline{A} = A/Z in \overline{G} such that \overline{G} = \overline{A} ⋊ \overline{C}. Since Fact 2.5 (iii) says that \overline{T} is central in \overline{C}, the definable subgroup AT is normal in G and the first paragraph gives G = AT. Thus we obtain \overline{C} = \overline{T}. Since G′ is torsion-free, Z(\overline{G}) ≤ F (\overline{G}) = \overline{G}′ is torsion-free too. In particular, since Z(\overline{G}) normalizes the Carter subgroup \overline{C}, we obtain Z(\overline{G}) ≤ \overline{C} ∩ F (\overline{G}) = \overline{T} ∩ F (\overline{G}) = 1. This proves that \overline{G} is a centerless 2-solvable group.

We show that \overline{G}′ has no nontrivial proper definable subgroup. Fact 6.4 (iii) gives G′′ ≤ Φ(G) ≤ M, so M is normal in MZ, and \overline{M} = MZ/Z is proper in \overline{G} by the first paragraph. Then \overline{M} is the unique maximal proper definable connected subgroup of \overline{G} containing \overline{T}. In particular \overline{G}′ has no decomposition of the form \overline{G}′ = \overline{A_{1}} \overline{A_{2}} for two proper definable connected subgroups \overline{A_{1}} and \overline{A_{2}} of \overline{G}′, with \overline{A_{1}} and \overline{A_{2}} normal in \overline{G}, otherwise \overline{M} would contain \overline{A_{i}} \overline{T} for i = 1, 2. Then Lemma 5.6 and Proposition 6.2 show that \overline{G} is definably linear over an interpretable field K. Thus, since K and \overline{ℚ} are isomorphic, definably in \overline{ℚ} (Fact 2.2), each closed subgroup of \overline{G} is definable, and the pseudo-tori of \overline{G} are tori. Moreover, in the pure field \overline{ℚ}, since G′ is torsion-free, \overline{G}′ is a \overline{ℚ}-vector space. Thus, in \overline{ℚ}, since \overline{T} is a torus acting on \overline{G}′, there is a basis (v_{1}, ⋯, v_{n}) of \overline{G}′, where n is the dimension of \overline{G}′ over \overline{ℚ}, such that \overline{T} normalizes \overline{ℚ} · v_{i} for each i. Then, by the uniqueness of \overline{M}, we obtain n = 1 and \overline{G}′ = \overline{ℚ} · v_{1}. Now Fact 2.3 says that \overline{G}′ has no nontrivial proper definable subgroup.

We show that G′/G′′ has no nontrivial proper definable subgroup. By Fact 6.1 (iii) and (iν), G/G′′ has a definable connected characteristic subgroup B/G′′ such that G/G′′ = B/G′′⋊CG′′/G′′. Since T is central in C (Fact 2.5 (iii)), the subgroup BT is normal in G and the first paragraph gives G = BT, so CG′′/G′′ = T G′′/G′′. In particular B contains G′. Since Fact 6.1 (iii) says that G = G′C, we obtain B = G′. Moreover, Z/G′′ = Z(G/G′′) normalizes CG′′/G′′ therefore, since CG′′/G′′ is a Carter subgroup of G/G′′ (Fact 6.1 (iii)), we obtain Z ≤ CG′′ (Fact 6.1 (ii)). Now Z/G′′ ∩G′/G′′ is trivial, and G′/G′′ is definably isomorphic to \overline{G}′. Thus G′/G′′ has no nontrivial proper definable subgroup. Since G is a solvable connected algebraic group, G′ is nilpotent, and we have Φ(G′) = G′′ (Fact 6.4 (iii)). Let x ∈ G′ \ G′′ and let X := d(x) be the smallest definable subgroup of G′ containing x. Then X is abelian and, since G′/G′′ has no nontrivial proper definable subgroup, we have G′ = XG′′ = XΦ(G′). Thus Fact 6.4 (i) gives G′ = X. In particular, G′ is abelian and G′′ = 1. Now G′ has no nontrivial proper definable subgroup. Since G = G′T and since F (G) contains G′, we have F (G) = G′(T ∩ F (G)). Consequently, either F (G) = T ∩ F (G), or T ∩ F (G) is a maximal proper definable subgroup of F (G). In the first case, we obtain G′ ≤ T and, since G = G′T, we have G = T and G is abelian, contradicting our choice of G. In the second case, T ∩ F (G) is normal in F (G). But, since T ∩ F (G) ≰ Z(G), we have T ∩ F (G) ≠ 1 and T ∩ Z(F (G)) is nontrivial. Hence, since we have G = G′T = F (G)T, the subgroup T ∩ Z(F (G)) is central in G, contradicting that G is centerless. This finishes the proof. □

7. Analysis of G/Z(G) and of G′

The purpose of this section is to prove that, if G is a connected algebraic group over \overline{ℚ}, then G/Z(G) is definably linear and G′ is definably affine, in the ACF - expansion of the pure group G (Theorems 7.7).

For any group G of finite Morley rank, we denote by S(G) the connected component of the intersection of the maximal pseudo-tori of G, and by T (G) the subgroup of G generated by its pseudo-tori.

Remark 7.1. –

• By the Zilber’s Indecomposability Theorem, T (G) is a connected definable subgroup of G, for any group G of finite Morley rank.

• If G is an ACF -group, then T (G) contains all the tori of G.

• Moreover, S(G) is central in G^{∘}.

Indeed, it is central in T (G), and the conjugacy of the maximal pseudo-tori of G (Fact 2.5 (i)) and a Frattini Argument provide G^{∘} = T (G)N_{G^{∘}}(T) for each maximal pseudo-tori T of G. So, by Fact 2.5 (ii), we have S(G) ≤ Z(G^{∘}).

In any group G of finite Morley rank, we denote by Q(G) the quasiunipotent radical, which is the largest normal connected definable nilpotent subgroup of G with no nontrivial radicable torsion subgroup. This subgroup was introduced in [2]. Ideally, in any ACF -group, this subgroup is the unipotent radical. However, this fails for some abelian connected groups, since the pure group \overline{ℚ}_{+} × \overline{ℚ}^{∗} is abstractly isomorphic to \overline{ℚ}^{∗}, so its quasiunipotent radical is trivial.

Lemma 7.2. – In any connected ACF_{0}-group G, we have

G/S(G) = Q(G/S(G)) · T (G)/S(G).

Proof – By Fact 2.5 (iν), we may assume S(G) = 1. Let K be an algebraically closed field of characteristic zero such that G is interpretable in K. We may assume that G is an algebraic group over K. In particular, T (G) is an algebraic subgroup of G containing each torus of G.

By Lemma 4.1, the algebraic group G is connected. We denote by G_{L} its largest connected linear algebraic subgroup. In particular, G/G_{L} is an abelian variety, so it is G/G_{L} is a pseudo-torus, and it is covered by T (G) (Fact 2.5 (iν)). Moreover, we have G_{L} = U ⋊S for S a maximal reductive subgroup of G_{L} and U its unipotent radical. Since S is generated by its tori, we obtain S ≤ T (G).

Now let Q := d(U) denote the smallest definable subgroup of G containing U. Since U is a unipotent group, Q is radicable and nilpotent, so Q is connected and nilpotent. We show that Q = U. By Fact 2.5 (iii), Q has a unique maximal pseudo-torus T. In particular T is normal in G and it is contained in each maximal pseudo-torus of G, so we have T ≤ S(G) = 1. Thus Q is quasiunipotent and it is contained in Q(G). Hence G is contained in Q(G)T (G), and we obtain the result. □

Proposition 7.3. – Let G be an algebraic group over \overline{ℚ}. Then, in the ACF - expansion of the pure group G, the quotient T (G)/S(G) is definably affine.

Proof – We may assume G = T (G) and, by Fact 2.5 (iν), we may assume S(G) = 1. Let T be a maximal pseudo-torus of G containing a maximal torus of G, and let B be a Borel subgroup of G, in the algebraic sense, with B containing T. Since S(B) is central in B (Remark 7.1), it is central in G too, and it is contained in S(G) = 1 by the conjugacy of the maximal pseudo-tori (Fact 2.5 (i)). Moreover, Fact 2.1 says that, for each minimal infinite definable T -normal section \overline{A} of B, the quotient T/C_{T} (\overline{A}) is definably affine. Thus, W (T)F (B)^{∘} is nilpotent, and since B′ is nilpotent and connected, W (T)F (B)^{∘} is contained in F (B). Now Theorem 6.3 implies that W (T) is central in B. Thus, by the conjugacy of the maximal pseudo-tori in B (Fact 2.5 (i)), the subgroup W (T)^{∘} is contained in S(B) = 1, and T is definably affine by Corollary 5.17. Now Theorem 5.9 yields the result. □

Lemma 7.4. – Let H be a normal connected definable subgroup of an ACF -group G. If H is definably affine, then G/C_{G}(H) is definably affine too.

Proof – Let K_{1}, …, K_{n} be interpretable fields such that H is definably affine over K_{1}, …, K_{n}, and such that K_{i} is not definably isomorphic to K_{j} for each i ≠ j. For each i, we denote by H_{i} the largest connected definable subgroup of H definably affine over K_{i} (Theorem 5.9). These subgroups are normal in G. By Proposition 5.11, H is generated by H_{1}, …, H_{n}. Hence, by Lemma 5.16, we may assume that H is definably affine over one interpretable field K. Moreover, we may assume that K is infinite.

Let L be an algebraically closed field such that G is interpretable in the pure field L. By Fact 2.2, the fields K and L are isomorphic, definably in L. We consider H⋊G/C_{G}(H), where G/C_{G}(H) acts on H by conjugation. It is isomorphic to an algebraic group over L, definably in L. Since K and L are isomorphic, definably in L, we find an isomorphism φ, definable in L, from H ⋊ G/C_{G}(H) to an algebraic group R over K. Since H is definably affine over K and since K and L are isomorphic, definably in L, the map φ_{|H} : H → φ(H) is definable in G. Moreover, since K and L are isomorphic, definably in L, the subgroup V = φ(G/C_{G}(H)) is algebraic over K, and it is definable in G. Now the semidirect product R = φ(H) ⋊ V is definable in G. Thus, since φ_{|H} is definable in G, the graph Δ = \{(\overline{g}, v) ∈ G/C_{G}(H) × V | ∀h ∈ H, φ_{|H}(\overline{g} · h) = v · φ_{|H}(h)\} of φ_{|G/C_{G}(H)} : G/C_{G}(H) → V is definable in G too. From now on, G/C_{G}(H) is definably isomorphic to an algebraic group V over K.

We verify that V is affine. Since the ACF -group H is connected, and since φ(H) and φ_{|H} are definable in G, then φ(H) is connected in G too, and φ(H) is connected in the pure field K. Thus V is an algebraic group acting faithfully over a connected algebraic group, so it is affine by the Rosenlicht’s Theorem [17, §5, Théorème 13]. This implies that G/C_{G}(H) is definably affine over K. □

Corollary 7.5. – Let G be an algebraic group over \overline{ℚ}. Then, in the ACF -expansion of the pure group G, the quotient G/C_{G}(T (G)) is definably affine.

Proof – By Corollary 5.13, we may assume that G is connected. In particular, S(G) is central in G. We consider C = C_{G}(T (G)/S(G)). Since [T (G), C] is contained in S(G), each maximal pseudo-torus of G is normalized by C and, by Fact 2.5 (ii), it is centralized by C^{∘}. This implies that C^{∘} centralizes T (G), so C/C_{G}(T (G)) is a finite group. By Proposition 7.3 and Lemma 7.4, the quotient G/C is definably affine. Then we obtain the result by Theorem 5.15. □

Lemma 7.6. – Let H be a normal definable subgroup of a connected ACF -group G. If G/C_{G}(H) is definably affine over the fields K_{1}, …, K_{n}, then [G, H] is definably affine over K_{1}, …, K_{n} too.

Proof – We consider the semi-direct product R = H ⋊ G/C_{G}(H), where G/C_{G}(H) acts on H by conjugation. Then R is an ACF -group, and the subgroup S of R generated by the conjugates of G/C_{G}(H) is definably affine over K_{1}, …, K_{n} (Theorem 5.9). But S contains the definable subgroup [G, H], so [G, H] is definably affine over K_{1}, …, K_{n}. □

Then we obtain a result on the abstract structure of the algebraic groups over \overline{ℚ}, and this one is fundamental for us.

Theorem 7.7. – Let G be a connected algebraic group over \overline{ℚ}. Then, in the ACF - expansion of the pure group G, the quotient G/Z(G) is definably linear, and G′ is definably affine.

Furthermore, if U/Z(G) (resp. V/Z(G)) is the largest connected subgroup of G/Z(G), definably linear over one interpretable field K (resp. L), and if K and L are not definably isomorphic, then [U, V] is trivial and U′ ∩ V′ is finite.

Proof – We show that G/Z(G) is definably affine. Let Q be the preimage of Q(G/S(G)) in G. Since S(G) is central in G, the group Q is nilpotent, and Q/Z(Q) is definably linear by Theorem 4.3. Moreover, by Corollary 7.5, the quotient Q/C_{Q}(T (G)) is definably affine, so QZ(G)/Z(G) is definably affine by Lemma 7.2. Since Proposition 7.3 says that T (G)/S(G) is definably affine, T (G)Z(G)/Z(G) is definably affine too. Hence G/Z(G) is definably affine by Theorem 5.9 and Lemma 7.2. Furthermore, Lemma 7.6 says that G′ is definably affine.

We show that G/Z(G) is definably linear. By Corollary 5.8, G/Z(G) has a finite normal subgroup E/Z(G) such that G/E is definably linear. In particular, G centralizes E/Z(G), and E is nilpotent. We consider a bounded exponent subgroup B_{0} of E covering E/Z(G). Then B_{0} is contained in a characteristic bounded exponent subgroup B_{1} of E, and B_{1} is finite since \overline{ℚ} is of characteristic zero. Hence B_{1} is central in G, and E/Z(G) is trivial. This proves that G/Z(G) is definably linear.

