Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers
aa r X i v : . [ m a t h . L O ] O c t DEFINABLE ENVELOPES OF NILPOTENT SUBGROUPS OFGROUPS WITH CHAIN CONDITIONS ON CENTRALIZERS
TUNA ALTINEL AND PAUL BAGINSKI
Abstract. An M C group is a group in which all chains of centralizers havefinite length. In this article, we show that every nilpotent subgroup of an M C group is contained in a definable subgroup which is nilpotent of the same nilpo-tence class. Definitions are uniform when the lengths of chains are bounded. Introduction
Chain conditions have played a central role in modern infinite group theory andone of the most natural chain conditions is one on centralizers. A group is said to be M C if all chains of centralizers of arbitrary subsets are finite. If there is a uniformbound d on the lengths of such chains, then G has finite centralizer dimension(fcd) and the least such bound d is known as the c -dimension of G , which wedenote dim( G ).The M C property has been studied by group theorists since many natural classesof groups possess this property. See [3] for a classic paper on the properties of M C groups. Many groups possess the stronger property of fcd, including abelian groups,free groups, linear groups, and torsion-free hyperbolic groups. Khukhro’s article onthe solvability properties of torsion fcd groups [8] compiles a lengthy list of groupswith fcd. Khukhro’s article, as well as several other foundational papers (see, forexample, [2, 4, 8, 9, 16]), have demonstrated that M C groups and groups with fcdare fairly well-behaved, for example by having Engel conditions closely linked tonilpotence.For model theorists, the interest in these groups derives from the well-studiedmodel-theoretic property of stability. A stable group must have fcd; in fact, itpossesses uniform chain conditions on all uniformly definable families of subsets.Stable groups have an extensive literature in model theory (see [12] or [15]), howeverthe properties of M C and fcd are appearing in other areas of the model theoryof groups, such as rosy groups with NIP [5] or Pınar U˘gurlu’s recent work onpseudofinite groups with fcd [14].The results of this paper reinforce Wagner’s work [4, 15, 16] in showing thatseveral basic properties of (sub)stable groups derive purely from these simple group-theoretic chain conditions, which force the left-Engel elements to be well-behaved.In contrast to Wagner’s generalizations, which revealed that M C sufficed for manygroup-theoretic properties of stable groups, we shall show that M C suffices for a logical property of stable groups, asserting the existence of certain definable groups. Date : October 30, 2018.
Key words and phrases. group theory, nilpotence, chains of centralizers, model theory,definability.
In the end, our results will yield an alternative path to a conclusion that also followsfrom Wagner’s analysis of left-Engel elements.It has been known for some time [12, Theorem 3.17], that if G is a stable groupand H is a nilpotent (or solvable) subgroup, then there exists a definable subgroup d ( H ) of G which contains H and has the same nilpotence class (derived length) as H . Such a subgroup d ( H ) is called a definable envelope of H . The existence ofdefinable envelopes allowed logicians to approximate arbitrary nilpotent subgroupsof stable groups with slightly “larger” nilpotent subgroups which were definable,i.e. manipulable with model-theoretic techniques.Our main theorem asserts the existence of definable envelopes of nilpotent sub-groups in M C groups and uniformly definable envelopes for groups with fcd. Defin-ability here always refers to formulas in the language L G of groups. These envelopesare N G ( H )-normal, meaning that if an element normalizes H , it also normalizesthe envelope. Main Theorem.
