Pseudofinite groups with NIP theory and definability in finite simple groups

Dugald Macpherson^{∗}, School of Mathematics, University of Leeds, Leeds LS2 9JT, England

h.d.macpherson@leeds.ac.uk

Katrin Tent^{†}, Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany tent@math.uni-muenster.de

July 27, 2011

Abstract: We show that any pseudofinite group with NIP theory and with a finite upper bound on the length of chains of centralisers is soluble-by-finite. In particular, any NIP rosy pseudofinite group is soluble-by-finite. This generalises, and shortens the proof of, an earlier result for stable pseudofinite groups. An example is given of an NIP pseudofinite group which is not soluble-by-finite. However, if C is a class of finite groups such that all infinite ultraproducts of members of C have NIP theory, then there is a bound on the index of the soluble radical of any member of C. We also survey some ways in which model theory gives information on families of finite simple groups, particularly concerning products of images of word maps.

1 Introduction

We consider in this paper groups G which are pseudofinite, that is, infinite groups which satisfy every first order sentence (in the language L_{g} of groups) which holds in all finite groups. Equivalently, G is elementarily equivalent to an infinite ultraproduct of finite groups. Or equivalently again, G is an infinite group with the finite model property: every sentence in the theory of the group has a finite model. We consider the structure of G, under the assumption that the first order theory Th(G) of G satisfies various generalisations of model theoretic stability.

It was shown in [21] that any stable pseudofinite group G has a definable soluble normal subgroup of finite index. This is not surprising; for by a classification due to Wilson [32] (with a slight strengthening due to Ryten – see [9, Proposition 2.14]) – any infinite pseudofinite simple group is a group of Lie type over a pseudofinite field, and in particular interprets a pseudofinite field [27, 5.2.4, 5.3.3, 5.4.3], and so has unstable theory by Duret [5]. However, an intricate argument with centralisers was needed in [21] to bound the derived length of soluble normal subgroups.

∗Research partially supported by EPSRC grant EP/H00677X/1

†Research partially supported by SFB 878

One generalisation of stability is the notion of simple theory. Pseudofinite fields (and certain difference fields, that is, fields equipped with a specified automorphism) are simple, in fact supersimple of finite rank, and it follows from Wilson’s classification that every simple pseudofinite group is interpretable in such a structure. Hence, every simple pseudofinite group has supersimple finite rank theory; this follows from the results of Hrushovski [12] and is made explicit in [9] (note that measurable structures are supersimple of finite rank – see e.g. [8, Corollary 3.7]). A satisfactory structure theory for pseudofinite groups with supersimple finite rank theory – under an additional assumption that ∃^{∞} is definable in T^{eq} – was initiated in [9]. The class of supersimple finite rank structures is sufficiently rich to include a lot of pseudofinite group theory, as indicated by, for example, [19, 4.11, 4.12]. Possible applications of the model theory of supersimple theories to finite simple groups are discussed in the final section of the present paper.

Another generalisation of stability of considerable current interest is that of NIP, or dependent theory. A formula ϕ(x̄, ȳ) has the independence property with respect to T if there is M ⊧ T and a set {ā_{i} : i ∈ ω} ⊂ M^{l(x̄)} such that for all S ⊆ ω there is b̄_{S} ∈ M^{l(ȳ)} such that for all i ∈ ω, M ⊧ ϕ(ā_{i}, b̄_{S}) if and only if i ∈ S. A theory T is NIP if no formula has the independence property with respect to T. Any simple or NIP theory is stable, and any theory which is both simple and NIP is stable. For groups, by the Baldwin-Saxl Theorem (see [2], or [6, Fact 0.17]) the NIP condition implies a useful chain condition: if G is an NIP group, then for every formula ϕ(x, ȳ) there is a natural number n_{ϕ} such that every finite intersection of ϕ-definable groups is an intersection of n_{ϕ} ϕ-definable groups. By Wilson’s theorem, there is no simple pseudofinite group with NIP theory, and we expected this, together with the above chain condition, to yield virtual solubility for pseudofinite groups with NIP theory. However, this is false, and in Section 3 below we give a construction of a pseudofinite group G with NIP theory which is not soluble-by-finite.

Our main theorem is the following. We say that a group G has the centraliser chain condition if there is a natural number n = n(G) such that there do not exist subsets F_{1}, …, F_{n+1} ⊂ G with

C_{G}(F_{1}) < … < C_{G}(F_{n+1}).

Theorem 1.1 Let G be a pseudofinite group with NIP theory, and suppose that G satisfies the centraliser chain condition. Then G has a soluble definable normal subgroup of finite index.

We obtain some information about finite groups just under an NIP assumption. Let us say that the class C of finite structures is an NIP class if every infinite ultraproduct of members of C has NIP theory. As a step in the proof of Theorem 1.1 we obtain the following result. Here, and throughout the paper, if G is a finite group we denote by R(G) its soluble radical, that is, the unique largest soluble normal subgroup of G.

Proposition 1.2 Let C be an NIP class of finite groups. Then there is d = d(C) ∈ ℕ such that |G : R(G)| ≤ d for every G ∈ C.

The notion of rosy theory is a common generalisation of the notions of ominimal theory and simple (and hence also of stable) theory. The concept was introduced by Onshuus in [22] and developed in Adler [1]. We omit the definition of rosiness, but note that by [6, Definition 0.3], a theory T is rosy if there is an independence relation | ⌣ on real and imaginary tuples which satisfies the following natural conditions :

(i) | ⌣ is automorphism invariant.

(ii) If c ∈ acl(aB) \ acl(B), then a |̸ ⌣_{B} c.

(iii) If a | ⌣_{B} C and B ∪ C ⊆ D, then there is a′ ∈ tp(a/BC) with a′ |̸ ⌣_{B} D. (iv) There is λ such that for any a, if (B_{i})_{i<α} are sets with B_{i} ⊂ B_{j} whenever

i < j and a |̸ ⌣_{B_{i}} B_{j} for i < j < α, then α < λ.

