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The Relatively Free Groups Satisfy Noncentral Commutative Transitivity
We prove that a free group, , relative to the variety, , of all groups simultaneously nilpotent of class at most and metabelian is such that the centralizer of every noncentral element is abelian. We relate that result to the model theory of such groups as well as a quest to find a relative analog in of a classical theorem of Benjamin Baumslag. We also touch briefly on similar considerations in the varieties of nilpotent groups.
1.1. Separation and Discrimination
Definition 1. Let be a group and a nonempty class of groups closed under isomorphism. separates provided to every nontrivial element ; there is a group and a homomorphism such that . If is a positive integer, then -separates provided to every ; there is a group and a homomorphism such that for all . discriminates provided -separates for every positive integer . Note that separation is the same as -separation. In the event that is the isomorphism class of a single group we may say that separates (-separates, discriminates) for separates (-separates, discriminates) .
Observe that separating is equivalent to being embeddable in a direct product of groups in . In particular, this is so if is isomorphic to a subdirect product of groups in (see ). Here is a subdirect product of the family provided that, for each fixed , the restriction of the projection onto the th coordinate remains surjective.
Definition 2. Let be a nonempty class of groups closed under isomorphism. is a prevariety if it is closed under taking subgroups and direct products.
B. Fine observed that is a prevariety if and only if it is a separating family in the sense that every group separated by must already lie in .
Definition 3. Let be a nonempty class of groups closed under isomorphism. For a given group let be the set . is a radical class provided, for every group , If is a radical class, then is the radical of relative to .
We observe that every prevariety is a radical class with Here .
Let us call the isomorphism class of the one element group trivial. Every nontrivial prevariety admits free objects of every rank. Indeed, if is a free group of rank , then is free of rank relative to .
Definition 4. One will say a group is freely separated (freely discriminated) provided it is separated (discriminated) by the class of free groups.
Definition 5. A group is commutative transitive or CT provided the binary relation , given by if and only if , is transitive.
Remark 6. Clearly is CT if and only if the centralizer in of any nontrivial element is abelian. This result was originally stated by Harrison .
Theorem 7 (Baumslag ). Let be a freely separated group. Then is freely discriminated if and only if it is CT.
Corollary 8 (Remeslennikov). is freely discriminated if and only if it is -freely separated.
Proof. It will suffice to show that every -freely separated group is CT. Suppose not. Let be -freely separated. If is not CT, then there are such that , , and , but . (Here, of course, denotes the commutator .) Then there is a free group and a homomorphism such that simultaneously and . But this is impossible since and free groups are CT. The contradiction shows that is CT and hence freely discriminated.
Our terminology differs from that of Benjamin Baumslag who used residually- which we use separated by . He also used -residually- where we use separated by and fully residually- which we use discriminated by . We prefer to reserve residually- for separation by a family of surjective homomorphisms into (equivalently being isomorphic to a subdirect product of groups in ) as well as using fully residually- for discrimination by a family of surjective homomorphisms into . We believe there are good historical grounds for this choice of terminology. Compare this to the discussion on page 101 of Cohn . B. Fine posed the question of analogues of Benjamin Baumslag’s theorem in varieties of metabelian and nilpotent groups. The authors in  tried to treat that question in the case of the metabelian variety. In retrospect it would have been more productive to not insist our homomorphisms be surjective. Although, because of the Nielsen-Schreier subgroup theorem, the concepts of residually free and freely separated coincide as do the concepts of fully residually free and freely discriminated, the analogue of the Nielsen-Schreier subgroup theorem can fail to hold in varieties of groups and the relativized concepts need not coincide.
1.2. Metabelian and Nilpotent Groups
Basically in this section, we fix notation. The left-normed or simple commutator is defined recursively by and if has already been defined, . If and are (not necessarily distinct) subgroups of a group , then is the subgroup of generated by all commutators as varies over the Cartesian product . The derived group or commutator subgroup of is the subgroup . The second derived group is . The group is metabelian provided it is an extension of an abelian group by an abelian group. Equivalently, is metabelian provided .
A nonincreasing sequence of subgroups normal in is a descending central series for provided that, for each positive integer , An example of such is the lower central series of defined recursively by and if has already been defined, . By Lemma 33.35 on page 86 of , may also be characterized as the subgroup of generated by the simple commutators as varies over the Cartesian power .
A nondescending sequence of subgroups normal in is an ascending central series for provided, for each positive integer ,
An example of such is the upper central series of defined recursively by and, if has already been defined, That is, is the complete preimage in of the center of . One can prove that, for an integer , if and only if . In that event, is said to be nilpotent of class at most . (The nilpotency class of a nilpotent group is the least such that .) More is true. If is nilpotent of class at most , then, for all integers , , .
1.3. Concluding Introductory Remarks
This paper is in good measure expository. However there is a genuinely new result expounded in the title. The exposition is intended to explain the result’s significance in a wider context. Another raiśon d’être of this paper is to ultimately list some unsolved problems as questions.
2. Preliminaries from Logic and Varieties of Groups
2.1. Quasivarieties and Varieties
We identify the set of nonnegative integers with the first limit ordinal. Adopting the von Neumann definition of a cardinal as an ordinal not in bijective correspondence with any prior ordinal, we also view as the first infinite cardinal .