Let G be an M C group and H ≤ G a nilpotent subgroup. Thenthere exists a subgroup D ≤ G definable in the language of groups with parametersfrom G , such that H ≤ D , D is N G ( H ) -normal, and D is nilpotent of the samenilpotence class as H .Moreover, for every pair of positive integers d and n , there exists a formula φ d,n ( x, y ) , where ℓ ( y ) = dn , such that for any group G of dimension d and any H ≤ G nilpotent of class n , there exists a tuple a ∈ G such that φ d,n ( G, a ) is anilpotent subgroup of G of class n which contains H and is N G ( H ) -normal. We hope that this result will prove useful in some of the current areas in logicwhere M C groups are appearing. This result may also open to the door to studyingsome of the logical properties of the non-elementary classes of groups with fcd listedin [8]. It is worth mentioning that [1] and [10] contain results on definable envelopesin elementary classes of groups whose theories are NIP or simple, respectively.We assume only a rudimentary knowledge of model theory and logic, namely thenotions of “definability” and the Compactness Theorem. Readers may consult anyintroductory text, such as [6] or [12], for explanations of these notions. Otherwise,the material will be primarily group-theoretic and self-contained.In the next section, we will define relevant terms from group theory and provesome fundamental lemmas about groups in general. In the following section, werestrict our focus to M C groups and prove our main theorem and some corollaries.2. Preliminaries
We write A ≤ G to denote that A is a subgroup of G and A ⊳ G to denote A is normal in G . If A ⊆ G then h A i denotes the subgroup generated by A . For anysubset A of G , the centralizer of A is C G ( A ) = { g ∈ G | ∀ a ∈ A ga = ag } , while thenormalizer of A is N G ( A ) = { g ∈ G | ∀ a ∈ A g − ag ∈ A } . If A and B are subgroupsof a group G , then A is N G ( B )-normal if N G ( B ) ≤ N G ( A ).Given g, h ∈ G , the commutator of g and h is [ g, h ] := g − h − gh . Iterated com-mutators are interpreted as left-normed, i.e., [ x, y, z ] will denote [[ x, y ] , z ]. When A, B ⊆ G , then we write [ A, B ] := h{ [ a, b ] | a ∈ A, b ∈ B }i . We define the lowercentral series of G as γ ( G ) := G and γ k +1 ( G ) := [ γ k ( G ) , G ]. A group G is nilpo-tent if γ n ( G ) = 1 for some n < ω ; the least n ≥ γ n +1 ( G ) = 1 is thenilpotence class of G . It is clear that a subgroup of a nilpotent group is nilpotentof equal or lesser nilpotence class. EFINABLE ENVELOPES IN M C GROUPS 3
The Hall-Witt identity relates the commutators of three elements: For all x, y, z ∈ G ,(2.1) 1 = [ x, y − , z ] y [ y, z − , x ] z [ z, x − , y ] x = [ x, y, z x ][ z, x, y z ][ y, z, x y ]The Hall-Witt identity is used to prove the well-known Three Subgroup Lemma,which we state in the needed level of generality. Lemma 2.1. [13, Three Subgroup Lemma, 5.1.10]
Let G be a group, N a subgroup,and K, L, and M subgroups of N G ( N ) . Then [ K, L, M ] ≤ N and [ L, M, K ] ≤ N together imply [ M, K, L ] ≤ N . This article shall be concerned with chains of centralizers. However, in order toanalyze them fully, we shall need a more general definition of iterated centralizers.
Definition 2.2.
Let P be a subgroup of G . We define the iterated centralizersof P in G as follows. Set C G ( P ) = 1 and for n ≥