By Lemma 7.6, the groups [G, U] and [G, V] are definably affine over K and L respectively, so they are definably linear over K and L respectively. By Proposition 5.11, this implies that U′ ∩ V′ ≤ [G, U] ∩ [G, V] and [U, V] ≤ [G, U] ∩ [G, V] are finite. Moreover, we have [U, V] = [U^{∘}Z(G), V] = [U^{∘}, V], so [U, V] is connected, and [U, V] is trivial, as desired. □

8. Canonical development of G

In this section, we prove Theorem 1.1 in the special case K = \overline{ℚ} (Lemma 8.10 and Theorem 8.14).

For the rest of this section, we fix a nontrivial connected algebraic group G over \overline{ℚ}, and we consider the ACF -expansion of the pure group G. By Theorem 7.7, we find interpretable fields K_{1}, …, K_{n} such that K_{i} is not definably isomorphic to K_{j} for each i ≠ j, and such that G/Z(G) is definably linear over K_{1}, …, K_{n}. By Lemma 5.6 and Proposition 5.11, if for each i we denote by G_{i}/Z(G) the largest connected subgroup of G/Z(G) definably linear over K_{i}, the quotient G/Z(G) is the direct product of the subgroups G_{i}/Z(G), and G′_{i} = (G_{i}^{∘})′ is definably linear over K_{i} for each i (Lemma 7.6). Moreover, Theorem 7.7 says that G is the central product of G_{1}, …, G_{n}, and we may assume that G_{i}/Z(G) is nontrivial for each i.

First we consider a purely algebraic lemma.

Lemma 8.1. – Let G̃ be a connected affine algebraic group over an algebraically closed field K of characteristic zero. Then, for each positive integer n, up to isomorphism of algebraic groups, there is a unique connected affine algebraic extension G^{∗} of G̃, with a normal finite subgroup X of exponent dividing n, and satisfying the following property:

for each connected affine algebraic group H over K, if H has a normal finite subgroup E of exponent dividing n such that G̃ and H/E are algebraically isomorphic, there is an algebraic surjective homomorphism γ : G^{∗} → H such that γ(X) = E.

Proof – In this proof, any couple (H, E), where H is a connected affine algebraic group over K and E a normal finite subgroup of H of exponent dividing n, is said to be an n-extension of G̃ if G̃ and H/E are isomorphic as algebraic groups. We consider the family ℰ of the n-extensions of G̃ modulo the equivalence relation R defined by the following assertion: “if (H_{1}, E_{1}) and (H_{2}, E_{2})) are two n-extensions of G̃, we denote by (H_{1}, E_{1})R(H_{2}, E_{2}) the existence of an algebraic isomorphism γ : H_{1} → H_{2} such that γ(E_{1}) = E_{2}.” We identify any n-extension of G̃ with its class modulo R. Moreover, we consider the order relation ≤ on ℰ defined by the following assertion: “if (H_{1}, E_{1}) and (H_{2}, E_{2})) are two elements of ℰ, we denote by (H_{1}, E_{1}) ≤ (H_{2}, E_{2}) the existence of an algebraic surjective homomorphism γ : H_{2} → H_{1} such that γ(E_{2}) = E_{1}.”

First we notice that, for each element (H, E) of ℰ, the order of E is at most n^{r}, where r denotes the Lie rank of the algebraic group G̃. Indeed, since E is finite and since G̃ and H/E are isomorphic as algebraic groups, the rank of H is r too. But the field K has characteristic zero, and E is finite and normal in H, so E is contained in any maximal torus of H. Consequently, since the exponent of E divides n, the order of E is at most n^{r}.

Now we have just to prove that, if (M, E) and (N, F) are two maximal elements of ℰ, then (M, E)R(N, F). Since M/E and N/F are isomorphic to G̃ as algebraic groups, we may consider a subgroup Δ of M × N such that Δ/(E × F) is the graph of an isomorphism of algebraic groups from M/E to N/F. We consider D = Δ^{∘} and A = D ∩(E ×F). Then D, M and N have the same dimensions over K, and D ∩M and D ∩ N are contained in A. We consider the projection maps ρ_{M} : M × N → M and ρ_{N} : M × N → N. Since D, M and N have the same dimensions, and since D ∩ M and D ∩ N are contained in A, the images of D by ρ_{M} and ρ_{N} are M and N respectively. Moreover, we have

D ∩ ρ_{M}^{−1}(E) = D ∩ (E × N) = D ∩ (Δ ∩ (E × N)) = D ∩ (E × F) = A

and, in the same way, D ∩ ρ_{N}^{−1}(F) = A. In particular, (D, A) is an n-extension of G̃ satisfying (M, E) ≤ (D, A) and (N, F) ≤ (D, A). By the maximalities of (M, E) and (N, F), we obtain (M, E)R(D, A) and (N, F)R(D, A). Finally, we find (M, E)R(N, F), and we can take (G^{∗}, X) = (M, E). □

For each i, we consider an algebraic complement C_{i}/G′_{i} of Z(G)^{∘}G′_{i}/G′_{i} in G_{i}^{∘}/G′_{i}. Since G is connected, the torsion part of Z(G) ∩ G′ is finite. Then, for each i, we denote by n_{i} the exponent of the torsion subgroup X_{i} of Z(G)^{∘} ∩G′_{i}, and by \overline{C_{i}}^{∗} the unique connected affine algebraic extension of \overline{C_{i}} := C_{i}/X_{i}, with a normal finite subgroup of exponent dividing n_{i} and satisfying the property of Lemma 8.1.

Moreover, we fix an algebraic complement Z to (Z(G)^{∘} ∩ G′)^{∘} in Z(G)^{∘}. We note that, since G is the central product of G_{1}, …, G_{n}, the group G′ is the central product of G′_{1}, …, G′_{n}, and G is also the central product of Z and C_{1}, …, C_{n}. Now we obtain the following structural result on G.

Lemma 8.2. – The group G has a central finite subgroup F_{0} such that G/F_{0} is the direct product of ZF_{0}/F_{0} and of ×_{k=1}^{n}C_{k}F_{0}/F_{0}.

Proof – Since G′_{j} is definably linear over K_{j} for each j, Lemma 5.6 and Corollary 5.8 imply that the Morley rank of Z(G) ∩ G′ = ∏_{j=1}^{n}(Z(G) ∩ G′_{j}) is ∑_{k=1}^{n} rk(Z(G) ∩ G′_{k}). Then we obtain

∑_{k=1}^{n} rk(C_{k}) = ∑_{k=1}^{n}(rk(G_{k}) − rk(Z(G)^{∘}) + rk(Z(G)^{∘} ∩ G′_{k}))
= ∑_{k=1}^{n}(rk(G_{k}/Z(G)) + rk(Z(G) ∩ G′_{k}))
= rk(G/Z(G)) + rk(Z(G) ∩ G′),

and rk(Z) + ∑_{k=1}^{n} rk(C_{i}) = rk(G/Z(G)) + rk(Z(G)) = rk(G). Thus, Z_{0} = Z ∩ (∏_{k=1}^{n} C_{k}) is finite, and for each j, the group E_{j} = C_{j} ∩ (∏_{k≠j} C_{k}) is finite too. Let F_{0} be the subgroup generated by Z_{0} and the subgroups E_{j}. Then F_{0} is finite and central in G, and G/F_{0} is the direct product of ZF_{0}/F_{0} and of ×_{k=1}^{n}C_{k}F_{0}/F_{0}. □

It is not clear that C_{i} is uniquely determined in the pure group G. However, we will see that \overline{C_{i}}^{∗} is uniquely determined in the pure group G, up to isomorphism of algebraic groups (Corollary 8.9). The proof need the following notion.

Definition 8.3. – A group A is said to be centrally indecomposable if there is no decomposition of A under the form of a central product of two proper subgroups having a finite intersection.

Lemma 8.4. – Let G be a connected affine algebraic group over an algebraically closed field K of characteristic zero. If G is a central product of two proper closed subgroups U and V with U ∩ V finite, then G has a nonstandard automorphism α centralizing U.

Furthermore, we may choose α such that, for any two infinite closed subgroups U_{1} and V_{1} of U and V respectively, the isomorphism β : U_{1}V_{1} → α(U_{1}V_{1}) induced by α is nonstandard.

Proof – First we notice that U and V are infinite since G = UV is connected, and since U and V are closed and proper in G. Let μ be a nontrivial field automorphism of K. Since U ∩ V is finite, we may choose μ as centralizing U ∩ V. We consider the automorphism α of G defined by α(uv) = μ(u)v for each u ∈ U and each v ∈ V. Then α centralizes U.

Let U_{1} and V_{1} be two infinite closed subgroups of U and V respectively. We assume toward a contradiction that the isomorphism β : U_{1}V_{1} → α(U_{1}V_{1}) induced by α is standard. Then we have β = γ ∘ ν for ν a field automorphism of K, and γ an isogeny. We obtain ν(v) = γ^{−1}(v) for each v ∈ V_{1}. Since V_{1} is infinite, this implies that ν is definable in the pure field K. Consequently, β is definable in the pure field K too. Now, since U_{1} is infinite and since we have β(u) = μ(u) for each u ∈ U_{1}, the automorphism μ is definable in the pure field K too. In particular, C_{K}(μ) is a definable subfield of K, so either μ is trivial, or C_{K}(μ) is finite. But the field K has characteristic zero, so μ centralizes an infinite subfield of K. Hence μ is trivial contradicting our choice of μ. □

Lemma 8.5. – For each i, we have C′_{i} = G′_{i} and Z(C_{i})^{∘} ≤ Z(G)^{∘} ∩ C_{i} ≤ C′_{i}.

Proof – Since G_{i} = G_{i}^{∘}Z(G) = C_{i}Z(G), we have C′_{i} = G′_{i}. In particular, since G_{i}^{∘}/G′_{i} is the direct product of C_{i}/G′_{i} and of Z(G)^{∘}G′_{i}/G′_{i}, we obtain Z(G)^{∘} ∩ C_{i} ≤ C′_{i}. Moreover, since G is the central product of G_{1}, …, G_{n}, we have also Z(C_{i})^{∘} ≤ Z(G)^{∘} ∩ C_{i}. □

Proposition 8.6. – For each i, the groups C_{i} and \overline{C_{i}} are centrally indecomposable.

Proof – We assume toward a contradiction that C_{i} is not centrally indecomposable. Let U and V be two proper subgroups of C_{i} having a finite intersection and such that C_{i} is the central product of U and V. Since C_{i} is connected, U and V are infinite. If U is abelian, then U centralizes G_{i}, and since G_{i} centralizes G_{j} for each j ≠ i by Theorem 7.7, we obtain U ≤ Z(G). Then Z(G)^{∘} ∩ U is an infinite subgroup of Z(G)^{∘} ∩ C_{i} ≤ C′_{i} = V′, contradicting that U ∩ V is finite. Hence U is not abelian. In the same way, V is not abelian. In particular, C_{C_{i}}(U) = Z(U)V and C_{C_{i}}(V) = UZ(V) are proper in C_{i}. Moreover, by the connectedness of C_{i}, we find C_{i} = C_{U}C_{V}, for C_{U} := C_{C_{i}}(V)^{∘} and C_{V} := C_{C_{i}}(U)^{∘}, and this product is central. Thus, C_{i} is the central product of two proper connected algebraic subgroups C_{U} and C_{V}.

We consider an algebraic complement R/C′_{U} of (C_{U} ∩ C_{V})^{∘}C′_{U}/C′_{U} in C_{U}/C′_{U}, and an algebraic complement S/C′_{V} of (R ∩ C_{V})^{∘}C′_{V} /C′_{V} in C_{V} /C′_{V}. Then C_{i} is the central product of R and S, and (R ∩ S)^{∘} is contained in C′_{U} ∩ C′_{V} ≤ U′ ∩ V′, therefore R ∩ S is finite. Since R ≤ C_{U} and S ≤ C_{V} are proper in C_{i}, they are infinite. In the same way that for U and V, the subgroups R and S are not abelian, and since they are connected, R′ and S′ are infinite.

By Lemma 8.4, the group C_{i} has a nonstandard automorphism α centralizing R(C_{i} ∩ F_{0}). Moreover, we may choose α such that the automorphism β of G′_{i} = C′_{i} = R′S′ induced by α is nonstandard. Since G is the central product of Z and C_{1}, …, C_{n}, and since C_{i} ∩ (Z ∏_{k≠i} C_{k}) is contained in C_{i} ∩ F_{0}, there is an automorphism μ of G centralizing Z ∏_{k≠i} C_{k} and whose restriction to C_{i} is α. In particular, the restriction of μ to C′_{i} is nonstandard. But C′_{i} = G′_{i} is definably linear over K_{i}, hence this contradicts Fact 3.6, and C_{i} is centrally indecomposable.

Finally, if \overline{C_{i}} is the central product of two proper subgroups U_{1}/X_{i} and V_{1}/X_{i} having a finite intersection, then U_{1} and V_{1} are two proper subgroups of C_{i} having a finite intersection, satisfying C_{i} = U_{1}V_{1}, and such that [U_{1}, V_{1}] ≤ X_{i} is finite and central in C_{i}. Let V_{2} be a subgroup of V_{1} such that C_{i} = U_{1}V_{2}, and such that |U_{1} ∩ V_{2}| is minimal among such subgroups. Then, for each u ∈ U_{1}, the map ad_{u} : V_{2} → X_{i}, defined for each v ∈ V_{2} by ad_{u}(v) = [u, v], is an homomorphism. Moreover, its kernel V_{3} has finite index in V_{2}. But \overline{ℚ} has characteristic zero and C_{i} is connected, so C_{i} has no proper subgroup of finite index. Hence we have C_{i} = U_{1}V_{3}, and V_{3} = V_{2} by the choice of V_{2}. Thus C_{i} is the central product of U_{1} and V_{2}, contradicting the previous paragraphs. Consequently, \overline{C_{i}} is centrally indecomposable. □

Proposition 8.7. – If A is any affine algebraic group over \overline{ℚ} abstractly isomorphic to \overline{C_{i}}, then A is isomorphic to \overline{C_{i}} as algebraic groups.