(v) If B ⊆ C ⊆ D, then a | ⌣_{B} D if and only if A | ⌣_{B} C and a | ⌣_{C} D.

(vi) C | ⌣_{A} B if and only if c | ⌣_{A} B for any finite c ⊆ C.

(vii) a | ⌣_{C} b if and only if b | ⌣_{C} a. A structure with an infinite descending chain of uniformly definable equivalence relations can never be rosy – see e.g. the proof of Proposition 1.3 in [6]. In particular, a field with a non-trivial definable valuation can never be rosy, and more generally a group with an infinite strictly descending chain of uniformly definable subgroups cannot be rosy. In combination with the consequence mentioned above of the Baldwin-Saxl Theorem this yields the following, for groups.

Proposition 1.3 [6, Corollary 1.8] Any group definable in an NIP rosy theory has the centraliser chain condition.

By Theorem 1.1, this yields immediately the following.

Corollary 1.4 Let G be a pseudofinite group with NIP rosy theory. Then G has a soluble definable normal subgroup of finite index.

We should not expect here to replace ‘soluble’ ‘by ‘nilpotent’, since examples (involving Chapuis, Simonetta, Khelif, and Zilber) are mentioned at the end of [21] of stable pseudofinite groups which are not nilpotent-by-finite.

Theorem 1.1 is proved in Section 2. In addition to Proposition 1.3, and the classification of simple pseudofinite groups, we use the following two results.

Theorem 1.5 [33, Wilson] There is a formula ψ(x) such that for every finite groups G, we have R(G) = {x ∈ G : G ⊧ ψ(x)}.

Theorem 1.6 [13, Khukhro] There is a function f : ℕ → ℕ, such that for any d ∈ ℕ, if G is a finite soluble group with no strictly descending chain of centralisers of length d + 1, then G has derived length at most f(d).

The final section of the paper is a discussion of some possible applications of model theory to structural questions on families of finite simple groups of fixed Lie rank. There are three main sources of applications: a generalisation of the Zilber Indecomposability Theorem for groups in supersimple theories; some still-unpublished work of Ryten showing that any family of finite simple groups is an ‘asymptotic class’, so that cardinalities of definable sets satisfy Lang-Weil-like uniformities; and results on generic types of groups in supersimple theories. No new results here are given, but the methods give, for example, a alternative approach to some recent results on word maps. For the Suzuki and Ree groups there is heavy dependence on a major result of Hrushovski [12].

Acknowledgement. We thank Sasha Borovik for drawing our attention to Theorem 1.6.

2 Proof of Theorem 1.1.

Proof of Proposition 1.2. Let C = {G_{i} : i ∈ ℕ} be a class of finite groups such that every non-principal ultraproduct of members of C has NIP theory. By Theorem 1.5, with ψ(x) the formula given in that theorem, for each i ∈ ω we have R(G_{i}) = {x ∈ G_{i} : G_{i} ⊧ ψ(x)}. By Łos’s Theorem, ψ defines a normal subgroup, denoted ψ(G), of any ultrapower G of members of C.

Write Ḡ_{i} := G_{i}/R(G_{i}), and let S_{i} := Soc(Ḡ_{i}), the direct product of the minimal normal subgroups of Ḡ_{i}. By the maximality of R(G_{i}), each minimal normal subgroup of Ḡ_{i} is non-abelian.

Claim 1. There is t ∈ ℕ such that each Ḡ_{i} has at most t distinct minimal normal subgroups.

Proof of Claim. Suppose that the claim is false. Then for each n ∈ ℕ there is i_{n} ∈ ℕ such that Ḡ_{i_{n}} has distinct minimal normal subgroups T_{i_{n},1}, …, T_{i_{n},n} (and possibly others). For each such n and j ∈ {1, …, n}, let x_{n,j}, y_{n,j} be noncommuting elements of T_{i_{n},j}. For such n and for any w ⊂ {1, …, n} let z_{n,w} be the product of the elements {x_{n,j} : j ∈ w} (the order of the product does not matter, as the x_{n,j} lie in distinct minimal normal subgroups, so commute). Now y_{n,j}z_{n,w} ≠ z_{n,w}y_{n,j} (in Ḡ_{i_{n}}) if and only if j ∈ w. It follows by Łos’s Theorem that a non-principal ultrafilter can be chosen on ℕ so that the formula χ(y, z) of form yz ≠ zy witnesses that Π_{i}Ḡ_{i}/U has the independence property. Thus, as Ḡ_{i} is uniformly interpretable in G_{i}, the infinite group Π_{i∈ℕ}G_{i}/U does not have NIP theory, a contradiction.

By Claim 1, after partitioning ℕ into finitely many sets if necessary, we may suppose that for each i, S_{i} is a direct product of exactly t distinct minimal normal subgroups of Ḡ_{i}, namely T_{i,1}, …, T_{i,t}. Each T_{i} is characteristically simple, so being non-abelian is a direct product of isomorphic non-abelian simple groups. Arguing as in the proof of Claim 1, we may suppose that for some r ∈ ℕ, each T_{i,j} is a direct product of at most r non-abelian simple groups. Thus, we may reduce to the case when each S_{i} is a direct product of exactly c minimal normal subgroups of S_{i}, namely S_{i} = R_{i,1} × … × R_{i,c}, where each R_{i,c} is non-abelian simple.

Claim 2. There is e ∈ ℕ such that any non-abelian simple subgroup of Ḡ_{i} has Lie rank at most e (where we define the Lie rank of the alternating group Alt_{n} to be n, and that of the sporadic simple groups to be 1).