1, let C nG ( P ) = ( x ∈ \ k Let G be a group and P a subgroup of G . Then [ γ i ( P ) , C kG ( P )] ≤ C k − iG ( P ) for all positive integers i and k such that i ≤ k . In particular, [ γ i ( G ) , Z k ( G )] ≤ Z k − i ( G ) . Bryant (Lemma 2.5 in [3]) used Hall’s lemma to determine conditions underwhich one could conclude a group and a subgroup have the same iterated centralizer.We shall pursue the same goal and restructure Bryant’s argument for our purposes.Th following technical lemma is the heart of the proof of our main theorem. Itsproof almost reproduces Bryant’s subtle argument. We include it not only forcompleteness, but also to clarify how our lemma and Bryant’s relate to each other,despite statements that differ considerably. Lemma 2.4. Let k ≥ be an integer, G be a group, and X ≤ P be two subgroupsof G satisfying the following conditions: (1) C iG ( X ) = C iG ( P ) for all i ∈ { , . . . , k − } ; (2) [ γ k ( P ) , C kG ( X )] = 1 ; (3) C G ( X ) = C G ( P ) . ALTINEL AND BAGINSKI Then C kG ( X ) = C kG ( P ) .Proof. The argument proceeds by induction on k , with k = 1 given by hypothesis(3). So we assume k > C iG ( X ) = C iG ( P ) for all 0 ≤ i ≤ k − 1, so set C i = C iG ( X ).We claim that the normalizers of these C i contain four groups important to thisproof. Namely, we claim that for all 1 ≤ i ≤ k − A ) X ∪ γ k − i ( P ) ∪ C kG ( X ) ∪ C kG ( P ) ⊆ k − \ j =0 N G ( C j ) . Since X normalizes all its iterated centers, we find X ≤ N G ( C j ) for all j ≤ k − P normalizes all its iterated centers, we find γ k − i ( P ) ≤ P ≤ N G ( C j )for all j ≤ k − 1. Lastly, by the definition of iterated centralizers, C kG ( X ) ≤ N G ( C j )and C kG ( P ) ≤ N G ( C j ) for all j ≤ k − B ) [ γ k − i ( P ) , C kG ( X )] ≤ C i for i = 0 , , . . . , k − . We shall prove ( B ) by induction on i for 0 ≤ i ≤ k − 1. Note that i = 0 isprecisely hypothesis (2), so assume i ≥ 1. We shall prove ( B ) using the ThreeSubgroup Lemma (Lemma 2.1) with γ k − i ( P ) , X, and C kG ( X ) relative to the group C i − G ( P ) = C i − . By ( A ), these three groups normalize C i − . Since X ≤ P , wehave by Lemma 2.3 and by induction on i that[ γ k − i ( P ) , X, C kG ( X )] ≤ [ γ k − i +1 ( P ) , C kG ( X )] ≤ C i − G ( P ) . By the definition of iterated centralizers, [ X, C kG ( X )] ≤ C k − G ( X ) = C k − and thuswe obtain the following chain of inequalities:[ X, C kG ( X ) , γ k − i ( P )] ≤ [ C k − , γ k − i ( P )] = [ C k − G ( P ) , γ k − i ( P )] ≤ C i − G ( P ) . The last inequality follows from Lemma 2.3. By the Three Subgroup Lemma(Lemma 2.1), we conclude that[ C kG ( X ) , γ k − i ( P ) , X ] ≤ C i − G ( P ) = C i − = C i − G ( X ) . Since [ C kG ( X ) , γ k − i ( P )] ≤ N G ( C j ) for all j ≤ i − A ), we conclude from thedefinition of iterated centralizers that [ C kG ( X ) , γ k − i ( P )] ≤ C iG ( X ) = C i , yieldingour claim ( B ).Setting i = k − B ) gives[ P, C kG ( X )] ≤ C k − = C k − G ( P ) . Yet ( A ) with i = k − C kG ( X ) ≤ N G ( C jG ( P )) for all j ≤ k − 1, so bythe definition of iterated centralizers, C kG ( X ) ≤ C kG ( P ). On the other hand, since X ≤ P , we find:[ X, C kG ( P )] ≤ [ P, C kG ( P )] ≤ C k − G ( P ) = C k − = C k − G ( X )Again, by ( A ) with i = k − 1, we find that C kG ( P ) ≤ N G ( C jG ( X )) for all j ≤ k − C kG ( P ) ≤ C kG ( X ) and we have equality. (cid:3) We shall also need a lemma relating the iterated centralizers of three nestedgroups. Lemma 2.5. Let A ≤ B ≤ C be groups and suppose that for all k < n , C kC ( A ) = C kC ( C ) . Then C jB ( A ) = C jC ( A ) ∩ B for all j ≤ n . EFINABLE ENVELOPES IN M C GROUPS 5 Proof. We shall prove this lemma by induction on 1 ≤ j ≤ n . For j = 1, this is justthe