Proof – We have to find an isomorphism between A and \overline{C_{i}}, definable in the pure field \overline{ℚ}. Let α be an abstract isomorphism between A and \overline{C_{i}}. For each isomorphism α_{0} : A → \overline{C_{i}}, we denote by \overline{α_{0}} the isomorphism between A/Z(A)^{∘} and \overline{C_{i}}/Z(\overline{C_{i}})^{∘}, induced by α_{0}. By Proposition 8.6, the group \overline{C_{i}} is centrally indecomposable, and since it is connected, then Theorem 7.7 implies that, in the ACF -expansion of the pure group \overline{C_{i}}, the quotient \overline{C_{i}}/Z(\overline{C_{i}}) is definably linear over one interpretable field. Then \overline{C_{i}}/Z(\overline{C_{i}})^{∘} is definably linear over one interpretable field too, by Proposition 5.14. Now Fact 3.6 says that the isomorphism \overline{α} : A/Z(A)^{∘} → \overline{C_{i}}/Z(\overline{C_{i}})^{∘} is standard. Thus, there is an automorphism δ of \overline{C_{i}}, induced by a field automorphism of \overline{ℚ}, such that \overline{α} ∘ \overline{δ}^{−1} = \overline{α ∘ δ^{−1}} is an isomorphism of algebraic groups. Let μ = α ∘ δ^{−1}, and let Δ be its graph. Then, if Z_{1} = Z(A)^{∘} × Z(\overline{C_{i}})^{∘}, the group ΔZ_{1} = ΔZ(A)^{∘} = ΔZ(\overline{C_{i}})^{∘} is definable in the pure field \overline{ℚ}. This implies that (ΔZ_{1})′ = Δ′ is definable in the pure field \overline{ℚ} too.

We consider an algebraic complement D/Δ′ of Z_{1}Δ′/Δ′ in ΔZ_{1}/Δ′. Then, since ΔZ_{1}/Z_{1} is the graph of \overline{μ}, we have D ∩ A ≤ ΔZ_{1} ∩ A = Z_{1}. So we obtain

D ∩ A ≤ (D ∩ Z_{1}) ∩ A ≤ Δ′ ∩ A ≤ Δ ∩ A.

Thus, since Δ is the graph of an isomorphism from A to \overline{C_{i}}, we find D ∩ A = 1. In the same way, D ∩ \overline{C_{i}} is trivial. If Z_{C}/X_{i} denotes the center of \overline{C_{i}}, then [C_{i}, Z_{C}] is contained in X_{i}, and it is connected since C_{i} is connected, so Z_{C} = Z(C_{i}). Hence Z(\overline{C_{i}})^{∘} is contained in \overline{C_{i}}′ by Lemma 8.5. This proves that Δ′ contains the graph of the restriction of μ to Z(A)^{∘}, and it implies that Δ′Z_{1} = Δ′Z(A)^{∘} = Δ′Z(\overline{C_{i}})^{∘}, so AD ≥ AΔ′ contains Z_{1}. Now AD contains Δ ≤ ΔZ_{1} = DZ_{1}, and it contains \overline{C_{i}} too since Δ is the graph of μ. Consequently, we have AD = A\overline{C_{i}} and, in the same way, \overline{C_{i}}D = A\overline{C_{i}}. Since D is an algebraic subgroup satisfying D ∩ A = D ∩ \overline{C_{i}} = 1 and AD = \overline{C_{i}}D = A\overline{C_{i}}, it is the graph of an isomorphism of algebraic groups from A to \overline{C_{i}}, and the proof is finished. □

Corollary 8.8. – Let A be a centrally indecomposable subgroup of G_{i} containing G′_{i}. If it is a complement of Z(G)^{∘}G′_{i}/G′_{i} in G_{i}^{∘}/G′_{i} then, up to isomorphism of algebraic groups, \overline{C_{i}} is the only affine algebraic group over \overline{ℚ} such that the abstract groups \overline{A} := A/X_{i} and \overline{C_{i}} are isomorphic.

Proof – The group (Z(G)^{∘} ∩ G′_{i})/X_{i} is definable in the pure group G, and it is torsion-free, so it is divisible and it has a complement S/X_{i} in Z(G)^{∘}/X_{i}. Then we have G_{i} = AS = C_{i}S, and A/(A ∩ S) is isomorphic to C_{i}/(C_{i} ∩ S) as algebraic groups. Moreover, the group A ∩ S is contained in A ∩ Z(G)^{∘} ≤ Z(G)^{∘} ∩ G′_{i} = X_{i}, so A ∩ S = X_{i}. In the same way, we have C_{i} ∩ S = X_{i}, so \overline{A} and \overline{C_{i}} are isomorphic as abstract groups, and the conclusion follows from Proposition 8.7. □

Corollary 8.9. – The algebraic groups \overline{C_{i}} and \overline{C_{i}}^{∗} depend just on the ACF -group G_{i}.

Proof – Since G is the central product of G_{1}, …, G_{n}, then we have Z(G_{i}) = Z(G), and the result follows from Corollary 8.8. □

From now on, we consider the direct product D(G) of the groups \overline{C_{1}}^{∗}, …, \overline{C_{n}}^{∗}, and of an abelian group T (G) which is \overline{ℚ}_{+} if Z(G)^{∘}G′/G′ is nontrivial and torsionfree, otherwise it is (\overline{ℚ}^{∗})^{r} where r is the Lie rank of Z(G)^{∘}G′/G′. We note that D(G) depends just on the pure group G by Corollary 8.9.

By the choice of T (G), there is an integer s such that Z(G)^{∘}G′/G′ is isomorphic to T (G) × \overline{ℚ}_{+}^{s} as algebraic groups, so T (G) × \overline{ℚ}_{+}^{s} is isomorphic to Z ≃ Z(G)^{∘}/(Z(G)^{∘} ∩ G′)^{∘} too. Consequently, since for each i there is a surjective algebraic homomorphism from \overline{C_{i}}^{∗} to C_{i} with finite kernel, there is a natural surjective algebraic homomorphism γ_{G} from D(G) × \overline{ℚ}_{+}^{s} to G, with finite kernel F by Lemma 8.2. Moreover, F is central in D(G)×\overline{ℚ}_{+}^{s} and, since \overline{ℚ}_{+}^{s} is torsion-free, F is contained in D(G). Since T (G) and T (G)/(T (G) ∩ F) are isomorphic as algebraic groups, we may choose F such that T (G) ∩ F is trivial.

We note that we have [D(G) × \overline{ℚ}_{+}^{s}, γ_{G}^{−1}(Z(G))] ≤ F, so γ_{G}^{−1}(Z(G)) = Z(D(G) × \overline{ℚ}_{+}^{s}) since D(G) × \overline{ℚ}_{+}^{s} is connected. Therefore we find, for each i,

γ_{G}^{−1}(G_{i}) = \overline{C_{i}}^{∗}Z(D(G) × \overline{ℚ}_{+}^{s}).

Lemma 8.10. – There is an abstract isomorphism from G to D(G)/F.

Furthermore, if G′ does not contain Z(G)^{∘}, then there is an abstract isomorphism μ : D(G)/F → D(G)/F × \overline{ℚ}_{+}^{t} for each integer t.

Proof – We may assume Z(G)^{∘} ≰ G′, and t ≥ 1. Let R be the torsion part of T (G). Then T (G) = R ×S, for a direct product S of countably many copies of ℚ_{+}. In particular, since \overline{ℚ}_{+}^{t} is a direct product of countably many copies of ℚ_{+} too, the groups T (G) and T (G) × \overline{ℚ}_{+}^{t} are abstractly isomorphic. More precisely, there is an abstract isomorphism α : T (G) → T (G) × \overline{ℚ}_{+}^{t} such that α(r) = r for each r ∈ R. This gives an abstract isomorphism α^{∗} : D(G) → D(G) × \overline{ℚ}_{+}^{t} such that α^{∗}(x) = x for each x ∈ (×_{i=1}^{n}\overline{C_{i}}^{∗}) × R. But F is finite, so it is contained in (×_{i=1}^{n}\overline{C_{i}}^{∗}) × R, and we obtain the desired isomorphism μ : D(G)/F → D(G)/F × \overline{ℚ}_{+}^{t}. □

However, by the following remark, F depends on the algebraic structure of G, and not just of the abstract structure of G. This is a serious problem for us.

Remark 8.11. – Let r ≥ 5 be a prime integer, and let G = H_{1} × H_{2} where H_{1} and H_{2} are two copies of the group H = \overline{ℚ}_{+} ⋊ \overline{ℚ}^{∗}, where the action of \overline{ℚ}^{∗} on \overline{ℚ}_{+} is defined by:

for each (a, x) ∈ \overline{ℚ}^{∗} × \overline{ℚ}_{+}, a · x = a^{r}x.

Let Δ be the graph in G of the identity automorphism of H, and let Z be the center of Δ. Now let δ be a field automorphism of \overline{ℚ} such that there exists t ∈ \overline{ℚ}^{∗} of order r and satisfying δ(t) ≠ t and δ(t) ≠ t^{−1}. This is possible since r ≥ 5. Let μ be the automorphism of G defined for each (h_{1}, h_{2}) ∈ H_{1} × H_{2} by μ(h_{1}, h_{2}) = (δ(h_{1}), h_{2}). Then μ induces an abstract isomorphism from G/Z to G/μ(Z). However, the algebraic groups G/Z and G/μ(Z) are not isomorphic.

Indeed, suppose toward a contradiction that there is an isomorphism f : G/Z → G/μ(Z) of algebraic groups. Then, for i = 1, 2, we have either f(H_{i}Z/Z) = H_{1}μ(Z)/μ(Z), or f(H_{i}Z/Z) = H_{2}μ(Z)/μ(Z). Let T be a maximal torus of Δ, and let T_{0} be a maximal torus of G containing T. Then T contains Z, and T/Z (resp. T_{0}/Z) is a maximal torus of Δ/Z (resp. G/Z). Moreover, T_{0}μ(Z)/μ(Z) is a maximal torus of G/μ(Z), and by the conjugacy of the maximal tori in G/μ(Z), we may assume f(T_{0}/Z) = T_{0}μ(Z)/μ(Z).

Now let S/μ(Z) = f(T/Z). Then S is of dimension one over \overline{ℚ}, and since S/μ(Z) is a torus, S^{∘} is a torus. We have T ∩ H_{i} ≤ Δ ∩ H_{i} = 1 for i = 1, 2, so T/Z ∩ H_{i}Z/Z = 1 for i = 1, 2, and S/μ(Z) ∩ H_{i}μ(Z)/μ(Z) = 1 for i = 1, 2. For i = 1, 2, we have Z ∩ H_{i} ≤ Δ ∩ H_{i} = 1 so, since μ is an abstract automorphism of G such that μ(H_{i}) = H_{i}, we have μ(Z) ∩ H_{i} = 1. Consequently S ∩ H_{i} ≤ μ(Z) ∩ H_{i} is trivial for i = 1, 2.

Since T_{0} is a maximal torus of G, it has the form T_{0} = T_{1} × T_{2}, where T_{i} is a maximal torus of H_{i} for i = 1, 2. Since S is a torus contained in T_{0}μ(Z) and since μ(Z) is finite, we have S^{∘} ≤ T_{0}. Now, since T_{1}, T_{2}, T and S^{∘} are of dimension one over \overline{ℚ}, the groups T and S^{∘} are the graphs of two isomorphisms α and β of algebraic groups from T_{1} to T_{2}. Moreover, since dim(T_{i}) = 1 for i = 1, 2, there are precisely two isomorphisms of algebraic groups from T_{1} to T_{2}, so we have either α = β or β(x) = α(x)^{−1} for each x ∈ T_{1}. In the first case, we have T = S^{∘}, so S contains Zμ(Z). But the existence of t ∈ \overline{ℚ}^{∗} of order r such that δ(t) ≠ t implies Z ≠ μ(Z), so Zμ(Z) has order r^{2} since r is a prime, and we have Zμ(Z) = Z(H_{1}) × Z(H_{2}). This contradicts S ∩ H_{i} = 1 for i = 1, 2. Hence we have β(x) = α(x)^{−1} for each x ∈ T_{1}.

Let Z_{−1} be the graph in G of the automorphism inversion of Z(H). Since we have β(x) = α(x)^{−1} for each x ∈ T_{1}, we have Z_{−1} ≤ S^{∘}. Moreover, the existence of t ∈ \overline{ℚ}^{∗} of order r such that δ(t) ≠ t^{−1} implies Z_{−1} ≠ μ(Z), so Z_{−1}μ(Z) has order r^{2} since r is a prime, and we have Z_{−1}μ(Z) = Z(H_{1}) × Z(H_{2}). But we have S ∩ H_{i} = 1 for i = 1, 2, so the latter implies that S does not contain Z_{−1}. Thus, there is no isomorphism of algebraic groups from G/Z to G/μ(Z).

Lemma 8.12. – Let X and Y be two finite subgroups of an algebraic torus T over \overline{ℚ}. If there is an isomorphism δ : X → Y, then there is a quasi-standard automorphism φ of T such that φ(x) = δ(x) for each x ∈ X.

Proof – Let X = X_{1} × ⋯ × X_{r} be a decomposition of X where, for each i, X_{i} is cyclic and, if i > 1, its order divides the one of X_{i−1}. For each i, we consider Y_{i} = δ(X_{i}). We show, by induction on r, that there exists a direct product T_{1} × ⋯ × T_{r} of subtori of T with dimension one, such that T_{i} contains X_{i} for each i. We may assume that T contains a direct product S = T_{1} × ⋯ × T_{r−1} of subtori with dimension one, such that T_{i} contains X_{i} for each i ≤ r − 1. For each prime p dividing |X_{r}|, there are p^{r−1} − 1 elements of order p in S, and since |X_{i}| divides |X_{i−1}| for each i > 1, there are p^{r−1} − 1 elements of order p in X_{1} × ⋯ × X_{r−1} too. Consequently, S ∩ X_{r} is trivial.