Proof of Claim. We argue as in the proof of Claim 1. It suffices to note that for any n, a sufficiently large alternating group contains a direct product of n copies of Alt_{4}. Likewise, non-abelian classical simple groups of large rank contain many commuting copies of PSL_{2}(q).

Claim 3. Let F be a family of finite simple groups of fixed Lie rank e. Then there is d = d(e) ∈ ℕ such that if K ∈ F and g, h ∈ K \ {1} then g is a product of at most d copies of h and h^{−1}.

Proof of Claim. This is well-known. It follows for example from the theorem in [25] that any non-principal ultraproduct of members of F is a group of the same Lie type over a pseudofinite field, and so is simple.

By Claims 2 and 3 we obtain the following: there is b ∈ ℕ such that for each i, c here is x_{i,c} ∈ R_{i,c} such that each element of R_{i,c} is a product of at most b R_{i,c}-conjugates of x_{i,c} and x_{i,c}^{−1}. As Ḡ_{i} normalises each T_{i,j}, and each T_{i,j} is a product of boundedly many groups of form R_{i,c}, it follows easily that the T_{i,j} are uniformly definable in the Ḡ_{i}. Hence also the R_{i,j} are uniformly definable in Ḡ_{i}.

To complete the proof of the proposition, it suffices to show that there is e ∈ ℕ such that |S_{i}| ≤ e for all i. For suppose this holds. Then C_{i} := CḠ_{i}(S_{i}) = 1, since otherwise, as S_{i} is centreless, the socle of C_{i} would be a product of minimal normal subgroups of Ḡ_{i} not contained in S_{i}, which is impossible. Thus, Ḡ_{i} embeds in Aut(S_{i}), so has order at most e!.

So suppose for a contradiction that there is no finite upper bound on |S_{i}|. Then by the classification of finite simple groups, there is some Lie type Chev (possibly twisted, but with the Lie rank fixed) such that the R_{i,j} include arbitrarily large finite simple groups of type Chev. Relabelling if necessary, we may suppose there is a subsequence (n_{i} : i ∈ ℕ) of ℕ such that each finite simple group R_{n_{i},1} has Lie type Chev, and |R_{n_{i},1}| → ∞ as i → ∞. We may suppose that R_{n_{i},1} is defined in Ḡ_{n_{i}} by the formula ϕ(x, ā_{i}).

Let U be a non-principal ultrafilter on ℕ containing N = {n_{i} : i ∈ ℕ}, and hence all cofinite subsets of N. Put G = Π_{i∈ℕ}G_{i}/U, and Ḡ := G/H, where H is the normal subgroup of G defined by ψ. Then there is ā ∈ Ḡ such that ϕ(x, ā) defines an infinite ultrapower of groups of type Chev, and hence, by [25], a group of Lie type Chev over a pseudofinite field. Such a subgroup has the independence property, by the results of Ryten and Duret mentioned above. It follows that Ḡ, and hence G, does not have NIP theory, a contradiction. □

The following lemma is standard.

Lemma 2.1 Let L be a countable language and M be a pseudofinite L-structure. Then there is an infinite class C of finite structures such that every infinite ultrapower of members of C is elementarily equivalent to M.

Proof. We may suppose that M = Π_{n∈ℕ}M_{i}/U where the M_{i} are finite with |M_{i}| → ∞ as i → ∞. Let {σ_{i} : i ∈ ℕ} list Th(M). Iteratively, we find a sequence U_{0} ⊃ U_{1} ⊃ … of members of U such that for each i ∈ ℕ, U_{i} contains the i smallest elements n_{i1} < … < n_{ii} of U_{i−1}, and such that for all i ∈ ℕ and j ∈ U_{i} with j > n_{ii}, M_{j} ⊧ σ_{i}. Put U := ⋂_{i∈ℕ} U_{i}. Then U is infinite, and by Łos’s Theorem, C := {M_{i} : i ∈ U} satisfies the lemma. □

Proof of Theorem 1.1. Let G be a pseudofinite group with NIP theory, such that every chain of centralisers has length at most e. Observe that there is a sentence τ_{e} in the language L_{g} of groups such that for every group H, H ⊧ τ_{e} if and only if every chain of centralisers in H has length at most e. By Lemma 2.1 there is a set C := {G_{i} : i ∈ ℕ} and an ultrafilter U on ℕ such that (after replacing G by an elementarily equivalent group if necessary) G = Π_{i∈ℕ}G_{i}/U, and every infinite ultraproduct of members of C is elementarily equivalent to G. It follows that C is an NIP class of finite groups, so by Proposition 1.2 there is d ∈ ℕ such that |G_{i} : R(G_{i})| ≤ d for all i ∈ ℕ. Also M_{i} ⊧ τ_{e} for cofinitely many i ∈ ℕ. Hence, by Theorem 1.6, R(G_{i}) has derived length at most f(e) for cofinitely many i ∈ ℕ. The property that the derived length is at most f(e) is first order expressible by a sentence asserting that a certain word vanishes on a group. Thus, by Łos’s Theorem, the normal subgroup R(G) := {x ∈ G : G ⊧ ψ(x)} is soluble of derived length at most f(e), and index at most d in G. □

3 A pseudofinite NIP group which is not solubleby-finite

We here prove the following theorem.

Theorem 3.1 There is a pseudofinite group G with NIP theory which is not soluble-by-finite.

Proof. Fix a prime p. It is well-known that the valued field ℚ_{p}, and hence its valuation ring ℤ_{p}, has NIP theory. Hence, the group H := SL_{2}(ℤ_{p}), which is interpretable in ℤ_{p}, also has NIP theory. Let M := pℤ_{p}, the maximal ideal of ℤ_{p}. For each k > 0 let H_{k} be the congruence subgroup of H consisting of matrices ( \begin{matrix}1 + a & b \\ c & 1 + d \end{matrix} ) which lie in H and satisfy a, b, c, d ∈ p^{k}M. Then H_{k}

is normal in H, and the quotient H̄_{k} := H/H_{k} is finite.