statement that C B ( A ) = C C ( A ) ∩ B . Assume the claim is true for some j < n .Then C ℓC ( A ) = C ℓC ( C ) = Z ℓ ( C ) for all ℓ ≤ j and thus by the induction hypothesis,we have for all ℓ ≤ j that N B ( C ℓB ( A )) = N B ( C ℓC ( A ) ∩ B ) = N B ( Z ℓ ( C ) ∩ B ).Because Z ℓ ( C ) is characteristic in C , N B ( Z ℓ ( C ) ∩ B ) = B . Thus, x ∈ C j +1 B ( A )if and only if x ∈ B and [ x, A ] ⊆ C jB ( A ) = C jC ( A ) ∩ B , with the latter equalitycoming from the induction hypothesis on j . For any x ∈ C j +1 C ( A ), we have bydefinition that [ x, A ] ∈ C jC ( A ). Since A ≤ B , we find C j +1 C ( A ) ∩ B ⊆ C j +1 B ( A ).On the other hand, C j +1 B ( A ) ≤ C = T jℓ =1 N C ( C ℓC ( C )) = T jℓ =1 N C ( C ℓC ( A )) and[ C j +1 B ( A ) , A ] ≤ C jB ( A ) = C jC ( A ) ∩ B . Thus C j +1 B ( A ) ⊆ C j +1 C ( A ) by definitionand we have C j +1 B ( A ) = C j +1 C ( A ) ∩ B . By induction, C jB ( A ) = C jC ( A ) ∩ B for all j ≤ n . (cid:3) Bounded chains of centralizers and definable envelopes As mentioned in the introduction, several well-studied classes of groups in grouptheory and model theory possess chain conditions on their centralizers. We restatethe definitions of these chain conditions precisely. Definition 3.1. A group G is has the chain condition on centralizers , denoted M C , if there exists no infinite sequence of subsets A n ⊆ G such that C G ( A n ) >C G ( A n +1 ) for all n < ω .A group G has finite centralizer dimension (fcd) if there is a uniform bound n ≥ G = C G (1) > C G ( A ) > . . . > C G ( A n ) of centralizers of subsets A i of G . The least bound (i.e. the length of the longest chain of centralizers) isknown as the c -dimension of G , which we will denote dim( G ).Note that since C G ( C G ( C G ( A ))) = C G ( A ) for all A ⊆ G , all descending chainsof centralizers are finite if and only if all ascending chains are finite. An immediateconsequence of the finite chain condition is the following observation: Proposition 3.2. Let G be an M C group. If A ⊆ G , then there is an A ′ ⊆ A finite such that C G ( A ) = C G ( A ′ ) . If G has centralizer dimension d , then A ′ can bechosen such that | A ′ | ≤ d . If dim( G ) = 1, then for any g ∈ G , we have G = C G (1) ≥ C G ( g ) ≥ C G ( G ) = Z ( G ), with at most one of these inequalities strict. Thus g ∈ Z ( G ) and G is abelian.Since centralizers relative to a subgroup H of G are calculated by simply intersectingwith H , it is clear that a subgroup of an M C group is M C . Furthermore, if G hasfcd and H ≤ G , then H has fcd and dim( H ) ≤ dim( G ). Clearly if G has fcd and H ≤ C G ( A ) for some A ⊆ G with A Z ( G ), then dim( H ) < dim( G ). In thenext lemma, valid for all groups, we revise this critical observation with necessaryconditions for a subgroup to be contained in a centralizer of a noncentral subset. Proposition 3.3. Let G be a group and H ≤ G . Then one of the following is true: (1) H ≤ Z ( G ) ; (2) there exists a subset A ⊆ G such that H ≤ C G ( A ) < G ; or (3) C G ( H ) = Z ( G ) , and hence Z ( H ) = Z ( G ) ∩ H , i.e. Z ( H ) ≤ Z ( G ) .Proof. Assume H Z ( G ), so C G ( H ) < G . If C G ( H ) > Z ( G ), then H ≤ C G ( C G ( H )) < G , so A = C G ( H ) witnesses (2). Thus, if (2) does not hold for ALTINEL AND BAGINSKI H , then C G ( H ) = Z ( G ) and so clearly Z ( H ) = C G ( H ) ∩ H = Z ( G ) ∩ H and Z ( H ) ≤ Z ( G ). (cid:3) While both M C and fcd are preserved under subgroups and finite direct products[9], they behave poorly under quotients. The quotient of an M