We note that X_{r} is contained in a subtorus R of T with dimension one. Indeed, otherwise, if we consider a subtorus R_{1} of T containg X and with minimal dimension, then R_{1} has a subtorus R_{2} such that R_{1}/R_{2} has dimension two. Let \overline{T_{1}} and \overline{T_{2}} be two subtori of R_{1}/R_{2} such that R_{1}/R_{2} = \overline{T_{1}} × \overline{T_{2}}. Let \overline{x} be the image in R_{1}/R_{2} of a generator x of X_{r}. By the minimality of dim(R_{1}), we have \overline{x} ∉ \overline{T_{i}} \ \{1\} for i = 1, 2, so we have \overline{x} = (\overline{t_{1}}, \overline{t_{2}}) with \overline{t_{i}} ∈ \overline{T_{i}} for i = 1, 2. Let μ be an isomorphism of algebraic groups from \overline{T_{1}} to \overline{T_{2}}. Then we find \overline{t} ∈ \overline{T_{1}} and (a, b) ∈ ℕ^{2} such that \overline{t_{1}} = \overline{t}^{a} and \overline{t_{2}} = μ(\overline{t})^{b}. Now we consider \overline{R} = \{(\overline{u}^{a}, μ(\overline{u})^{b}) | \overline{u} ∈ \overline{T_{1}}\}. Then \overline{R} is the image of a nonzero algebraic homomorphism from \overline{T_{1}} to R_{1}/R_{2}, so \overline{R} is a subtorus with dimension one. But \overline{R} contains \overline{x}, hence its preimage R_{0} is a subtorus of R_{1} containing X_{r}. Since we have dim(R_{0}) = dim(R_{1}) − 1, we have a contradiction with the choice of R_{1}, and X_{r} is contained in a subtorus R of T with dimension one.

Now SR is a subtorus of T, and we find a subtorus V of SR such that SR = S×V. we have x = (s, v) for s ∈ S and v ∈ V \ {1}. Let c_{1} and d_{1} be the orders of s and v respectively. Then c_{1} and d_{1} divides |X_{r}|, and we consider c := |X_{r}|/c_{1}. The previous paragraph applied with ⟨s⟩ yields a subtorus S_{1} of S with dimension one and containing s. Let γ : S_{1} → V be an isomorphism of algebraic groups, and let y ∈ S_{1} of order |X_{r}| such that y^{c} = s. Then there exists an integer e dividing |X_{r}| such that γ(y)^{e} = v. Since |X_{i}| divides |X_{i−1}| for each i > 1, any element of S with order dividing |X_{r}| belongs to X_{1} × ⋯ × X_{r−1}. In particular, we have y ∈ X_{1} × ⋯ × X_{r−1}. Let Q := {(u^{c}, γ(u)^{e}) | u ∈ S_{1}}. Since Q is the image of a nontrivial algebraic homomorphism from S_{1} to S × V, it is a subtorus with dimension one. Moreover, Q contains x, and since S ∩ X_{r} = 1, no element of S ∩ Q has order dividing |X_{r}|. But we have S ∩ Q = S_{1} ∩ Q = {u^{c} | u ∈ S_{1}, u^{e} = 1}, and e divides |X_{r}|, hence each element of S ∩ Q has order dividing e and |X_{r}|. This proves that S ∩ Q = 1, and we may choose T_{r} to be Q. Thus, as claimed, there exists a direct product T_{1} × ⋯ × T_{r} of subtori of T with dimension one, such that T_{i} contains X_{i} for each i.

In the same way, there exists a direct product T′_{1} × ⋯ × T′_{r} of subtori of T with dimension one, such that T′_{i} contains Y_{i} for each i. Let T_{0} (resp. T′_{0}) be an algebraic complement of T_{1} × ⋯ × T_{r} (resp. T′_{1} × ⋯ × T′_{r}) in T. Then T_{0} and T′_{0} are two tori of dimension dim(T) − r, so there is an isomorphism of algebraic groups φ_{0} : T_{0} → T′_{0}. For each i = 1, …, r, let φ_{i} : T_{i} → T′_{i} be an isomorphism of algebraic groups. Since T = T_{0} × T_{1} × ⋯ × T_{r} = T′_{0} × T′_{1} × ⋯ × T′_{r}, the map φ′ : T → T, defined by φ′(u_{i}) = φ_{i}(u_{i}) for each i and each u_{i} ∈ T_{i}, is an algebraic automorphism of T. Moreover, for each i, the subgroup X_{i} (resp. Y_{i}) is the only subgroup of T_{i} (resp. T′_{i}) of order |X_{i}| = |Y_{i}|, so we have φ′(X_{i}) = Y_{i} for each i.

For each i = 1, …, r, let x_{i} be a generator of X_{i}. Then, for each i, φ′(x_{i}) and δ(x_{i}) are two elements of order |X_{i}| in T′_{i}, so there is a field automorphism β_{i} of \overline{ℚ} such that (φ′ ∘ β_{i})(x_{i}) = δ(x_{i}). Let φ be the automorphism of T, defined by φ(u_{0}) = φ′(u_{0}) for each u_{0} ∈ T_{0}, and by φ(u_{i}) = (φ′ ∘ β_{i})(u_{i}) for each i = 1, …, r. Thus φ is a quasi-standard automorphism of T, and it satisfies φ(x) = δ(x) for each x ∈ X, as desired. □

Lemma 8.13. – For each i, the exponent of Y_{i} := γ_{G}^{−1}(X_{i}) ∩ \overline{C_{i}}^{∗}Z(D(G))^{∘} is n_{i}.

Moreover, Y_{i} is finite and central in D(G), and we have Y_{i} = (γ_{G}^{−1}(X_{i}) ∩ \overline{C_{i}}^{∗})(γ_{G}^{−1}(X_{i}) ∩ T (G)).

Proof – Since X_{i} ≤ Z(G) is finite and central in G, and since the kernel F of γ_{G} is finite, then Y_{i} ≤ γ_{G}^{−1}(X_{i}) is finite and normal in D(G). Moreover, since D(G) is connected, then Y_{i} is central in D(G).

By the choice of \overline{C_{i}}^{∗}, the exponent of Y_{i}^{C} := γ_{G}^{−1}(X_{i}) ∩ \overline{C_{i}}^{∗} is n_{i}. Since we have X_{i} ≤ C_{i} = γ_{G}(\overline{C_{i}}^{∗}), we obtain X_{i} = γ_{G}(Y_{i}^{C}). Now we consider Y_{i}^{T} := γ_{G}^{−1}(X_{i}) ∩ T (G). Since T (G) ∩ F is trivial and since the exponent of X_{i} is n_{i}, the exponent of Y_{i}^{T} divides n_{i}, and we have just to prove that Y_{i}^{C}Y_{i}^{T} contains Y_{i}.

Let u ∈ Y_{i}. Then there exist c ∈ \overline{C_{i}}^{∗} and z ∈ Z(D(G))^{∘} such that u = cz. Since we have γ_{G}(\overline{C_{i}}^{∗}) = C_{i} and γ_{G}(Z(D(G))^{∘}) = γ_{G}(Z(D(G)))^{∘} = Z(G)^{∘}, we find γ_{G}(c) ∈ C_{i} and γ_{G}(c) = γ_{G}(u)γ_{G}(z^{−1}) ∈ X_{i}Z(G)^{∘} = Z(G)^{∘}. Thus we obtain γ_{G}(c) ∈ C_{i} ∩ Z(G)^{∘} = G′_{i} = (G_{i}^{∘})′ = C′_{i}, and there exists c′ ∈ (\overline{C_{i}}^{∗})′ such that γ_{G}(c) = γ_{G}(c′). Now we have u = (cc^{′−1})(c′z) with cc^{′−1} ∈ \overline{C_{i}}^{∗} and c′z ∈ (\overline{C_{i}}^{∗})′Z(D(G))^{∘}. Since we have γ_{G}(cc^{′−1}) = γ_{G}(c)γ_{G}(c′)^{−1} = 1 ∈ X_{i}, we find cc^{′−1} ∈ Y_{i}^{C}. Consequently, we have just to prove that c′z belongs to Y_{i}^{C}Y_{i}^{T}.

Since D(G) is the direct product of \overline{C_{1}}^{∗}, …, \overline{C_{n}}^{∗} and of T (G), the subgroup Z(D(G))^{∘} is the direct product of Z(\overline{C_{1}}^{∗})^{∘}, …, Z(\overline{C_{n}}^{∗})^{∘} and of T (G). Moreover Lemma 8.5 implies that (\overline{C_{i}}^{∗})′ contains Z(\overline{C_{i}}^{∗})^{∘}, therefore (\overline{C_{i}}^{∗})′Z(D(G))^{∘} is the direct product of (\overline{C_{i}}^{∗})′, T (G) and of the groups Z(\overline{C_{k}}^{∗})^{∘} for k ≠ i. But, for each k ≠ i, the torsion part of Z(\overline{C_{k}}^{∗})^{∘} is contained in the maximal torus T_{k} of Z(\overline{C_{k}}^{∗})^{∘}, and T_{k} is trivial since Z(\overline{C_{k}}^{∗})^{∘} is contained in (\overline{C_{k}}^{∗})′ by Lemma 8.5. Hence the torsion part of (\overline{C_{i}}^{∗})′Z(D(G))^{∘} is contained in (\overline{C_{i}}^{∗})′ × T (G).

Since X_{i} and F are finite, then γ_{G}^{−1}(X_{i}) is finite too, and since γ_{G}(c′z) = γ_{G}(c)γ_{G}(z) = γ_{G}(u) belongs to X_{i}, the element c′z is of finite order. Thus c′z belongs to (\overline{C_{i}}^{∗})′ × T (G), and there exist d ∈ (\overline{C_{i}}^{∗})′ and t ∈ T (G) such that c′z = dt. Moreover, since c′z has finite order, the elements d and t have finite orders too. Now we find γ_{G}(t) = γ_{G}(d)^{−1}γ_{G}(c′z) ∈ C′_{i}X_{i} = C′_{i}, and γ_{G}(t) ∈ C′_{i} ∩ γ_{G}(T (G)) ≤ C′_{i} ∩ Z(G)^{∘}, so γ_{G}(t) is a torsion element of C′_{i} ∩ Z(G)^{∘} and γ_{G}(t) belongs to X_{i}. This proves that t belongs to Y_{i}^{T}. Furthermore, this proves that γ_{G}(d) = γ_{G}(c′z)γ_{G}(t)^{−1} belongs to X_{i}, so we obtain d ∈ Y_{i}^{C} and c′z = dt belongs to Y_{i}^{C}Y_{i}^{T}, as desired. □

Theorem 8.14. – Let H be an affine algebraic group over \overline{ℚ}. If H and G are abstractly isomorphic, then there is a quasi-standard automorphism α of D(G), and an isomorphism of algebraic groups between H and

• D(G)/α(F) × \overline{ℚ}_{+}^{t} for an integer t if G′ does not contains Z(G)^{∘};

• D(G)/α(F) if G′ contains Z(G)^{∘}.

Proof – Let δ_{0} : G → H be an abstract isomorphism. Since H and G are abstractly isomorphic, and since G is connected, then H is connected too and, in the ACF -expansion of the pure group H, we find interpretable fields L_{1}, …, L_{n} such that L_{i} is not definably isomorphic to L_{j} for each i ≠ j, and such that H/Z(H) is definably linear over L_{1}, …, L_{n}. For each i, we denote by H_{i}/Z(H) the largest connected subgroup of H/Z(H) definably linear over L_{i}. We may assume H_{i} = δ_{0}(G_{i}) for each i. Moreover, since H ≃ G, we may assume D(H) = D(G), and even that T (H) = T (G) and that the groups \overline{C_{1}}^{∗}, …, \overline{C_{n}}^{∗} for H are the same that for G. Let t be the integer such that Z(H)^{∘}H′/H′ and T (H)×\overline{ℚ}_{+}^{t} are isomorphic as algebraic groups. Then we consider the natural surjective algebraic homomorphism γ_{H} : D(H) × \overline{ℚ}_{+}^{t} → H. It satisfies γ_{H}^{−1}(H_{i}) = \overline{C_{i}}^{∗} Z(D(H) × \overline{ℚ}_{+}^{t}) for each i, and its kernel E is a central finite subgroup of D(H) = D(G), and it verifies E ∩T (H) = 1.

By Lemma 8.10, the quotient D(G)/F (resp. D(G)/E) is abstractly isomorphic to G (resp. H), so there is an abstract isomorphism δ : D(G)/F → D(G)/E. We notice that, if G′ contains Z(G)^{∘}, then we have s = 0, and H′ contains Z(H)^{∘} too, so t = 0. Hence, in all the cases, we have just to prove that there is a quasi-standard automorphism α of D(G) such that D(G)/E and D(G)/α(F) are isomorphic as algebraic groups. In particular, since G and D(G)/F (resp. H and D(H)/E) are abstractly isomorphic by Lemma 8.10, we may assume G = D(G)/F (resp. H = D(H)/E), and δ = δ_{0}. Thus we have s = 0 and t = 0, and H_{i} = δ(G_{i}) for each i. From now on, the main difficulty for the proof is to find a quasi-standard automorphism α of D(G) such that D(G)/Z(D(G))^{∘}α(F) and D(G)/Z(D(G))^{∘}E are isomorphic as algebraic groups.

For each i, we consider P_{i} := \overline{C_{i}}^{∗}Z(D(G)) = γ_{G}^{−1}(G_{i}) = γ_{H}^{−1}(H_{i}). Then, for each i, we have G_{i} = P_{i}/F, H_{i} = P_{i}/E, and δ(P_{i}/F) = P_{i}/E. Let Z_{F} /F := Z(G) = Z(D(G)/F) and Z_{E}/E := Z(H) = Z(D(G)/E). Then [D(G), Z_{F}] is a connected subgroup of the finite subgroup F, so Z_{F} is central in D(G), and since it contains Z(D(G)), we obtain Z_{F} = Z(D(G)). In the same way, for each central finite subgroup Y of D(G), we have Z(D(G)/Y) = Z(D(G))/Y. In particular, we obtain Z_{E} = Z(D(G)). Now the algebraic groups P_{i}/Z(D(G)), G_{i}/Z(G) and H_{i}/Z(H) are isomorphic. Moreover, since G/Z(G) is the direct product of the groups G_{1}/Z(G), …, G_{n}/Z(G), we have

D(G)/Z(D(G)) = P_{1}/Z(D(G)) × ⋯ × P_{n}/Z(D(G)).