Let U be a non-principal ultrafilter on ω, and let G be the ultraproduct ΠH̄_{k}/U. Then G is a pseudofinite group, and is NIP by the following lemma, since the groups H̄_{k} are uniformly interpretable in an NIP theory.

If M is a structure and ϕ(x_{1}, …, x_{m}, y_{1}, …, y_{n}) is a formula which does not have the independence property in M, then there is a greatest natural number d such that there are distinct ā_{1}, …, ā_{d} ∈ M such that for each S ⊆ {1, … d} there is b̄_{S} ∈ M^{n} with, for all i ∈ {1, …, d}, M ⊧ ϕ(ā_{i}, b̄_{S}) ⇔ i ∈ S. Such d is called the Vapnik-Chervonenkis dimension, or VC-dimension, of the family of definable sets in the x̄-variables determined by ϕ (or just of the formula ϕ).

Lemma 3.2 Let L, L′ be first order languages, and let M be an L-structure with NIP theory. Suppose that {M_{i} : i ∈ I} is a set of L′-structures which is uniformly interpretable in M (with I an interpretable set of M). Let J be an infinite subset of I and V a non-principal ultrafilter on J. Then the ultraproduct N = Π_{j∈J}M_{j}/V has NIP theory.

Proof of Lemma 3.2. It suffices to observe that the VC-dimension of any L′-formula ϕ(x̄, ȳ) is uniformly bounded across the class of structures M_{i}. We leave the details as an exercise. □

Note that if a group is soluble-by-finite, then so are all its subgroups and quotients. Therefore, in order to show that G is not soluble-by-finite, we first prove the following claim.

Claim 1. The group G has a normal subgroup N such that G/N ≅ SL_{2}(ℤ_{p}). Proof of Claim. We view the groups H̄_{k} and G as structures in the language

L^{+} := L_{g} ∪ {P_{i} : i < ω} where the P_{i} are unary predicates. In H̄_{k}, P_{i} is interpreted by H_{i}/H_{k} for i ≤ k and by 1 = H_{k}/H_{k} for i > k. Thus, the P_{i} are interpreted by a descending chain of normal subgroups of H̄_{k}. The group G has by Łos’s Theorem a corresponding strictly descending chain P_{0}^{G} > P_{1}^{G} > … consisting of normal subgroups of G. Put N := ⋂_{i∈ω} P_{i}^{G}. Compactness together with ω-saturation of G (viewed as an L^{+}-structure) yields that G/N ≅ SL_{2}(ℤ_{p}).

To complete the proof of Theorem 3.1 we now note:

Claim 2. The group SL_{2}(ℤ_{p}) is not soluble-by-finite.

Proof of Claim. This must be well-known: if it were soluble-by-finite, then so would be SL_{2}(ℤ) < SL_{2}(ℤ_{p}) and its quotient PSL_{2}(ℤ), which is a free product of a cyclic group of order two and a cyclic group of order three, and clearly not soluble-by-finite (see [26], Section 6.2). □

Remark 3.3 Let G be a pseudofinite NIP group which is not soluble-by-finite. By Lemma 2.1 G ≅ H for some ultraproduct H = Π_{i∈ℕ}H_{i}/U, such that every infinite ultraproduct of the H_{i} is elementarily equivalent to G.

The formula ψ(x) defines a non-soluble normal subgroup ψ(H) of finite index in H. By the methods of Section 2, it can be shown that ψ(H) has subgroups N_{1} < N_{2} which are normal in H, such that ψ(H)/N_{2} is pro-soluble (an inverse limit of soluble groups) but not soluble, and N_{1} is the union of a chain of soluble groups but is not soluble. We have not investigated the possible structure of N_{2}/N_{1}. In fact, these conclusions can be shown to hold for any infinite NIP group which is a non-principal ultraproduct of distinct finite groups and is not soluble-by-finite.

4 Model theory of finite simple groups

In this section we make some remarks about possible applications of model theory to finite group theory, via pseudofinite groups. As mentioned in the introduction, one generalisation of the notion of stable first order theory is that of simple theory. This notion was introduced by Shelah in [30] and developed in the 1990s in [14] and [15] and further in other papers. Many ideas first appeared in [4] and in early versions of [11]. A convenient source, mainly used below, is [31]. Simplicity theory is a context for an abstract theory of independence, given by ‘non-forking’, which is less powerful than the corresponding independence theory in stability theory, but stronger than that in rosy theories. In stable theories, over a suitable base, the first order type of tuples ā and b̄, combined with the knowledge that they are independent, determines the type of āb̄, but this is false in general in simple theories.

We emphasise the distinction between the group-theoretic notion of simple group and the model-theoretic notion of group definable in a simple theory. We also stress that our methods below only seem to have applications for families of finite simple groups of fixed Lie rank.

Among the simple theories are the supersimple ones, for which there are global model-theoretic notions of rank or dimension for definable sets. We shall only deal with supersimple finite rank theories, in which all the main notions of model-theoretic rank coincide on any definable set (though not on types). Below, we shall work with SU-rank. The SU-rank on types is an ordinal-valued rank defined by transfinite induction: for any type p over A, SU(p) ≥ α + 1 if there is B ⊃ A such that p has a forking extension q over B with SU(q) ≥ α, and for limit ordinals δ, SU(p) ≥ δ if SU(p) ≥ β for all ordinals β < δ. If X is a set defined by a formula ϕ(x, ā) with ā from A, then SU(X) is the supremum (which will be the maximum in the finite rank theories considered here) of the SU(p) for types p over A containing the formula ϕ(x, ā).