C group, even by itscenter, may fail to be M C (see [3]). This is the principal complication in the proofof our main theorem.From a logical perspective, the class of M C groups is not elementary: indeed,it is consistent to have chains of arbitrarily long length, so by the CompactnessTheorem there would be an elementary extension of an M C group with an infinitechain of centralizers; this extension is clearly not M C . Conversely, if all groupselementarily equivalent to a group G are M C , then there must be a uniform boundon the lengths of chains of centralizers in G , i.e., G has fcd. Indeed, having cen-tralizer dimension d can be expressed by a first-order formula in the language L G of groups: x , . . . , x d +1 , C (1) > C ( x ) > C ( x , x ) > . . . > C ( x , x , . . . , x d +1 ).Therefore, groups with fcd have the advantage of being analyzable using model the-oretic methods since for a fixed dimension d they form an elementary class. Manyclassic families of groups in model theory, such as stable groups, have fcd and arenot simply M C .Our goal in this section is to demonstrate the existence of definable envelopesof nilpotent subgroups of M C groups. Roughly speaking, if G is an M C groupand H is a nilpotent subgroup, we show that there is a definable nilpotent group D which contains H but is not much “larger”, in the sense that it has the samenilpotence class. In model theory, the existence of such envelopes allows one toreplace an arbitrary nilpotent group with a similar one which can be manipulatedusing model theoretic techniques. While one might perhaps expect such envelopesto exist in the elementary class of groups with a fixed centralizer dimension, wehave succeeded in showing such envelopes exist even for the non-elementary classof M C groups. The advantage of fcd in this case is uniform definability of theseenvelopes in terms of the dimension of the ambient group and the nilpotence classof the subgroup. We are now ready to prove our main theorem. Theorem 3.4. Let G be an M C group and H ≤ G a nilpotent subgroup. Thenthere exists a subgroup D ≤ G definable in the language of groups with parametersfrom G , such that H ≤ D , D is N G ( H ) -normal, and D is nilpotent of the samenilpotence class as H .Moreover, for every pair of positive integers d and n , there exists a formula φ d,n ( x, y ) , where ℓ ( y ) = dn , such that for any group G of dimension d and any H ≤ G nilpotent of class n , there exists a tuple a ∈ G such that φ d,n ( G, a ) is anilpotent subgroup of G of class n which contains H and is N G ( H ) -normal.Proof. Let G be a M C group and H be a nilpotent subgroup of G of class n . Forall 1 ≤ k ≤ n , we will construct a descending chain of definable subgroups ( E k ) nk =1 having the following properties:(1) each E k is definable;(2) each E k contains H ;(3) for each j ∈ { , . . . , k } and each subgroup H ≤ P ≤ E k , C jE k ( H ) = C jE k ( P ) = C jE k ( E k ) = Z j ( E k ) ;(4) each E k is N G ( H )-normal. EFINABLE ENVELOPES IN M C GROUPS 7 By the chain condition, the collection of all centralizers of G which contain H hasa least element, C G ( A ), which must be N G ( H )-normal. By Proposition 3.2, A canbe taken to be a finite set A = { x , , . . . , x ,m } , with m ≤ dim( G ) if G has fcd.Set E = C G ( A ). Since C G ( A ) is N G ( H )-normal, N G ( H ) ≤ N G ( HZ ( C G ( A ))).On the other hand, any N G ( HZ ( C G ( A )))-conjugate of E still contains H . Bythe minimal choice of C G ( A ), we conclude that E = C G ( A ) is N