But, for each i, in the ACF -expansion of the pure group G, the quotient G_{i}/Z(G) is definably linear over K_{i}, so Fact 3.6 says that the isomorphism from G_{i}/Z(G) to H_{i}/Z(H) induced by δ is a standard isomorphism. Hence, for each i, since P_{i}/Z(D(G)), G_{i}/Z(G) and H_{i}/Z(H) are isomorphic as algebraic groups, the automorphism \overline{δ_{i}^{∗}} of P_{i}/Z(D(G)) induced by δ, is a standard automorphism. In other words, for each i, there is a field automorphism φ_{i} of \overline{ℚ}, such that the automorphism \overline{δ_{i}^{φ}} := \overline{δ_{i}^{∗}} ∘ φ_{i}^{−1} of P_{i}/Z(D(G)) is algebraic over \overline{ℚ}.

We fix i, and we consider the preimage Ω of the graph of \overline{δ_{i}^{φ}} in P_{i} × Q_{i}. It is an algebraic subgroup and, since P_{i}/Z(D(G)) ≃ G_{i}/Z(G) is connected, then Ω/(Z(D(G)) × Z(D(G))) is connected too, and we have Ω = Ω^{∘}(Z(D(G)) × Z(D(G))).

We consider the group Y_{i}^{H} := γ_{H}^{−1}(X_{i}^{H})∩\overline{C_{i}}^{∗}Z(D(G))^{∘} = γ_{H}^{−1}(X_{i}^{H})∩P_{i}^{∘}, where X_{i}^{H} is the torsion part of Z(H)^{∘} ∩ H′_{i}. Then we have δ(Y_{i}F/F) = δ(γ_{G}^{−1}(X_{i})/F ∩ \overline{C_{i}}^{∗}Z(D(G))^{∘}F/F) = δ(γ_{G}^{−1}(X_{i})/F) ∩ δ(\overline{C_{i}}^{∗}Z(D(G))^{∘}F/F). But we have G = D(G)/F, so γ_{G}^{−1}(X_{i})/F is precisely X_{i}, that is the torsion part of Z(G)^{∘} ∩ G′_{i}. Hence δ(γ_{G}^{−1}(X_{i})/F) is the torsion part of Z(H)^{∘} ∩ H′_{i}, that is δ(γ_{G}^{−1}(X_{i})/F) = X_{i}^{H}. Moreover, we have P_{i} = \overline{C_{i}}^{∗}Z(D(G)), so we find δ(\overline{C_{i}}^{∗}Z(D(G))^{∘}F/F) = δ(P_{i}^{∘}F/F) = P_{i}^{∘}E/E, and we obtain δ(Y_{i}F/F) = X_{i}^{H} ∩ P_{i}^{∘}E/E = (γ_{H}^{−1}(X_{i}^{H})/E) ∩ (P_{i}^{∘}E/E) = (γ_{H}^{−1}(X_{i}^{H}) ∩ P_{i}^{∘})E/E = Y_{i}^{H}E/E. Thus, the isomorphism δ induces an isomorphism δ_{i}^{∗} : P_{i}^{∘}F/Y_{i}F → P_{i}^{∘}E/Y_{i}^{H}E and, since we have P_{i}^{∘} ∩ F ≤ Y_{i} ≤ P_{i}^{∘} and P_{i}^{∘} ∩ E ≤ Y_{i}^{H} ≤ P_{i}^{∘}, the isomorphism δ induces an isomorphism δ_{i}^{0} :

P_{i}^{∘}/Y_{i} → P_{i}^{∘}/Y_{i}^{H}. We consider the isomorphism δ_{i} := δ_{i}^{0} ∘ φ_{i}^{−1} from P_{i}^{∘}/Y_{i} to P_{i}^{∘}/Y_{i}^{H}. Let \overline{δ_{i}} be the isomorphism from P_{i}^{∘}/Z(D(G))^{∘}Y_{i} to P_{i}^{∘}/Z(D(G))^{∘}Y_{i}^{H} induced by δ_{i}. Since Z(D(G))^{∘}Y_{i} and Z(D(G))^{∘}Y_{i}^{H} have finite index in Z(D(G)), then the graph of \overline{δ_{i}} has finite index in Ω/(Z(D(G))^{∘}Y_{i} × Z(D(G))^{∘}Y_{i}^{H}), because Ω/(Z(D(G)) × Z(D(G))) is the graph of \overline{δ_{i}^{φ}}. Since \overline{ℚ} is an algebraically closed field and of characteristic zero, the connected algebraic groups over \overline{ℚ} have no proper subgroup of finite index. In particular, P_{i}^{∘}/Z(D(G))^{∘}Y_{i} has no proper subgroup of finite index, so the graph of \overline{δ_{i}} has no proper subgroup of finite index. Moreover, by considering Ω_{Y} := Ω^{∘}(Y_{i} × Y_{i}^{H}) and Z_{Y} := Z(D(G))^{∘}Y_{i} × Z(D(G))^{∘}Y_{i}^{H}, then the group Ω_{Y} /Z_{Y} is connected, so Ω_{Y} /Z_{Y} has no proper subgroup of finite index. This implies that the graph of \overline{δ_{i}} is Ω_{Y} /Z_{Y}. In particular, \overline{δ_{i}} is an isomorphism of algebraic groups.

Let Θ denote the preimage in P_{i}^{∘} × P_{i}^{∘} of the graph of δ_{i}. Then we have Ω_{Y} = Θ Z_{Y}, and Ω′_{Y} (Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) = Θ′(Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) is the graph of the restriction of δ_{i} to (P_{i}^{∘})′/Y_{i}. Moreover, since Z_{Y} Ω′_{Y} /Ω′_{Y} (Y_{i} × Y_{i}^{H}) is a connected algebraic subgroup of the connected abelian group Ω_{Y} /Ω′_{Y} (Y_{i} × Y_{i}^{H}), it has an algebraic complement W_{0}/Ω′_{Y} (Y_{i} × Y_{i}^{H}).

(1) The algebraic groups W := W_{0}^{∘} and \overline{C_{i}}^{∗} are isomorphic, and W ∩ (Z_{i} × P_{i}^{∘}) and W ∩ (P_{i}^{∘} × Z_{i}) are trivial. Moreover, the image of the first (resp. the second) projection of W (Z_{i} × Z_{i}) in P_{i}^{∘} × P_{i}^{∘} is surjective.

For each k, let Z_{k} := T (G) × (×_{j≠k}Z(\overline{C_{j}}^{∗})^{∘}). Then we have Z(D(G))^{∘} = Z(\overline{C_{k}}^{∗})^{∘} × Z_{k} for each k. Let Z_{W} := Z_{i}Y_{i} × Z_{i}Y_{i}^{H} ≤ P_{i}^{∘} × P_{i}^{∘}. Since Lemma 8.5 gives Z(\overline{C_{i}}^{∗})^{∘} ≤ (\overline{C_{i}}^{∗})′, we obtain Z_{Y} ≤ ((P_{i}^{∘})′ × (P_{i}^{∘})′)Z_{W}. Then W_{0}((P_{i}^{∘})′ × (P_{i}^{∘})′)Z_{W} contains Ω^{∘} ≤ Ω_{Y} and, since Ω_{Y} /Z_{Y} is the graph of the isomorphism \overline{δ_{i}} : P_{i}^{∘}/Z(D(G))^{∘}Y_{i} → P_{i}^{∘}/Z(D(G))^{∘}Y_{i}^{H}, its first (resp. second) projection in P_{i}^{∘} × P_{i}^{∘} is surjective. Moreover, since Ω′_{Y} (Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) = Θ′(Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) is the graph of the restriction to (P_{i}^{∘})′Y_{i}/Y_{i} of the isomorphism δ_{i} : P_{i}^{∘}/Y_{i} → P_{i}^{∘}/Y_{i}^{H}, the first (resp. second) projection of W_{0}Ω′_{Y} Z_{W} in P_{i}^{∘} × P_{i}^{∘} is surjective too. But W_{0} contains Ω′_{Y}, so we have (W_{0}Ω′_{Y} Z_{W})^{∘} = (W_{0}Z_{W})^{∘} ≤ W Z_{W}, and since P_{i}^{∘} is connected, the first (resp. second) projection of W Z_{W} in P_{i}^{∘} × P_{i}^{∘} is surjective.

We verify that rk(W) = rk(\overline{C_{i}}^{∗}). We have

rk(W) = rk(W_{0})
= rk(W_{0}/Ω′_{Y} (Y_{i} × Y_{i}^{H})) + rk(Ω′_{Y} (Y_{i} × Y_{i}^{H}))
= (rk(Ω_{Y} /Ω′_{Y} (Y_{i} × Y_{i}^{H})) − rk(Z_{Y} Ω′_{Y} /Ω′_{Y} (Y_{i} × Y_{i}^{H})))
+rk(Ω′_{Y} (Y_{i} × Y_{i}^{H})),

and since Y_{i}×Y_{i}^{H} is finite, we obtain rk(W) = rk(Ω_{Y})−rk(Z_{Y} Ω′_{Y} /Ω′_{Y}) = rk(Ω_{Y})− (rk(Z_{Y}) − rk(Z_{Y} ∩ Ω′_{Y})). Moreover, by definition of Ω, we have

rk(Ω_{Y}) = rk(Ω)
= rk(Ω/(Z(D(G)) × Z(D(G)))) + rk(Z(D(G)) × Z(D(G)))
= rk(P_{i}/Z(D(G))) + rk(Z_{Y})
= rk(P_{i}^{∘}/Z(D(G))^{∘}Y_{i}) + rk(Z_{Y})
= rk(\overline{C_{i}}^{∗}Z(D(G))^{∘}/Z(D(G))^{∘}) + rk(Z_{Y})
= rk(\overline{C_{i}}^{∗}/(\overline{C_{i}}^{∗} ∩ Z(D(G))^{∘})) + rk(Z_{Y})
= rk(\overline{C_{i}}^{∗}) − rk(\overline{C_{i}}^{∗} ∩ Z(D(G))^{∘}) + rk(Z_{Y}),

and since D(G) is the direct product of \overline{C_{i}}^{∗} by another connected algebraic group, we obtain rk(Ω_{Y}) = rk(\overline{C_{i}}^{∗}) − rk(Z(\overline{C_{i}}^{∗})^{∘}) + rk(Z_{Y}). Since for each central finite subgroup Y of D(G) we have Z(D(G)/Y) = Z(D(G))/Y, then we have Z(D(G)/Y_{i}) = Z(D(G))/Y_{i}. Since D(G) is the central product of the groups P_{1}, …, P_{n}, we have Z(P_{i}^{∘}/Y_{i}) = Z(D(G)/Y_{i}) ∩ P_{i}^{∘}/Y_{i}. Therefore, since P_{i} contains Z(D(G)), we obtain

Z(P_{i}^{∘}/Y_{i})^{∘} = (Z(D(G)/Y_{i})∩P_{i}^{∘}/Y_{i})^{∘} = (Z(D(G))/Y_{i}∩P_{i}^{∘}/Y_{i})^{∘} = Z(D(G))^{∘}Y_{i}/Y_{i}.

In the same way, we have Z(P_{i}^{∘}/Y_{i}^{H})^{∘} = Z(D(G))^{∘}Y_{i}^{H}/Y_{i}^{H}, so

δ_{i}(Z(D(G))^{∘}Y_{i}/Y_{i}) = Z(D(G))^{∘}Y_{i}^{H}/Y_{i}^{H}.

Since Ω′_{Y} (Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) = Θ′(Y_{i} × Y_{i}^{H})/(Y_{i} × Y_{i}^{H}) is the graph of the restriction to (P_{i}^{∘})′Y_{i}/Y_{i} of δ_{i}, the latter proves that (Z_{Y} ∩ Ω′_{Y})Y_{i}/Y_{i} is the graph of the restriction to (Z(D(G))^{∘}Y_{i} ∩ (P_{i}^{∘})′Y_{i})/Y_{i} of δ_{i}. Now, since Y_{i} × Y_{i}^{H} is finite, we have rk(Z_{Y} ∩ Ω′_{Y}) = rk(Z(D(G))^{∘}Y_{i} ∩ (P_{i}^{∘})′Y_{i}) = rk(Z(D(G))^{∘} ∩ (P_{i}^{∘})′) = rk(Z(D(G))^{∘} ∩ (\overline{C_{i}}^{∗})′) = rk(Z(\overline{C_{i}}^{∗})^{∘} ∩ (\overline{C_{i}}^{∗})′), and since Lemma 8.5 implies that (\overline{C_{i}}^{∗})′ contains Z(\overline{C_{i}}^{∗})^{∘}, we obtain rk(Z_{Y} ∩ Ω′_{Y}) = rk(Z(\overline{C_{i}}^{∗})^{∘}), and

rk(W) = rk(Ω_{Y}) − rk(Z_{Y}) + rk(Z_{Y} ∩ Ω′_{Y})
= (rk(\overline{C_{i}}^{∗}) − rk(Z(\overline{C_{i}}^{∗})^{∘}) + rk(Z_{Y})) − rk(Z_{Y}) + rk(Z(\overline{C_{i}}^{∗})^{∘})
= rk(\overline{C_{i}}^{∗}),

as claimed.