It can be shown that any family of finite simple groups of fixed Lie rank is uniformly interpretable in a family of finite fields, or (in the case of Suzuki and Ree groups) in a family of finite difference fields, that is, fields equipped with an automorphism. In fact, by [27, Ch. 5], if parameters are allowed then the groups are uniformly bi-interpretable in the (difference) fields. Thus, the groups PSL_{2}(q) are uniformly parameter bi-interpretable in the fields 𝔽_{q}, the Ree and Suzuki groups ^{2}F_{4}(2^{2k+1}) and ^{2}B_{2}(2^{2k+1}) are uniformly parameter biinterpretable with the difference fields (F_{2^{2k+1}}, x ↦ x^{2^{k}}), and the Ree groups 2G_{2}(3^{2k+1}) are uniformly parameter bi-interpretable with the difference fields (𝔽_{3^{2k+1}}, x ↦ x^{3^{k}}). Now infinite ultraproducts of finite fields have supersimple SU-rank 1 theory – that is, the set defined by the formula x = x has SU-rank 1 – by for example [3]. The ultraproducts of the corresponding difference fields also have supersimple SU-rank one theory, by the results of Hrushovski [12] and of Ryten (see e.g. [27, Theorem 3.5.8]). For the difference fields this rests on deep work from the 1990s in [12], and Hrushovski was clearly aware then of the supersimplicity of pseudofinite simple groups, and applications similar to some of those below.

We mention three possible lines of application to finite simple groups.

1. Zilber Indecomposability. The Irreducibility Theorem for linear algebraic groups was reworked by Zilber for groups of finite Morley rank. Other modeltheoretic versions have appeared, but for us the following result of Wagner is convenient. See [31, 4.5.6], or, for the guise below, [10, Remark 2.5].

Theorem 4.1 (Indecomposability Theorem) Let G be a group interpretable in a supersimple finite SU-rank theory, and let {X_{i} : i ∈ I} be a collection of definable subsets of G. Then there exists a definable subgroup H of G such that:

(i) H ≤ ⟨X_{i} : i ∈ I⟩, and there are n ∈ ℕ, ϵ_{1}, …, ϵ_{n} ∈ {−1, 1}, and i_{1}, …, i_{n} ∈ I, such that H ≤ X_{i_{1}}^{ϵ_{1}} … X_{i_{n}}^{ϵ_{n}}.

(ii) X_{i}/H is finite for each i ∈ I.

If the collection of X_{i} is setwise invariant under some group Σ of definable automorphisms of G, then H may be chosen to be Σ-invariant.

This has the following almost immediate application to finite simple groups. The result below can also be deduced from [20, Theorem 1], in combination with Theorem 4.4 below.

Theorem 4.2 Let C = {G_{i} : i ∈ ℕ} be a family of finite simple groups of fixed Lie type τ, and let ϕ(x, y_{1}, …, y_{m}) be a formula in the language of groups. Then there is a positive integer d = d(ϕ, τ) with the following property: if ā_{i} ∈ G_{i}^{m} for each i ∈ ℕ, and X_{i} := ϕ(G_{i}, ā_{i}) with |X_{i}| > 1 and |X_{i}| → ∞ as i → ∞, then each G_{i} is a product of at most d conjugates of the set X_{i} ∪ X_{i}^{−1}.

Proof. Suppose that this is false. Then there is a decreasing sequence of infinite subsets (I_{j} : j ∈ ℕ) of ℕ with infinite intersection I such that for any d ∈ ℕ, and for all but finitely many j ∈ I_{d}, G_{j} is not a product of at most d conjugates of X_{j} ∪ X_{j}^{−1}. Choose a non-principal ultrafilter U on ℕ which contains the set I. Let G := Π_{j∈ℕ}G_{i}/U and X := Π_{j∈ℕ}G_{i}/U. Then G is a simple pseudofinite group so has supersimple finite SU-rank theory, and, by Łos’s Theorem, X is an infinite definable subset of G such that for each d ∈ ℕ, G is not a product of at most d conjugates of X ∪ X^{−1}. By Theorem 4.1 (including the final assertion), G has a definable normal subgroup H which is infinite (as X/H is infinite) and is contained in a product of a bounded number of conjugates of X ∪ X^{−1}. This is a contradiction, since by simplicity of G, H = G. □

Other applications of Theorem 4.1 were found in [19]. In particular, it was shown in Corollary 4.11 that certain maximal subgroups (those which are not ‘subfield subgroups’) of finite simple groups are uniformly definable in the groups, and hence, if also unbounded in order, they are ‘uniformly maximal’ [19, Proposition 4.2(ii)].

2. Asymptotic classes. The following definition is due to Elwes [7].

Definition 4.3 A class C of finite first order structures is, for some positive integer N, an N-dimensional asymptotic class, if the following holds.

(i) For every L-formula ϕ(x̄, ȳ) where l(x̄) = n and l(ȳ) = m, there is a finite set of pairs D ⊆ ({0, …, Nn} × ℝ^{>0}) ∪ {(0, 0)} and for each (d, μ) ∈ D a collection Φ_{(d,μ)} of pairs of the form (M, ā) where M ∈ C and ā ∈ M^{m}, so that {Φ_{(d,μ)} : (d, μ) ∈ D} is a partition of {(M, ā) : M ∈ C, ā ∈ M^{m}}, and

||ϕ(M^{n}, ā)| − μ|M|^{\frac{d}{N}} | = o(|M|^{\frac{d}{N}})

as |M| → ∞ and (M, ā) ∈ Φ_{(d,μ)}.

(ii) Each Φ_{(d,μ)} is ∅-definable, that is to say {ā ∈ M^{m} : (M, ā) ∈ Φ_{(d,μ)}} is uniformly ∅-definable across C.

By the main theorem of [3], the class of finite fields is a 1-dimensional asymptotic class, and by Theorem 3.5.8 of [27] the classes of difference fields (F_{2^{2k+1}}, x ↦ x^{2^{k}}) and (𝔽_{3^{2k+1}}), x ↦ x^{3^{k}}) also form 1-dimensional asymptotic classes. The bi-interpretability results of Ryten mentioned above now yield the following.