G ( HZ ( C G ( A )))-normal. Thus, without loss of generality we may replace H with HZ ( C G ( A )). ByProposition 3.3, we conclude C E ( H ) = Z ( H ) = Z ( E ) = C E ( E ). In fact, bythis proposition, for any group P with H ≤ P ≤ E , we have C P ( H ) = Z ( H ) = Z ( P ) = C E ( P ). So we have successfully constructed E .Suppose now that the E j have been constructed for 1 ≤ j < k and consider P such that H ≤ P ≤ E k − . By property (3) and Lemma 2.5, we have C jP ( H ) = C jE k − ( H ) ∩ P ( • )for all 1 ≤ j ≤ k . Before defining E k , we first define an intermediate definablesubgroup for every h ∈ C kE k − ( H ): E k ( h ) = { x ∈ E k − | [ x, h ] ∈ Z k − ( E k − ) } . Since Z k − ( E k − ) ⊳ E k − , E k ( h ) is a subgroup of E k − . It is definable in E k − with h as parameter. Since by induction on k , C k − E k − ( H ) = Z k − ( E k − ) and H ≤ E k − , H ≤ E k ( h ). Clearly, h ∈ E k ( h ) as well.Since h ∈ E k ( h ), we find that [ E k ( h ) , h ] ≤ Z k − ( E k − ) ∩ E k ( h ) ≤ Z k − ( E k ( h )).As a result, h ∈ Z k ( E k ( h )). By Lemma 2.3, [ γ k ( E k ( h )) , Z k ( E k ( h ))] = 1, we con-clude that [ γ k ( E k ( h )) , h ] = 1 . ( •• )By the M C condition on the ambient group, there exist x k, , . . . , x k,m k ∈ C kE k − ( H )for some m k ∈ N ∗ such that C E k − ( C kE k − ( H )) = C E k − ( x k, , . . . , x k,m k ). We thendefine E k = m k \ i =1 E k ( x k,i ) . The subgroup E k is definable in E k − with parameters x k, , . . . , x k,m k and contains H . If G has dimension d , then we may take m k = d , so this definition of E k interms of E k − is uniform in terms of d parameters. Moreover, H ≤ E k ≤ E k − .By the choice of the x i , ( •• ) implies that[ γ k ( E k ) , C kE k − ( H )] = 1 . ( • • • )By property (3) of E k − , we have C iE k − ( H ) = C iE k − ( E k ) for all 0 ≤ i ≤ k − k ≥ C E k − ( H ) = Z ( E k − ) = C E k − ( E k ). By Lemma 2.4, C kE k − ( H ) = C kE k − ( E k ). Therefore, for all h ∈ C kE k − ( H ), [ E k , h ] ⊆ C k − E k − ( E k ) = Z k − ( E k − ) by property (3) of E k − . In other words, E k ≤ E k ( h ) for all h ∈ C kE k − ( H ) and hence E k = \ h ∈ C kEk − ( H ) E k ( h ) . By property (4) of E k − , we conclude that C kE k − ( H ) and Z k − ( E k − ) are N G ( H )-normal and thus so is E k , establishing the fourth property for E k .We are left with establishing the third property for E k . Let P be any sub-group such that H ≤ P ≤ E k ≤ E k − . The statement ( • ) demonstrates that ALTINEL AND BAGINSKI C jE k ( H ) = C jE k − ( H ) ∩ E k for all 1 ≤ j ≤ k . Together with ( • • • ), we concludethat [ γ k ( E k ) , C kE k ( H )] = 1, and hence [ γ k ( P ) , C kE k ( H )] = 1, which is condition(2) in Lemma 2.4. Condition (3) of this lemma follows from the fact in the firstparagraph that C E k ( E k ) = C E k ( P ) = C E k ( H ) = Z ( E k ) = Z ( H ).It remains to show that C jE k ( H ) = C jE k ( P ) = C jE k ( E k ) = Z j ( E k ) for 1 ≤ j ≤ k inorder to complete the induction on k . Note that we always have C nG ( G ) = Z n ( G ), sothe first two equalities are the only ones we need to establish. If we have succeededin showing C jE k ( H ) = C jE k ( P ) = C jE k ( E k ) = Z j ( E k ) for all 1 ≤ j < k , thenLemma 2.4 guarantees that C kE k ( H ) = C kE k ( P ) = C kE k ( E k ). So it remains for usto show that C jE k ( H ) = C jE k ( P ) = C jE k ( E k ) = Z j ( E k ) for all 1 ≤ j < k . Weshall prove this by induction on j < k . We have already noted that for j = 1, C E k ( E k ) = C E k ( P ) = C E k ( H ) = Z ( E k ) = Z ( H ). Assume it is true for all ℓ < j .We know from ( • ) and property (3) for E k − that C jE k ( H ) = C jE k − ( H ) ∩ E k = Z j ( E k − ) ∩ E k . Similarly, C jE k ( P ) = Z j ( E k − ) ∩ E k . Yet Z j ( E k − ) ∩ E k ≤ Z j ( E k ), so C jE k ( P ) and C jE k ( H ) are both subgroups of Z j ( E k ). By the induction hypothesis on j − H, Z j ( E k )] ≤ [ P, Z j ( E k )] ≤ [ E k , Z j ( E k )] ≤ Z j − ( E k ) = C j − E k ( H ) = C j − E k ( P ) . By the induction hypothesis on ℓ < j , we know N E k ( C ℓE k ( H )) = N E k ( C ℓE k ( P )) = N E k ( Z ℓ ( E k )) = E k , so by the definition of iterated centralizers, we determine Z j ( E k ) ≤ C jE k ( H ) ∩ C jE k ( P ). Therefore C jE k ( E k ) = Z j ( E k ) = C jE k ( P ) = C jE k ( H ),as desired.This completes the induction on k constructing the descending chain of sub-groups E k . The definable envelope of H is Z k ( E k ), where k is the nilpotence classof H . Indeed, H ≤ C kE k ( H ) = Z k ( E k ) by construction. As noted during the proof,if G has dimension d , the iterative construction of the E k is uniform and requires d parameters at each stage for a total of dn parameters from G . Thus for groupswith a fixed centralizer dimension, the definable envelopes of nilpotent subgroupsare uniformly definable. (cid:3) Corollary 3.5. Let G be an M C group and H ⊳ G be a normal nilpotent subgroup.Then H ≤ D ≤ G for some normal, definable subgroup D of G which is nilpotentof the same class as H .For every pair of positive integers d and n , there is a formula φ d,n ( x, y ) suchthat if G is a group of dimension d and H ⊳ G is a normal nilpotent subgroup ofnilpotence class n , then for some a ∈ G , φ d,n ( G, a ) is a normal nilpotent subgroupof class n which contains H .Proof. Let H ⊳ G be a normal nilpotent subgroup. By Theorem 3.4, there is adefinable nilpotent subgroup D of G which is nilpotent of the same class as H suchthat H ≤ D ≤ G . This group D is N G ( H )-normal, and hence normal in G . (cid:3) The subgroup generated by all normal nilpotent subgroups of G is called the Fitting group F ( G ) of G . We shall say the definable Fitting group dF ( G )of G is the subgroup generated by all definable normal nilpotent subgroups of G .Clearly, for any group G we have dF ( G ) ≤ F ( G ) and Corollary 3.5 indicates thatin a M C group, F ( G ) = dF ( G ). For completeness, we mention that Wagner [15,Definition 1.1.7] defined a similar concept known as the definable generalized Fitting EFINABLE ENVELOPES IN M C GROUPS 9 subgroup , F ∗ ( G ). We have the following relationship between these three notions: dF ( G ) ≤ F ( G ) ≤ F ∗ ( G ) in all groups G . None of these groups need be definable,though it may occur that in a particular model of Th( G ), these groups coincidewith definable groups. For dF ( G ), such a situation is often not a coincidence. Lemma 3.6. Let G be a group and suppose the definable Fitting subgroup dF ( G ) coincides with a definable nilpotent subgroup φ ( G, a ) . Then dF ( G ) is ∅ -definable in T h ( G ) , the theory of G in the language L G of groups.Proof. Suppose dF ( G ) = φ ( G, a ) for some L G formula φ ( x, y ) and a ∈ G . Suppose dF ( G ) has nilpotence class n . In any elementary extension G ′ (cid:23) G , φ ( G ′ , a ) is anormal group of nilpotence class n , and thus contained in dF ( G ′ ). For any formula θ ( x, y ) in the language of groups and any integer k ≥ 1, the following sentence istrue in G : G | = ∀ y ( ( θ ( x, y ) is a normal nilpotent group