We show that W/(W ∩ (Y_{i} × Y_{i}^{H})) and \overline{C_{i}} are isomorphic as algebraic groups. Indeed, there is an isomorphism of algebraic groups between \overline{C_{i}} and

γ_{G}^{−1}(C_{i})/γ_{G}^{−1}(X_{i}) = \overline{C_{i}}^{∗}γ_{G}^{−1}(X_{i})/γ_{G}^{−1}(X_{i}) ≃ \overline{C_{i}}^{∗}/(γ_{G}^{−1}(X_{i}) ∩ \overline{C_{i}}^{∗}),

and since we have P_{i}^{∘} = \overline{C_{i}}^{∗}Z(D(G))^{∘} = \overline{C_{i}}^{∗} ×Z_{i}, the groups P_{i}^{∘}/(γ_{G}^{−1}(X_{i})∩\overline{C_{i}}^{∗})Z_{i} and \overline{C_{i}} are isomorphic as algebraic groups. Thus, since Z_{i} contains γ_{G}^{−1}(X_{i}) ∩ T (G) ≤ T (G), the algebraic groups \overline{C_{i}} and P_{i}^{∘}/Y_{i}Z_{i} are isomorphic by Lemma 8.13, so \overline{C_{i}} and (P_{i}^{∘} × P_{i}^{∘})/(Y_{i}Z_{i} × P_{i}^{∘}) are isomorphic too. Now, since the first projection of W Z_{W} in P_{i}^{∘} ×P_{i}^{∘} is surjective, that is W Z_{W} ({1}×P_{i}^{∘}) = P_{i}^{∘} ×P_{i}^{∘}, the algebraic groups \overline{C_{i}} and W Z_{W} /(W Z_{W} ∩ (Y_{i}Z_{i} × P_{i}^{∘})) are isomorphic. Hence, since Z_{W} is contained in Y_{i}Z_{i} × P_{i}^{∘}, the algebraic groups \overline{C_{i}} and W/(W ∩ (Y_{i}Z_{i} × P_{i}^{∘})) = W/(W ∩(W Z_{W} ∩(Y_{i}Z_{i}×P_{i}^{∘}))) are isomorphic. In particular, since we have rk(W) = rk(\overline{C_{i}}^{∗}) = rk(\overline{C_{i}}) by the previous paragraph, the intersection W ∩ (Y_{i}Z_{i} × P_{i}^{∘}) is finite. Since Ω_{Y} /Z_{Y} is the graph of \overline{δ_{i}}, we have Ω_{Y} ∩ (Z(D(G))^{∘}Y_{i} × P_{i}^{∘}) = Z_{Y} and W ∩ (Y_{i}Z_{i} × P_{i}^{∘}) ≤ Ω_{Y} ∩ (Z(D(G))^{∘}Y_{i} × P_{i}^{∘}) is contained in W ∩ Z_{Y} ≤ W_{0} ∩ Z_{Y} ≤ Ω′_{Y} (Y_{i} × Y_{i}^{H}), so we find

W ∩ (Y_{i}Z_{i} × P_{i}^{∘}) ≤ W ∩ Z_{Y}
≤ Z_{Y} ∩ Ω′_{Y} (Y_{i} × Y_{i}^{H}))
≤ (Z(D(G))^{∘}Y_{i} ∩ (P_{i}^{∘})′Y_{i}) × (Z(D(G))^{∘}Y_{i}^{H} ∩ (P_{i}^{∘})′Y_{i}^{H}).

Let W_{P} (resp. W_{Q}) be the image of the finite group W ∩ (Y_{i}Z_{i} × P_{i}^{∘}) by the first (resp. the second) projection in P_{i}^{∘} × P_{i}^{∘}. Then W_{P} F/F is a finite subgroup of

(Z(D(G))^{∘}Y_{i} ∩ (P_{i}^{∘})′Y_{i})F/F = (Z(D(G))^{∘}Y_{i}F/F) ∩ ((P_{i}^{∘})′Y_{i}F/F)
= (γ_{G}(Z(D(G)))^{∘}γ_{G}(Y_{i})) ∩ ((γ_{G}(P_{i})^{∘})′γ_{G}(Y_{i}))
≤ (Z(G)^{∘}X_{i}) ∩ ((G_{i}^{∘})′X_{i})
= Z(G)^{∘} ∩ G′_{i},

so W_{P} F/F is contained in X_{i}, and W_{P} is contained in γ_{G}^{−1}(X_{i}) ∩ P_{i}^{∘} = Y_{i}. In the same way we obtain W_{Q} ≤ Y_{i}^{H}, so we have W ∩ (Y_{i}Z_{i} × P_{i}^{∘}) ≤ Y_{i} × Y_{i}^{H}. Thus W/(W ∩ (Y_{i} × Y_{i}^{H})) and \overline{C_{i}} are isomorphic as algebraic groups.

We prove that W and \overline{C_{i}}^{∗} are isomorphic as algebraic groups, and that W ∩(Z_{i}× P_{i}^{∘}) and W ∩ (P_{i}^{∘} × Z_{i}) are trivial. Since the exponent of Y_{i} × Y_{i}^{H} is n_{i} by Lemma 8.13, and since W/(W ∩(Y_{i} ×Y_{i}^{H})) and \overline{C_{i}} are isomorphic as algebraic groups, there is an isogeny ι_{1} from \overline{C_{i}}^{∗} to W. Since the first projection of W Z_{W} in P_{i}^{∘} × P_{i}^{∘} is surjective, and since P_{i}^{∘} is connected, the first projection of W (Z_{i} × Z_{i}) in P_{i}^{∘} × P_{i}^{∘} is surjective too. Thus we have P_{i}^{∘} × P_{i}^{∘} = (W (Z_{i} × Z_{i}))({1} × P_{i}^{∘}) = W (Z_{i} × P_{i}^{∘}). Then, since we have P_{i}^{∘} = \overline{C_{i}}^{∗} ×Z_{i}, the algebraic groups W (Z_{i} ×P_{i}^{∘})/(Z_{i} ×P_{i}^{∘}) and \overline{C_{i}}^{∗} are isomorphic, so W/(W ∩ (Z_{i} × P_{i}^{∘})) and \overline{C_{i}}^{∗} are isomorphic too. Therefore, since we have rk(W) = rk(\overline{C_{i}}^{∗}), there is an isogeny ι_{2} from W to \overline{C_{i}}^{∗} with kernel W ∩ (Z_{i} × P_{i}^{∘}). Now ι_{2} ∘ ι_{1} is an isogeny from \overline{C_{i}}^{∗} to \overline{C_{i}}^{∗}. Since ker(ι_{2} ∘ ι_{1}) is finite and since \overline{C_{i}}^{∗} is connected, the preimage of Z(\overline{C_{i}}^{∗}) by ι_{2} ∘ ι_{1} is precisely Z(\overline{C_{i}}^{∗}). In particular, if X_{C} denotes the torsion part of Z(\overline{C_{i}}^{∗}), then X_{C} contains ker(ι_{2} ∘ ι_{1}), and X_{C}/ ker(ι_{2} ∘ ι_{1}) is isomorphic to X_{C}. But Lemma 8.5 implies that (\overline{C_{i}}^{∗})′ contains Z(\overline{C_{i}}^{∗})^{∘}, so Z(\overline{C_{i}}^{∗})^{∘} has no nontrivial torus, and it is torsion-free. Hence X_{C} is finite and ker(ι_{2} ∘ ι_{1}) is trivial. Thus ι_{2} ∘ ι_{1} is an automorphism of \overline{C_{i}}^{∗}, and ι_{2} is an isomorphism from W to \overline{C_{i}}^{∗}. In particular, W and \overline{C_{i}}^{∗} are isomorphic as algebraic groups, and since the kernel of ι_{2} is W ∩ (Z_{i} × P_{i}^{∘}), we have W ∩ (Z_{i} × P_{i}^{∘}) = 1. In the same way, we obtain W ∩ (P_{i}^{∘} × Z_{i}) = 1.

(2) P_{i}^{∘} is the direct product of Z_{i} by R_{i} (resp. S_{i}), the first (resp. the second) projection in P_{i}^{∘} × P_{i}^{∘} is surjective, and W is the graph of an isomorphism of algebraic groups ω_{i} : R_{i} → S_{i}.

Since we have W ∩ ({1} × S_{i}) ≤ W ∩ (Z_{i} × P_{i}^{∘}) = 1 and W ∩ (R_{i} × {1}) ≤ W ∩ (P_{i}^{∘} × Z_{i}) = 1, the algebraic group W is the graph of an isomorphism ω_{i} : R_{i} → S_{i} of algebraic groups. Moreover, since we have W ∩ (Z_{i} × P_{i}^{∘}) = 1 (resp. W ∩ (P_{i}^{∘} × Z_{i}) = 1), then R_{i} ∩ Z_{i} (resp. S_{i} ∩ Z_{i}) is trivial. Therefore, since the first (resp. the second) projection of W (Z_{i} × Z_{i}) in P_{i}^{∘} × P_{i}^{∘} is surjective, then P_{i}^{∘} is the direct product of R_{i} by Z_{i} (resp. S_{i} by Z_{i}).

(3) D(G) is the direct product of T (G) and of the groups R_{k} (resp. S_{k}) for k ∈ {1, …, n}.

We note that, for each k, since we have P_{k}^{∘} = \overline{C_{k}}^{∗} × Z_{k} = R_{k} × Z_{k}, we find (\overline{C_{k}}^{∗})′ = R′_{k}, Z(\overline{C_{k}}^{∗}) = Z(P_{k}^{∘}) ∩ \overline{C_{k}}^{∗} and Z(R_{k}) = Z(P_{k}^{∘}) ∩ R_{k}. Therefore, since (\overline{C_{k}}^{∗})′ contains Z(\overline{C_{k}}^{∗})^{∘} (Lemma 8.5), then R_{k} ≥ (\overline{C_{k}}^{∗})′ contains Z(\overline{C_{k}}^{∗})^{∘}, and Z(R_{k}) contains Z(\overline{C_{k}}^{∗})^{∘}. But we have Z(P_{k}^{∘}) = Z(\overline{C_{k}}^{∗}) × Z_{k} = Z(R_{k}) × Z_{k}, hence we obtain Z(R_{k})^{∘} = Z(\overline{C_{k}}^{∗})^{∘} for each k. In particular, the product T (G)(∏_{j=1}^{n} R_{j}) contains Z_{k} = T (G) × (×_{j≠k}Z(R_{j})^{∘}) for each k, and since P_{k}^{∘} is the direct product of R_{k} by Z_{k} for each k, this product contains \overline{C_{k}}^{∗} for each k. Thus we have D(G) = T (G)(∏_{j=1}^{n} R_{j}). Since D(G) is the central product of T (G) and of the groups P_{j} = \overline{C_{j}}^{∗}Z(D(G)) for j ∈ \{1, …, n\}, and since R_{j} is contained in P_{j} for each j, then D(G) the central product of T (G) and of the groups R_{j} for j ∈ {1, …, n}. This implies that that Z(D(G)) is the product of the groups T (G) and Z(R_{j}) for j ∈ \{1, …, n\}. Moreover, since the algebraic groups R_{k} and \overline{C_{k}}^{∗} are isomorphic for each k, and since D(G) is the direct product of T (G) and of \overline{C_{j}}^{∗} for j ∈ \{1, …, n\}, we have rk(D(G)) = rk(T (G))+∑_{j=1}^{n} rk(\overline{C_{j}}^{∗}) = rk(T (G))+∑_{j=1}^{n} rk(R_{j}). Therefore, for each k, the intersection I_{k} := R_{k} ∩ (T (G) ∏_{j≠k} R_{j}) is finite. Since D(G) is the central product of T (G) and of the groups R_{j} for j ∈ {1, …, n}, then I_{k} is central in R_{k}.

For each k, since (\overline{C_{k}}^{∗})′ contains Z(\overline{C_{k}}^{∗})^{∘} by Lemma 8.5, the maximal torus of Z(\overline{C_{k}}^{∗})^{∘} is trivial, and Z(\overline{C_{k}}^{∗})^{∘} is torsion-free. Consequently, for each k, the torsion part J_{k}^{C} of Z(\overline{C_{k}}^{∗}) is finite, and Z(\overline{C_{k}}^{∗}) is the direct product of J_{k}^{C} by Z(\overline{C_{k}}^{∗})^{∘}. But, for each k, the algebraic groups R_{k} and \overline{C_{k}}^{∗} are isomorphic, hence Z(R_{k}) is the direct product of its torsion part J_{k} ≃ J_{k}^{C} by Z(R_{k})^{∘}. Since D(G) is the direct product of T (G) ≤ Z(D(G))^{∘} and of the groups \overline{C_{j}}^{∗} for j ∈ \{1, …, n\}, then Z(D(G)) is the direct product of T (G) and of the groups Z(\overline{C_{j}}^{∗}) for j ∈ {1, …, n}. Thus Z(D(G)) is the direct product of Z(D(G))^{∘} and of the groups J_{j}^{C} for j ∈ {1, …, n}. In particular, its Morley degree is ∑_{j=1}^{n} |J_{j}^{C}| = ∑_{j=1}^{n} |J_{j}|. Since Z(D(G)) is the product of the groups T (G) and Z(R_{j}) for j ∈ {1, …, n}, and since Z(R_{j}) is the direct product of J_{j} by Z(R_{j})^{∘} for each j, then Z(D(G)) is the product of Z(D(G))^{∘} and of the groups J_{j} for j ∈ {1, …, n}. But its Morley degree is ∑_{k=1}^{n} |J_{j}|, hence Z(D(G)) is the direct product of Z(D(G))^{∘} and of the groups J_{j} for j ∈ {1, …, n}.

Now we consider k ∈ {1, …, n} and x ∈ I_{k}. Then there are u ∈ T (G) and x_{j} ∈ R_{j} for each j ≠ k such that x = u ∏_{j≠k} x_{j}. Let x_{k} := x^{−1}. Then we have u ∏_{j=1}^{n} x_{j} = 1 and, for each j, we find x_{j} = u^{−1} ∏_{l≠j} x_{l} ∈ I_{j}. For each j, since I_{j} is a finite central subgroup of R_{j}, we have I_{j} ≤ J_{j}. But we have u ∈ T (G) ≤ Z(D(G))^{∘}, and Z(D(G)) is the direct product of Z(D(G))^{∘} and of the groups J_{j} for j ∈ {1, …, n}. Hence we find u = 1 and x_{j} = 1 for each j. In particular, x and I_{k} are trivial. Thus D(G) is the direct product of T (G) and of the groups R_{k} for k ∈ {1, …, n}, as desired. In the same way, D(G) is the direct product of T (G) and of the groups S_{k} for k ∈ {1, …, n}.

(4) Determination of the quasi-standard automorphism α of D(G).

Since, for each j, the group Z(\overline{C_{j}}^{∗})^{∘} is torsion-free, and since Z(D(G))^{∘} is the direct product of T (G) and of the groups Z(\overline{C_{j}}^{∗})^{∘} for j ∈ \{1, …, n\}, then the torsion part S of T (G) is the one of Z(D(G))^{∘}. Since we have Z_{F} = Z_{E} = Z(D(G)), then SF/F (resp. SE/E) is the torsion part of Z(D(G)/F)^{∘} (resp. Z(D(G)/E)^{∘}), and we obtain δ(SF/F) = SE/E. Moreover, since F ∩ T (G) = E ∩ T (G) = 1, the automorphism δ induces an automorphism δ_{S} of S such that δ(xF) = δ_{S}(x)E for each x ∈ S.