Theorem 4.4 [27, Ryten] If C is a family of finite simple groups of fixed Lie type, then C is an N-dimensional asymptotic class for some N.

Remark 4.5 Let C = {G_{i} : i ∈ ℕ} be an asymptotic class of finite simple groups as above, and let G^{∗} := Π_{i∈ℕ}G_{i}/U be an infinite ultraproduct of members of C. Let ϕ(x̄, ȳ) be a formula with l(x̄) = m and l(ȳ) = n, let ā ∈ (G^{∗})^{n} with ā = (ā_{i})/U, and suppose that there is U ∈ U such that for all i ∈ U, ϕ(G_{i}^{m}, ā_{i}) has size approximately μ|G_{i}|^{d} (in the sense of asymptotic classes). Then it follows that SU(ϕ((G^{∗})^{m}, ā)) = d. SU(G^{∗}). This can be deduced from [7, 5.4], since C is parameter-bi-interpretable with a 1-dimensional asymptotic class (of fields or difference fields).

3. Word maps. Any non-trivial group word w(x_{1}, …, x_{d}) defines, in any group G, a map w : G^{d} → G, the word map corresponding to w. It is shown in [16] that there is a function f such that if G is a finite simple group, w is a non-trivial word, and ϵ > 0, then |w(G)| ≥ |G|^{1−ϵ} for sufficiently large G. In fact (and this could also be deduced from the last statement using Theorem 4.4), we have: if C is a family of finite simple groups of fixed Lie type, and w is a non-trivial word, then there is μ > 0 such that if G ∈ C is sufficiently large then |w(G)| ≥ μ|G|.

Theorem 4.6 ([28]) For every non-trivial word w there is N = N(w) such that if G is a finite simple group with |G| ≥ N then w(G)^{3} = G.

This result is extended in various further directions in [17]. For example, it follows from [17, Theorem 1.7] that if C is a family of finite simple groups of fixed Lie type (other than the Suzuki or Ree groups) and w_{1}, w_{2} are non-trivial words then there is N, dependent on the Lie type and w_{1}, w_{2}, such that if G ∈ C with |G| > N then w_{1}(G)w_{2}(G) = G.

We mention a possible alternative approach, which fairly quickly yields Theorem 4.6 for families of finite simple groups of fixed Lie type together with a little further information, weaker than that in the last paragraph. It rests on the above-stated result of Larsen from [16], and some general model theory of groups in (super)simple theories. An advantage may be that Suzuki and Ree groups can be treated simultaneously with other families of finite simple groups with no extra work, though this rests on the major work of Hrushovski in [12], in combination with [27].

First, for groups definable in simple theories there is a theory of generic types, analogous to that in stable theories, developed by Pillay [23] and described in [31, Sections 4.3–4.5]. We shall consider a simple theory T, such that in any M ⊧ T there is an ∅-definable infinite group. Let κ be an infinite cardinal, sufficiently large relative to |T | and to the size of any parameter sets we consider, and let M̄ be a κ-saturated model of T, with an ∅-definable group G. Any parameter sets A, B, C will be subsets of M̄ of much smaller size than κ, and M will always denote an elementary submodel of M̄ of much smaller cardinality than κ. Let | ⌣ denote the relation of non-forking (i.e. independence) in simple theories: for subsets A, B, C of M̄, A | ⌣_{C} B denotes that A and B are independent over C in the sense of non-forking, that is, for any ā from A, tp(ā/B ∪ C) does not fork over C. If A is a set of parameters in M̄, then S_{G}(A) denotes the set of types over A which contain the formula x ∈ G; that is the set of maximal consistent (with T) sets of formulas in the variable x, with parameters from A, which include the formula x ∈ G. Following [31] (see Definition 4.3.2 and also Lemma 4.3.4) a type p ∈ S_{G}(A) is generic if for any b ∈ G and a realising p with a | ⌣_{A} b, we have ba | ⌣_{A} b. The group G has a certain subgroup G_{A}^{o} (the ‘connected component over A’), which is the intersection of the A-type-definable subgroups of G of bounded index (that is, index less than κ), and a generic type over A is principal if it is realised in G_{A}^{o}. Part (i) of the following result was first proved in [24, Proposition 2.2], and (ii) is an immediate consequence.

Theorem 4.7 Let T be a simple theory over a countable language, M̄ an ω_{1}- saturated model of T with a countable elementary substructure M, and G an ∅-definable group in M̄. Let p_{1}, p_{2}, p_{3} be three principal generic types of G over M.

(i) There are g_{1}, g_{2} ∈ M̄ such that g_{i} ⊧ p_{i} for i = 1, 2, g_{1} | ⌣_{M} g_{2}, and g_{1}g_{2} ⊧ p_{3}.

(ii) If r ∈ S_{G}(M) has realisations in G_{M}^{o} then there are a_{i} ∈ G with a_{i} ⊧ p_{i} (for i = 1, 2, 3) such that a_{1}a_{2}a_{3} ⊧ r.

Proof. (i) See for example [31, Proposition 4.5.6], though as phrased above one must use ω_{1}-saturation to find the g_{i} in M̄.

(ii) Choose a_{3}, b ∈ M̄ such that a_{3} ⊧ p_{3}, b ⊧ r, and a_{3} | ⌣_{M} b, and put c_{3} := a_{3}^{−1}b. Let p′_{3} := tp(c_{3}/M). Then p′_{3} is a generic type of G over M. Indeed (repeatedly using 4.3.2 and 4.3.4 of [31]), tp(a_{3}^{−1}/M) is generic, so as a_{3}^{−1} | ⌣_{M} b, we find tp(a_{3}^{−1}/Mb) is generic, so tp(a_{3}^{−1}b/Mb) is generic. As tp(a_{3}^{−1}/M) is generic and a_{3}^{−1} | ⌣_{M} b we also get a_{3}^{−1}b | ⌣_{M} b, and this forces that tp(a_{3}^{−1}b/M) is generic. Also, by the assumptions on p_{3} and r, p′_{3} has realisations in G_{M}^{o} so is principal.