of class k ) →∀ x ( θ ( x, y ) → φ ( x, a ) )) . Therefore the same sentence is true in G ′ , so that φ ( G ′ , a ) contains all definablenormal nilpotent subgroups of G ′ . By definition of the definable Fitting group, φ ( G ′ , a ) = dF ( G ′ ).We now claim that dF ( G ) = φ d,n ( G, a ) can be defined without parameters in L G , the language of groups. If not, by Svenonius’ Theorem [11, Thm 9.2], there isan elementary extension ( G ′ , a ′ ) (cid:23) ( G, a ) and an L G -automorphism σ of G ′ whichdoes not preserve φ d,n ( G ′ , a ′ ). Yet a and a ′ have the same L -type, so φ d,n ( G ′ , a ′ )contains all definable normal nilpotent subgroups of G ′ , and by the same argumentas above, φ d,n ( G ′ , a ′ ) = dF ( G ′ ). Since dF ( G ′ ) is characteristic, it is preservedby all automorphisms of G ′ , a contradiction. So the definable Fitting subgroup is ∅ -definable in T h ( G ). (cid:3) We now address the issue of the definability of the Fitting group F ( G ) directly.Of great importance will be the following result: Lemma 3.7 (Theorem 1.2.11, [15]) . Let G be M C . Then the Fitting subgroup F ( G ) is nilpotent. Using this result and the work of Bludov [2], Wagner has already showed that inan M C group, F ( G ) is ∅ -definable. In fact, his result states that F ( G ) = L ( G ), theset of bounded left Engel elements ([16, Corollary 2.5]). The nilpotence of F ( G )(Lemma 3.7), then implies that F ( G ) consists of those elements x ∈ G which satisfythe n th left Engel condition for a fixed n . We shall arrive at the ∅ -definability of F ( G ) by using the avenue of definable envelopes instead. Corollary 3.8. Let G be an M C group. The Fitting subgroup F ( G ) coincides witha ∅ -definable, nilpotent subgroup of G . If G has fcd, F ( G ) is itself ∅ -definable in T h ( G ) .Proof. Let G be an M C group. The Fitting subgroup F ( G ) is nilpotent by Lemma3.7. As already mentioned, Corollary 3.5 implies that F ( G ) = dF ( G ). Therefore dF ( G ) is ∅ -definable in T h ( G ) by Lemma 3.6 and F ( G ) thus coincides with an ∅ -definable nilpotent subgroup of G . If G is fcd, then any elementary extension G ′ of G has the same c -dimension. In particular, G ′ is M C , so F ( G ′ ) = dF ( G ′ ) byCorollary 3.5. Thus the Fitting group and definable Fitting group are both givenby the same ∅ -definable formula over T h ( G ). (cid:3) At this point, the following question is natural: Is the solvable analogue of Theorem 3.4 true in an M C -group? Our “normalized” construction yields the following very partial answer to this ques-tion: Corollary 3.9. Let G be an M C group and H ≤ G a solvable subgroup. If thereexist nilpotent subgroups A, B ≤ H such that A ⊳ H and H = AB , then H iscontained in a definable solvable subgroup of G . Acknowledgements The authors thank Alexandre Borovik for his stimulating questions and C´edricMilliet for a careful reading of an earlier draft. References [1] R. de Aldama. Chaˆınes et d´ependance . Ph.D. dissertation, Universit´e Lyon-1, Lyon, France,2009.[2] V. V. Bludov. On locally nilpotent groups with the minimality condition for centralizers, Algebra Logika (1998), 270–278; English transl. in Algebra and Logic (1998), 151–156.[3] R. Bryant. Groups with the Minimal Condition on Centralizers, J. Alg. (1979), 371–383.[4] J. Derakhshan and F. O. Wagner. Nilpotency in groups with chain conditions. Quart. J.Math. Oxford Ser. (2) (1997), no. 192, 453–466.[5] C. Ealy, K. Krupinski and A. Pillay. 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