Let X := T (G) ∩ D(G)′F. Since we have T (G) ∩ D(G)′ = T (G) ∩ (\overline{C_{1}}^{∗} × ⋯ × \overline{C_{n}}^{∗})′ = 1, then X is a finite subgroup of T (G), and it is contained in S ≤ T (G). If S = 1, we denote by φ_{X} the identity map of T (G). Otherwise T (G) is a torus, and Lemma 8.12 provides a quasi-standard automorphism φ_{X} of T (G) such that φ_{X}(x) = δ_{S}(x) for each x ∈ X. We consider the automorphism α of D(G) defined as follows: for each j, for each x ∈ R_{j}, α(x) = (ω_{j} ∘ φ_{j})(x); for each x ∈ T (G), α(x) = φ_{X}(x). Thus α is a quasi-standard automorphism of D(G).

(5) D(G)/Z(D(G))^{∘}α(F) and D(G)/Z(D(G))^{∘}E are isomorphic as algebraic groups.

Let \overline{α} : D(G)/F → D(G)/α(F) be the abstract isomorphism induced by α, and let ν = δ ∘ \overline{α}^{−1}. For each i, we have

\overline{α}(P_{i}/F) = α(P_{i})/α(F) = α(R_{i}Z(D(G)))/α(F) = S_{i}Z(D(G))/α(F) = P_{i}/α(F),

and ν(P_{i}/α(F)) = δ(P_{i}/F) = P_{i}/E. We show that the map

\overline{ν} : D(G)/Z(D(G))^{∘}α(F) → D(G)/Z(D(G))^{∘}E

induced by ν is an isomorphism of algebraic groups. We consider the abstract isomorphism \overline{α}^{Z} : D(G)/Z(D(G))^{∘}F → D(G)/Z(D(G))^{∘}α(F) induced by α, and let \overline{δ} : D(G)/Z(D(G))^{∘}F → D(G)/Z(D(G))^{∘}E be the abstract isomorphism induced by δ. For each i, we denote by \overline{ν}_{i} : P_{i}^{∘}α(F)/Z(D(G))^{∘}α(F) → P_{i}^{∘}E/Z(D(G))^{∘}E the restriction of \overline{ν} to P_{i}^{∘}α(F)/Z(D(G))^{∘}α(F). For each i, since the subgroup R_{i} (resp. S_{i}) covers P_{i}^{∘}F/Z(D(G))^{∘}F (resp. P_{i}^{∘}α(F)/Z(D(G))^{∘}α(F)), and since ω_{i} : R_{i} → S_{i} satisfies

ω_{i}(R_{i} ∩ Z(D(G))^{∘}F) = ω_{i}(R_{i}) ∩ α(Z(D(G))^{∘}F) = S_{i} ∩ Z(D(G))^{∘}α(F),

then ω_{i} induces an isomorphism of algebraic groups

\overline{ω}_{i} : P_{i}^{∘}F/Z(D(G))^{∘}F → P_{i}^{∘}α(F)/Z(D(G))^{∘}α(F).

Thus, for each i and each \overline{x} ∈ P_{i}^{∘}α(F)/Z(D(G))^{∘}α(F), we have \overline{ν}_{i}(\overline{x}) = (\overline{δ} ∘ (\overline{α}^{Z})^{−1})(\overline{x}) = (\overline{δ} ∘ φ_{i}^{−1} ∘ \overline{ω}_{i}^{−1})(\overline{x}), so we obtain

\overline{ν}_{i} = (\overline{δ} ∘ φ_{i}^{−1}) ∘ \overline{ω}_{i}^{−1}.

But, for each i, we have

Y_{i} = γ_{G}^{−1}(X_{i}) ∩ \overline{C_{i}}^{∗}Z(D(G))^{∘} ≤ γ_{G}^{−1}(Z(G)^{∘}) ∩ P_{i}^{∘} = Z(D(G))^{∘}F ∩ P_{i}^{∘},

and F ∩P_{i}^{∘} ≤ γ_{G}^{−1}(X_{i})∩\overline{C_{i}}^{∗}Z(D(G))^{∘} = Y_{i}, so we have Z(D(G))^{∘}Y_{i} = Z(D(G))^{∘}F ∩ P_{i}^{∘}, and in the same way, Z(D(G))^{∘}Y_{i}^{H} = Z(D(G))^{∘}E ∩ P_{i}^{∘}. Hence the isomorphism \overline{δ} ∘ φ_{i}^{−1} : P_{i}^{∘}F/Z(D(G))^{∘}F → P_{i}^{∘}E/Z(D(G))^{∘}E is induced by \overline{δ_{i}}, and it is algebraic. Since \overline{ω}_{i} is an isomorphism of algebraic groups too, this proves that \overline{ν}_{i} is an isomorphism of algebraic groups. Since D(G) is generated by the subgroups P_{i}^{∘} for i = 1, …, n, the map \overline{ν} is an isomorphism of algebraic groups, as desired.

(6) Final argument.

Let Δ be the preimage of the graph of ν in D(G) × D(G), and let Z_{D} = Z(D(G))^{∘}α(F)×Z(D(G))^{∘}E. Then ΔZ_{D}/Z_{D} is the graph of \overline{ν}, so it is an algebraic group and, since D(G)/Z(D(G))^{∘}α(F) is connected, ΔZ_{D}/Z_{D} is connected. Moreover, since Z_{D}/(α(F) × E) is connected too, ΔZ_{D}/(α(F) × E) is connected. We have (ΔZ_{D}/(α(F)×E))′ = Δ′(α(F)×E)/(α(F)×E), so Δ′(α(F)×E)/(α(F)×E) is a connected algebraic subgroup of ΔZ_{D}/(α(F) × E).

We consider the diagonal Δ_{0} of T (G) × T (G), and let ρ be the isomorphism from T (G)α(F)/α(F) to T (G)E/E whose graph is Δ_{0}(α(F) × E)/(α(F) × E). Let \overline{u} ∈ D(G)′α(F)/α(F) ∩ T (G)α(F)/α(F). Then there exists x ∈ X such that \overline{α}^{−1}(\overline{u}) = xF. Hence we have ν(\overline{u}) = δ(xF) = δ_{S}(x)E = φ_{X}(x)E = α(x)E, and

(\overline{u}, ν(\overline{u})) = (α(x)α(F), α(x)E) ∈ Δ_{0}(α(F) × E)/(α(F) × E).

Consequently, for each \overline{u} ∈ D(G)′α(F)/α(F) ∩ T (G)α(F)/α(F), we have ν(\overline{u}) = ρ(\overline{u}), and we can define an isomorphism

ν_{ρ} : D(G)′T (G)α(F)/α(F) → D(G)′T (G)E/E

by ν_{ρ}(\overline{x}) = ν(\overline{x}) for each \overline{x} ∈ D(G)′α(F)/α(F), and ν_{ρ}(\overline{x}) = ρ(\overline{x}) for each \overline{x} ∈ T (G)α(F)/α(F). Then the graph of ν_{ρ} is

Δ_{1}/(α(F) × E) := Δ′Δ_{0}(α(F) × E)/(α(F) × E).

Moreover, since Δ′(α(F) × E)/(α(F) × E) and Δ_{0}(α(F) × E)/(α(F) × E) are two algebraic groups, Δ_{1} is algebraic, and ν_{ρ} is an isomorphism of algebraic groups.

Since Δ_{1}/(α(F) × E) is the graph of an isomorphism of algebraic groups between two connected algebraic groups, it is connected. Since ΔZ_{D}/(α(F) × E) is connected, then ΔZ_{D}/Δ_{1} is connected too. But Δ_{1} contains Δ′, so ΔZ_{D}/Δ_{1} is abelian. Hence, since ΔZ_{D}/Δ_{1} and Z_{D}Δ_{1}/Δ_{1} are connected, there is an algebraic complement Δ_{2}/Δ_{1} of Z_{D}Δ_{1}/Δ_{1} in ΔZ_{D}/Δ_{1}. Now, since ΔZ_{D}/Z_{D} is the graph of \overline{ν}, we have ΔZ_{D}/Z_{D} ∩ (D(G) × Z(D(G))^{∘}E)/Z_{D} = 1 and

Δ_{2} ∩ (D(G) × E) ≤ ΔZ_{D} ∩ (D(G) × E)
= Z_{D} ∩ (D(G) × E)
= Z(D(G))^{∘}α(F) × E.

Consequently we obtain

Δ_{2} ∩ (D(G) × E) = Δ_{2} ∩ (Z(D(G))^{∘}α(F) × E)
≤ Δ_{2} ∩ Z_{D}
≤ Δ_{1}.

Finally, , since Δ_{1}/(α(F) × E) is the graph of an isomorphism of algebraic groups from D(G)′T (G)α(F)/α(F) to D(G)′T (G)E/E, we find

Δ_{2} ∩ (D(G) × E) = Δ_{1} ∩ (D(G) × E) = α(F) × E.

In the same way, we obtain Δ_{2} ∩ (α(F) × D(G)) = α(F) × E.

Since Lemma 8.5 implies that (\overline{C_{i}}^{∗})′ contains Z(\overline{C_{i}}^{∗})^{∘} for each i, then Z(D(G))^{∘} is contained in D(G)′T (G). Therefore, for each \overline{x} ∈ Z(D(G))^{∘}F/F = Z(D(G)/F)^{∘}, there exist \overline{d} ∈ (D(G)′F/F)∩(Z(D(G))^{∘}F/F) and \overline{u} ∈ T (G)F/F such that \overline{x} = \overline{d} \overline{u}, and we have

ν_{ρ}(\overline{x}) = ν_{ρ}(\overline{d})ν_{ρ}(\overline{u}) = ν(\overline{d})ρ(\overline{u}) ∈ Z(D(G)/E)^{∘} · T (G)E/E = Z(D(G))^{∘}E/E.

The latter proves that ν_{ρ}(Z(D(G))^{∘}F/F) = Z(D(G))^{∘}E/E. Thus, since the graph of ν_{ρ} is Δ_{1}/(α(F) × E), the group (Δ_{1} ∩ Z_{D})/(α(F) × E) is the graph of the restriction of ν_{ρ} to Z(D(G))^{∘}F/F. Consequently, we obtain

rk(Δ_{1} ∩ Z_{D}) = rk((Δ_{1} ∩ Z_{D})/(α(F) × E)) = rk(Z(D(G))^{∘}F/F) = rk(Z(D(G))).

This yields

rk(Z_{D}Δ_{1}) = rk(Z_{D}) + rk(Δ_{1}) − rk(Δ_{1} ∩ Z_{D})
= 2rk(Z(D(G))) + rk(Δ_{1}) − rk(Z(D(G)))
= rk(Z(D(G))) + rk(Δ_{1}).

Moreover, since ΔZ_{D}/Z_{D} is the graph of \overline{ν}, we have

rk(ΔZ_{D}/Z_{D}) = rk(D(G)/Z(D(G))^{∘}α(F)) = rk(D(G)) − rk(Z(D(G)))

and

rk(ΔZ_{D}) = (rk(D(G)) − rk(Z(D(G)))) + rk(Z_{D}) = rk(D(G)) + rk(Z(D(G))).

Now, since Δ_{2}/Δ_{1} is a complement of Z_{D}Δ_{1}/Δ_{1} in ΔZ_{D}/Δ_{1}, we find

rk(Δ_{2}) = rk(ΔZ_{D}/Δ_{1}) − rk(Z_{D}Δ_{1}/Δ_{1}) + rk(Δ_{1})
= rk(ΔZ_{D}) − rk(Z_{D}Δ_{1}) + rk(Δ_{1})
= (rk(D(G)) + rk(Z(D(G)))) − (rk(Z(D(G))) + rk(Δ_{1})) + rk(Δ_{1})
= rk(D(G)).

Since we have Δ_{2} ∩ (D(G) × E) = α(F) × E and Δ_{2} ∩ (α(F) × D(G)) = α(F) × E by the previous paragraph, the latter implies that Δ_{2}/(α(F) × E) is the graph of an isomorphism β from D(G)/α(F) to D(G)/E. Moreover, β is algebraic since Δ_{2} is an algebraic group.

Finally, D(G)/α(F) and D(G)/E are isomorphic as algebraic groups, as desired. □

We note that Theorem 8.14 together with Lemma 8.10 provide Theorem 1.1 in the case K = \overline{ℚ}.

9. The final argument

In this section, we fix a connected affine ACF_{0}-group M = (G, · ,^{−1} , 1, ⋯), interpretable in the pure field \overline{ℚ}, and an elementary substructure M^{∗} = (G^{∗}, ⋯) of M. We note that the structures M and M^{∗} have finite Morley rank, so any infinite field, interpretable in M or M^{∗}, is algebraically closed by a theorem of A. Macintyre [4, Theorem 8.1].

Lemma 9.1. – Let U^{∗}/V^{∗} be a definable section of G^{∗}, and let U/V be its canonical extension to G. If U/V is definably isomorphic to K_{+} (resp. K^{∗}) for an infinite interpretable field K, then U = U^{∗}V.

Proof – In this case, there is an infinite field L, interpretable in M^{∗}, and a definable morphism α from U^{∗} to L_{+} (resp. L^{∗}), with kernel V^{∗}. Then we have L = M_{0}/R_{0} for M_{0} a definable subset of (G^{∗})^{n}, where n is an integer, and R_{0} a definable equivalence relation over M_{0}. We consider the extensions L_{1}, M_{1}, R_{1} and α_{1} to G of L, M_{0}, R_{0} and α respectively. Since L_{1} is an infinite field, interpretable in \overline{ℚ}, then Fact 2.2 says that L_{1} is isomorphic to \overline{ℚ}. Moreover, L is infinite and interpretable in M^{∗}, so it is algebraically closed. Since \overline{ℚ} has no proper algebraically closed subfield, this implies that the canonical embedding of L in L_{1} is an isomorphism: the subset M_{0} of M_{1} covers L_{1} = M_{1}/R_{1}. Thus, for each u ∈ U, there is u^{∗} ∈ U^{∗} such that α_{1}(u) = α_{1}(u^{∗}). Since α_{1} is a morphism with kernel V, this yields the result. □

We note that, since G is connected and since its ground field is algebraically closed and of characteristic zero, there is no proper subgroup of finite index in G.