It follows by (i) that there are a_{1}, a_{2} ∈ M̄ such that a_{1} ⊧ p_{1}, a_{2} ⊧ p_{2}, and a_{1}a_{2} = c_{3}. Hence a_{1}a_{2}a_{3} = b ⊧ r. □

Lemma 4.8 [31] In the context above, assume that T is a finite SU-rank supersimple theory (so all definable sets have finite SU-rank), and let A be a parameter set. Then

(i) If p ∈ S_{G}(A) then p is generic if and only if SU(p) = SU(G).

(ii) If X is an A-definable subset of G, then SU(G) = SU(X) if and only if some generic type p ∈ S_{G}(A) contains a formula defining X.

Proof. (i) See [31, P.168].

(ii) Immediate from (i) and the definition of SU-rank for types and formulas. □

Theorem 4.9 Let C be a family of finite simple groups of fixed Lie type, and let w_{i}(x_{1}, …, x_{d_{i}}) be non-trivial words, for i = 1, 2, 3.

(i) There is N = N(C, w_{1}, w_{2}, w_{3}) ∈ ℕ such that if H ∈ C with |H| > N then w_{1}(H)w_{2}(H)w_{3}(H) = H.

(ii) |H \ w_{1}(H)w_{2}(H)| = o(|H|) for sufficiently large H ∈ C.

(iii) |w_{1}(H)w_{2}(H)|/|H| → 1 as |H| → ∞, for H ∈ C.

Proof. (i) Suppose (i) is false. Then there is an infinite ultraproduct G^{∗} of members of C such that w_{1}(G^{∗})w_{2}(G^{∗})w_{3}(G^{∗}) is a proper subset of G^{∗}. We view G^{∗} as an elementary submodel of a κ-saturated model T, in the manner described above. Then G^{∗} is ω_{1}-saturated, and has a countable elementary substructure M. By the result of Larsen [16] mentioned above, there is μ > 0 such that if H ∈ C is sufficiently large then |w_{i}(H)| ≥ μ|H| for i = 1, 2, 3. It follows from Remark 4.5 that SU(w_{i}(G^{∗})) = SU(G^{∗}) for each i (and likewise, SU(w_{i}(G)) = SU(G)). Hence, by Lemma 4.8, there is for each i = 1, 2, 3 a generic type p_{i} of G over M containing the formula x ∈ w_{i}(G). As all models of T := Th(G) are simple in the group-theoretic sense, a very saturated model of T cannot have a proper subgroup of bounded index, so G = (G)_{M}^{o} and all generic types of G over M are principal.

Let r be any type over M realised in G^{∗} \ w_{1}(G^{∗})w_{2}(G^{∗})w_{3}(G^{∗}). Then by Theorem 4.7(ii), and ω_{1}-saturation of G^{∗}, there are a_{1}, a_{2}, a_{3}, b ∈ G^{∗} such that a_{i} ⊧ p_{i} and b ⊧ r and a_{1}a_{2}a_{3} = b. In particular, a_{i} ∈ w_{i}(G^{∗}), so b ∈ w_{1}(G^{∗})w_{2}(G^{∗})w_{3}(G^{∗}), which is a contradiction.

(ii) Again, suppose this is false. Then by Theorem 4.4 there is ν > 0 and infinitely many groups H ∈ C such that |H \ w_{1}(H)w_{2}(H)| > ν|H|. Then, by Remark 4.5, we may choose an infinite ultraproduct G^{∗} of members of C such that SU(G^{∗}\w_{1}(G^{∗})w_{2}(G^{∗})) = SU(G^{∗}). Again let M be a countable elementary substructure of G^{∗}. By Lemma 4.8 for i = 1, 2, 3 there are generic types p_{i} of G^{∗} over M such that p_{1} contains the formula x ∈ w_{1}(G^{∗}), p_{2} contains the formula x ∈ w_{2}(G^{∗}), and p_{3} contains the formula x ∈ G^{∗} \ w_{1}(G^{∗})w_{2}(G^{∗}). As in (i), the p_{i} are all principal. By ω_{1}-saturation and Theorem 4.7(i) there are a_{1}, a_{2} ∈ G^{∗} such that a_{1} ⊧ p_{1}, a_{2} ⊧ p_{2}, and a_{3} := a_{1}a_{2} ⊧ p_{3}. In particular, a_{i} ∈ w_{i}(G^{∗}) for i = 1, 2 so a_{3} ∈ w_{1}(G^{∗})w_{2}(G^{∗}), which is a contradiction.

(iii) This is immediate from (ii). □

Remark 4.10 1. Part (i) above is of course just a restatement of a special case of Theorem 4.6. Part (iii) was proved in [29]. We do not know whether these model-theoretic methods can yield the stronger assertion that if C is a family of finite simple groups of fixed Lie rank and w_{1}, w_{2} are non-trivial words, then w_{1}(G)w_{2}(G) = G for sufficiently large G ∈ C. The latter (already proved except in the case of Suzuki and Ree groups in [17]) has recently been proved in [18] with C replaced by the class of all non-abelian finite simple groups.

2. It should be possible to strengthen the asymptotic statements in (ii), (iii), by working with tighter error terms in the definition of ‘asymptotic class’, in the manner of [3] rather than with the o-notation. More precisely, Theorem 4.4 should still hold if, in Definition 4.3, the condition

||ϕ(M^{n}, ā)| − μ|M|^{\frac{d}{N}} | = o(|M|^{\frac{d}{N}})

is replaced by, for some constant c,

||ϕ(M^{n}, ā)| − μ|M|^{\frac{d}{N}} | ≤ c|M|^{\frac{d}{N} − \frac{1}{2}} .