Proposition 9.2. – G = G^{∗}Z(G)^{∘}.

Proof – Since there is no proper subgroup of finite index in G, we have just to prove that G = G^{∗}Z(G). Let B^{∗} be a Borel subgroup of G^{∗}, that is a maximal solvable connected definable subgroup, and let B be its extension to G. Then B is a maximal solvable connected definable subgroup of G, so it is an algebraic Borel subgroup of G by [4, Corollary 5.38] and Lemma 4.1. In particular, we have Z(B) = Z(G), so Z(B^{∗}) = Z(G^{∗}). Moreover, since B is the extension of B^{∗}, we have B = d(B^{∗}). Let \overline{D} = D/Z(G) be the largest connected definable subgroup of \overline{B} = B/Z(G) contained in \overline{B^{∗}} = B^{∗}Z(G)/Z(G). Then \overline{D} is normal in \overline{B^{∗}}, so \overline{D} is normal in \overline{B}.

If \overline{D} = \overline{B^{∗}}, then we have \overline{B^{∗}} = \overline{B}. Since the conjugates of B cover G, the ones of B^{∗} in G^{∗} cover G^{∗}. In particular, \overline{G^{∗}} = G^{∗}Z(G)/Z(G) is generated by the conjugates of \overline{D} in \overline{G^{∗}}. Now, since \overline{D} is definable in \overline{G} and connected, \overline{G^{∗}} is a connected definable subgroup of \overline{G}. Since G^{∗} is an elementary substructure of G, this proves that \overline{G^{∗}} = \overline{G}, and G = G^{∗}Z(G). Hence we may assume \overline{D} ≠ \overline{B^{∗}}.

Let D_{0}^{∗}/Z(G^{∗}) be the largest connected definable subgroup of B^{∗}/Z(G^{∗}), with D_{0}^{∗} contained in D, and let D_{0} be its extension to G. Then D_{0}^{∗} is normal in B^{∗}, and D_{0}/Z(G) is a normal connected definable subgroup of B/Z(G), and D_{0} is contained in D. Let U^{∗}/D_{0}^{∗} be a B^{∗}-minimal section of B^{∗}, and let U be the extension of U^{∗} to G. Then U/D_{0} is a B-minimal section of B. By choice of D_{0}^{∗}, the subgroups U^{∗} and U are not contained in D, and U^{∗}/Z(G^{∗}) and U/Z(G) are connected.

If U/D_{0} is definably isomorphic to K_{+} or K^{∗} for an infinite interpretable field K, then Lemma 9.1 gives U = U^{∗}D_{0}. But this implies that U is contained in B^{∗}D, hence U/Z(G) is contained in \overline{B^{∗}}, contradicting the maximality of \overline{D}.

If the Fitting subgroup F (B) covers U/D_{0}, then U/D_{0} is definably isomorphic to a section of F (B)/Z(B). But F (B)/Z(B) is a Ũ-group by Fact 2.16, hence U/D_{0} is a Ũ-group by Fact 2.14. Since \overline{ℚ} is algebraically closed of characteristic zero, there is no infinite definable group of bounded exponent in F (B)/Z(B), and Fact 2.12 shows that F (B)/Z(B) is torsion-free. In particular, U/D_{0} is torsion-free. Consequently, by B-minimality of U/D_{0}, Fact 2.12 provides an interpretable algebraically closed field L of characteristic zero such that U/D_{0} is definably isomorphic to L_{+}. This contradicts the previous paragraph, so F (B) does not cover U/D_{0}, and (F (B) ∩ U)D_{0}/D_{0} is finite. But D_{0} contains Z(B) and F (B)/Z(B) is torsion-free, so F (B) avoids U/D_{0}.

Since B is a connected solvable algebraic group, B/F (B)^{∘} is abelian, and F (B) is a maximal nilpotent subgroup of B, so F (B) is the intersection of the centralizers of the B-minimal sections of B. Then we find finitely many B-minimal sections R_{1}/S_{1}, …, R_{n}/S_{n} of B such that F (B) is the intersection of their centralizers in B. Now, for each i = 1, …, n, either B centralizes R_{i}/S_{i}, or B/C_{B}(R_{i}/S_{i}) is definably isomorphic to K_{i}^{∗} for an interpretable algebraically closed field K_{i} (Fact 2.1). Consequently, for each i = 1, …, n such that C_{B}(R_{i}/S_{i}) does not cover U/D_{0}, there is a finite subgroup F_{i}/D_{0} of U/D_{0} such that U/F_{i} is definably isomorphic to an infinite definable subgroup of K_{i}^{∗}. Morover, in \overline{ℚ}, the fields K_{i} and \overline{ℚ} are definably isomorphic (Fact 2.2), so K_{i}^{∗} has Morley rank one (in \overline{ℚ}). This implies that U/F_{i} is definably isomorphic to K_{i}^{∗}.

Now, if i and j are two elements of {1, …, n} such that the subgroups C_{B}(R_{i}/S_{i}) and C_{B}(R_{j}/S_{j}) do not cover U/D_{0}, then U/F_{i}F_{j} is definably isomorphic to K_{i}^{∗} and K_{j}^{∗}, so K_{i} and K_{j} are definably isomorphic (Lemma 5.4). Moreover, since F (B) avoids U/D_{0}, the intersection of the subgroups F_{i} of this form is D_{0}. Thus, there is i ∈ {1, …, n} such that U/D_{0} is definably isomorphic to a subgroup of (K_{i}^{∗})^{n}. By minimality of U/D_{0}, this implies that U/D_{0} is definably isomorphic to K_{i}^{∗}, contradicting that U/D_{0} is definably isomorphic to no group of the form K^{∗} for an infinite interpretable field K. This finishes the proof. □

Corollary 9.3. – If G′ contains Z(G)^{∘}, then G = G^{∗}, otherwise G is abstractly isomorphic to G^{∗} × \overline{ℚ}_{+}.

Proof – In the first case, Proposition 9.2 gives G′ = (G^{∗})′, so G^{∗} ≥ G′ contains Z(G)^{∘}, and again Proposition 9.2 provides the result.

In the second case, we note that, since the ground field of G is of characteristic zero, there are finitely many elements of order n in Z(G) for each n ∈ ℕ. Hence Z(G^{∗}) contains the torsion of Z(G). Moreover, since the ground field of G is algebraically closed and of characteristic zero, Z(G)^{∘} is divisible, so Z(G^{∗})^{∘} is divisible too. Therefore Z(G^{∗})^{∘} = G^{∗} ∩ Z(G)^{∘} has a complement D in Z(G)^{∘}, and D is divisible and torsion-free since Z(G)^{∘} is divisible and since Z(G^{∗}) contains the torsion of Z(G). Now Proposition 9.2 gives G = G^{∗} × D. Since D is divisible and torsion-free, it is a direct product of countably many copies of ℚ, and \overline{ℚ}_{+} is isomorphic to \overline{ℚ}_{+} ×D. But Z(G)^{∘} is not contained in G′, hence Lemma 8.10 shows that G is abstracly isomorphic to

G × \overline{ℚ}_{+} ≃ G^{∗} × D × \overline{ℚ}_{+} ≃ G^{∗} × \overline{ℚ}_{+},

and the result follows. □

Corollary 9.4. – Let H be another connected affine algebraic group over \overline{ℚ}. If the pure groups G and H are elementarily equivalent, then they are abstractly isomorphic.

Proof – We assume that M = (G, · ,^{−1} , 1), and we consider the structure N = (H, · ,^{−1} , 1). Let T be the theory of M and N. By [15, Theorem 4.2.20], there is an elementary substructure M_{0} = (G_{0}, · ,^{−1} , 1) of M (resp. N_{0} = (H_{0}, · ,^{−1} , 1) of N), where M_{0} (resp. N_{0}) is a prime model of T. Then M_{0} and N_{0} are isomorphic by [15, Corollary 4.2.16]. Now Corollary 9.3 says that,

• either G′ contains Z(G)^{∘}, therefore H′ contains Z(H)^{∘}, and we have G = G_{0} ≃ H_{0} = H;

• or G′ does not contain Z(G)^{∘}, therefore H′ does not contain Z(H)^{∘}, and

we have

G ≃ G^{∗} × \overline{ℚ}_{+} ≃ H^{∗} × \overline{ℚ}_{+} ≃ H.

□

From now on, we are ready for the proof of Theorem 1.1 and, simultaneously, for the one of Theorem 1.5 in the affine case. We notice that, for the proof below, it is essential that the automorphism α of D in Theorem 8.14 is quasi-standard: the transition from the ground field \overline{ℚ} to any algebraically closed field requires this property.

Proof of Theorem 1.1 in the general case, and of Theorem 1.5 for affine groups – Let K be an algebraically closed field of characteristic zero, and let G be a connected affine algebraic group over K. We consider an elementary substructure K_{1} of the pure field K, with K_{1} isomorphic to \overline{ℚ}. Let G_{1} be the elementary substructure of G in K_{1}. Then Lemma 8.10 and Theorem 8.14 provide the existence of a connected affine algebraic group D_{1} over K_{1} and of a finite central subgroup F of D_{1}, such that the affine algebraic groups over K_{1} abstractly isomorphic to G_{1} are the following ones, up to isomorphism of algebraic groups:

• D_{1}/α_{1}(F) if G′_{1} contains Z(G_{1})^{∘};

• D_{1}/α_{1}(F) × (K_{1})_{+}^{s} if Z(G_{1})^{∘} is not contained in G′_{1},

where α_{1} is a quasi-standard automorphism of D_{1}, and s is an integer. We may assume that there is an integer s such that G_{1} and D_{1}/F × (K_{1})_{+}^{s} are isomorphic as algebraic groups, and we may assume that D_{1} is constructed as D(G_{1}) in §8. We consider the elementary extension D of D_{1} to K. We note that, since the extension of F to K is F, the algebraic groups G and D/F × K_{+}^{s} are isomorphic.

Firstly, if G′ contains Z(G)^{∘}, then for each quasi-standard automorphism α of D, the groups D/F and D/α(F) are abstractly isomorphic. Secondly, if Z(G)^{∘} is not contained in G′, we consider an integer r and a quasi-standard automorphism α of D, and we show that D/α(F) × K_{+}^{r} is abstractly isomorphic to G. Since Z(G)^{∘} is not contained in G′, then Z(G_{1})^{∘} is not contained in G′_{1}, and the subgroup T_{1} of D_{1}, corresponding to T (G_{1}) in §8, is either (K_{1})_{+}, or (K_{1}^{∗})^{k} for a positive integer k. Hence D has an algebraic subgroup A such that we have either D = A × K_{+}, or D = A × (K^{∗})^{k} for a positive integer k. Since K has characteristic zero, then for each integer l, the groups K_{+} (resp. (K^{∗})^{k}) and K_{+} × K_{+}^{l} (resp. (K^{∗})^{k} × K_{+}^{l}) are abstractly isomorphic. This implies that D/F is abstractly isomorphic to D/F ×K_{+}^{l} for each integer l. In particular, the groups G, D/F and D/F × K_{+}^{r} are abstractly isomorphic. Since D/F and D/α(F) are abstractly isomorphic too, we obtain the abstract isomorphy between G and D/α(F) × K_{+}^{r}.

Now we consider a connected affine algebraic group H over K, and we assume that the pure groups G and H are elementarily equivalent. Let H_{1} be the elementary substructure of H in K_{1}. Then the pure groups G_{1} and H_{1} are elementarily equivalent, and Corollary 9.4 shows that G_{1} and H_{1} are abstractly isomorphic. Consequently, there exists a quasi-standard automorphism α_{1} of D_{1} such that:

• either G′_{1} contains Z(G_{1})^{∘}, and H_{1} and D_{1}/α_{1}(F) are isomorphic as algebraic groups;

• or G′_{1} does not contain Z(G_{1})^{∘}, and there is an integer r such that H_{1} and

D_{1}/α_{1}(F) × (K_{1})_{+}^{r} are isomorphic as algebraic groups. Since α_{1}(F) is finite, its extension to K is α_{1}(F), and one of the following two conditions is satisfied:

• either Z(G)^{∘} ≤ G′, and H and D/α_{1}(F) are isomorphic as algebraic groups;

• or Z(G)^{∘} ≰ G′, and H and D/α_{1}(F) × K_{+}^{r} are isomorphic as algebraic

groups. Moreover, since α_{1} is a quasi-standard automorphism of D_{1}, there is two decompositions D_{1} = R_{1}^{∗}×⋯×R_{n}^{∗} and D_{1} = S_{1}^{∗}×⋯×S_{n}^{∗} of D_{1}, where R_{1}^{∗}, …, R_{n}^{∗}, S_{1}^{∗}, …, S_{n}^{∗} are some algebraic subgroups of D_{1} such that α_{1}(R_{i}^{∗}) = S_{i}^{∗} for each i, and such that the isomorphism α_{1}^{i} : R_{i}^{∗} → S_{i}^{∗} induced by α_{1} is standard for each i. Thus, for each i = 1, …, n, there is a field automorphism δ_{i}^{∗} of K_{1} and an isomorphism μ_{i}^{∗} : R_{i}^{∗} → S_{i}^{∗} of algebraic groups such that α_{1}^{i} = μ_{i}^{∗} ∘ δ_{i}^{∗}. Now we consider the extensions R_{1}, …, R_{n}, S_{1}, …, S_{n} to K of R_{1}^{∗}, …, R_{n}^{∗}, S_{1}^{∗}, …, S_{n}^{∗} respectively, and the ones μ_{1}, …, μ_{n} of μ_{1}^{∗}, …, μ_{n}^{∗} respectively. Moreover, for each i, we consider an extension δ_{i} to K of the automorphism δ_{i}^{∗} of K_{1}. Then, for each i, the isomorphism α^{i} = μ_{i} ∘ δ_{i} from R_{i} to S_{i} is standard, and it satisfies α^{i}(x) = α_{1}^{i}(x) for each x ∈ R_{i}^{∗}. Now the automorphism α of D defined by α(x) = α^{i}(x) for each x ∈ R_{i} is quasi-standard, and it satisfies α(F) = α_{1}(F). Finally, H has the desired form. □

Acknowledgements

I would like to thank Tuna Altınel for his valuable comments that contributed to improve this article, and his encouragement.

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