We have not checked this.

3. If w(x_{1}, …, x_{d}) is a non-trivial word, and C is a class of finite simple groups of fixed Lie type, then w defines the word map w : G^{d} → G for G ∈ C.

Theorem 4.4 is applicable in the class C to the formula ϕ(x_{1}, …, x_{d}, y) which says w(x_{1}, …, x_{d}) = y, and hence yields information on the distribution of the solution sets.

References

[1] H. Adler, ‘A geometric introduction to forking and thorn-forking’, J. Math. Logic 9 (2009), 1–20.

[2] J. Baldwin, J. Saxl, ‘Logical stability in group theory’, J. Austral. Math. Soc. 21 (1976), 267–276.

[3] Z. Chatzidakis, L. van den Dries, A.J. Macintyre, ‘Definable sets over finite fields’, J. Reine Angew. Math. 427 (1992), 107–135.

[4] G. Cherlin, E. Hrushovski, Finite structures with few types, Annals of Mathematics Studies No. 152, Princeton University Press, Princeton, 2003.

[5] J-L. Duret, ‘Les corps faiblement algébraiquement clos non séparablement clos ont la propriété d’indépendance’, Model theory of Algebra and Arithmetic (Eds. L. Pacholski et al.), Springer Lecture Notes vol. 834, 1980, pp. 135–157.

[6] C. Ealy, K. Krupinski, A. Pillay, ‘Superrosy dependent groups having finitely many satisfiable generics’, Ann. Pure Appl. Logic 151 (2008), 1– 21.

[7] R. Elwes, ‘Asymptotic classes of finite structures’, J. Symb. Logic 72 (2007), 418–438.

[8] R. Elwes, H.D. Macpherson, ‘A survey of asymptotic classes and measurable structures’, in Model Theory with Applications to Algebra and Analysis, Vol. 2 (Eds. Z. Chatzidakis, H.D. Macpherson, A. Pillay, A.J. Wilkie), London Math. Soc. Lecture Notes 350, Cambridge University Press, 2008, pp. 125–159.

[9] R. Elwes, E. Jaligot, H.D. Macpherson, M.J. Ryten, ‘Groups in supersimple and pseudofinite theories’, Proc. London Math. Soc., to appear.

[10] R. Elwes, M. Ryten, ‘Measurable groups of low dimension’, Math. Logic Quarterly 54 (2008), 374-386.

[11] E. Hrushovski, ‘Pseudofinite fields and related structures’, in Model theory and applications (Eds. L. Bélair, Z. Chatzidakis, P. D’Aquino, D. Marker, M. Otero, F. Point, A. Wilkie), Quaderni di Matematica, vol. 11, Caserta, 2005, 151–212.

[12] E. Hrushovski, ‘The elementary theory of the Frobenius automorphism’, preprint, arXiv:math/0406514v1 [math.LO].

[13] E.I. Khukhro, ‘On solubility of groups with bounded centralizer chains’, Glasgow Math. J. 51 (2009), 49–54.

[14] B. Kim, ‘Forking in simple unstable theories’, J. London Math. Soc. 57 (1998), 257–267.

[15] B. Kim, A. Pillay, ‘Simple theories’, Ann. Pure Appl. Logic 88 (1997) 149–164.

[16] M. Larsen, ‘Word maps have large image’, Isr. J. Math. 139 (2004), 149– 156.

[17] M. Larsen, A. Shalev, ‘Word maps and Waring type problems’, J. Amer. Math. Soc 22 (2009), 437–466.

[18] M. Larsen, A. Shalev, P. Tiep, ‘Waring problem for finite simple groups’, Ann. Math., to appear.

[19] M.W. Liebeck, H.D. Macpherson, K. Tent, ‘Primitive permutation groups of bounded orbital diameter’, Proc. London Math. Soc. (3) 100 (2010), 216–248.

[20] M.W. Liebeck, N. Nikolov, A. Shalev, ‘Product decompositions in finite simple groups’, arXiv:1107.1528v1.

[21] H.D. Macpherson, K. Tent, ‘Stable pseudofinite groups, J. Alg. 312 (2007), 550–561.

[22] A. Onshuus, ‘Properties and consequences of thorn-independence’, J. Symb. Logic 71 (2006), 1–21.

[23] A. Pillay, ‘Definability and definable groups in simple theories’, J. Symb. Logic 63 (1998), 788–796.

[24] A. Pillay, T. Scanlon, F. Wagner, ‘Supersimple fields and division rings’, Math. Research Letters 5 (1998), 473–483.

[25] F. Point, ‘Ultraproducts and Chevalley groups’, Arch. Math. Logic 38 (1999) 355–372.

[26] D.J.S. Robinson, A course in the theory of groups, 2nd edition, Springer, 1996.

[27] M.J. Ryten, Results around asymptotic and measurable groups, Ph.D. thesis, University of Leeds, 2007. http://www.logique.jussieu.fr/modnet/ preprint no. 123.

[28] A. Shalev, ‘Word maps, conjugacy classes, and a non-commutative Waring-type theorem, Ann. Math. 170 (2009), 1383–1416.

[29] A. Shalev, ‘Mixing and generation in simple groups’, J. Alg. 319 (2008), 3075–3086.

[30] S. Shelah, ‘Simple unstable theories’, Ann. Pure Appl. Logic 19 (1980), 177–203.

[31] F. Wagner, Simple theories, Kluwer, Dordrecht, 2000.

[32] J.S. Wilson, ‘On pseudofinite simple groups’, J. London Math. Soc.) 51 (1995), 471–490.

[33] J.S. Wilson, ‘First-order characterization of the radical of a finite group’, J. Symb. Logic 74 (2009), 1429–1435.