Let be the first order language with equality containing a binary operation symbol , a unary operation symbol −1, and a constant symbol . Thus, an -structure is a set provided with a binary operation , , a unary operation , , and a distinguished element . We may view every group as an -structure.
Suppose that is the set of variables of . We remark that being first order means that, in the interpretation of the variables in any -structure , they are allowed to roam over individual elements of only but never, for example, over subsets of nor functions defined on . The phrase “with equality” means that contains the logical symbol which is always to be interpreted as the identity relation in any -structure.
A universal sentence of is one of the form where is a tuple of distinct variables and is a formula of containing no quantifiers and containing occurrences of (at most) the variables in . Since vacuous quantifications are permitted, quantifier free sentences of are considered special cases of universal sentences. For example, is viewed as a universal sentence of . Less trivial examples are the group axioms; namely, Henceforth, we will suppress the symbol and indicate the binary operation by juxtaposition. Furthermore, in all that follows, we will tacitly assume that all sets of sentences under consideration are consistent and contain the group axioms above so that we are restricting our -structures to groups. That allows us to write every term of as a word on the free monoid on (here ), where the empty word is identified with the constant symbol . Modulo the group axioms we make some definitions.
Definition 9. A quasi-identity or quasilaw of is a universal sentence of the form A quasivariety is the model class of a set of quasi-identities.
Remark 10. Quasi-identities are easily seen to be preserved under taking subgroups and direct products. Thus, every quasivariety is a prevariety. The converse is false. An example of a prevariety which is not a quasivariety is the class of abelian groups containing no nonzero divisible subgroup.
Definition 11. A universal sentence of of the form is a law or identity. A variety is the model class of a set of laws.
Remark 12. Generally we omit the universal quantifiers and write for the law .
Remark 13. The law is equivalent to the quasi-identity: Hence, every variety is a quasivariety. The converse is false. An example of a quasivariety that is not a variety is the class of torsion-free groups.
Remark 14. A classic theorem of Garrett Birkhoff asserts that a nonempty class of groups is a variety if and only if it is closed under taking subgroups, homomorphic images, and direct products.
Definition 15. Let and be varieties of groups. Then the product class is the class of all groups which are extensions of a group by a group .
One can show that is a variety called the product variety of by (see ). We now give some examples of varieties of groups.(1)The isomorphism class of the trivial group is the trivial variety. All other varieties of groups are nontrivial. is determined by the law .(2)The class of all groups is a variety. It is determined by the law .
Remark 16. Multiplication of varieties is associative. is the identity element and is a zero. That is, for any variety , and .(3)The class of all abelian groups is a variety. It is determined by the law .(4)The product variety is the class of all metabelian groups. Since a group is metabelian if and only if commutators commute, is determined by the law .(5)For each integer , the class of all groups nilpotent of class at most is a variety. It is determined by the law .(6)Since the intersection of any family of varieties is again a variety, we have, for each fixed integer , that the class of groups simultaneously nilpotent of class at most and metabelian is a variety. is determined by the laws and .
Since every variety is a prevariety, we have that every nontrivial variety admits free objects (a.k.a. relatively free groups) of all ranks . Here is an absolutely free group of rank .
Definition 17. If is a variety of groups and is a group, then is the verbal subgroup of corresponding to .
If one is given a set such that is a set of laws determining the variety , then one has a good description of as the subgroup of generated by all elements of the form as varies over and varies over tuples from .
Remark 18. A variety can be determined by more than one set of laws. If we wish a unique such set we can take to be the set of all such that every group in satisfies the law , but this set of laws is unnecessarily large.
We now give some examples of verbal subgroups.(1).(2).(3).(4).(5).(6).
Definition 19. Let be a nontrivial variety of groups and a cardinal. Let the -generator group lie in . Let be the group free in on the set . Let be a surjective homomorphism. If is the normal closure of in , then is a presentation of relative to . If and are finite, then is said to be finitely presented relative to .
Remark 20. Of course, if is finitely presented relative to the variety of all groups, then is finitely presented relative to .
We next paraphrase a definition due to G. Baumslag, B. H. Neumann, H. Neumann, and P. M. Neumann [N].
Definition 21. Let be a nontrivial variety of groups. A group discriminates provided that discriminates its free group of countably infinite rank.
Definition 22. Let be a nontrivial variety of groups and a positive integer. is -discriminated provided that is discriminated by its free group of rank .
Remark 23. It follows from Benjamin Baumslag’s original paper  that the variety of all groups is -discriminated.
Definition 24. Let be a nontrivial variety of groups and a group in . is -freely separated provided to each ; there is a group free in and a homomorphism such that . If is a positive integer, is --freely separated provided for every there is a group free in and a homomorphism such that for all . is -freely discriminated provided it is --freely separated for every positive integer .
2.2. Universal Theories
Observe that commutative transitivity is rendered by the universal sentence Every nonabelian group with a nontrivial center must violate this property for if and , then commutes with which commutes with , but and do not commute. We introduce noncentral commutative transitivity, abbreviated NZCT, as the universal property: asserting that the centralizer of a noncentral element be abelian. Our main result, proven in the next section, is that the relatively free groups satisfy NZCT.
Remark 25. NZCT was first introduced by Levin and Rosenberger in  where it was called commutation transitivity. However, Remeslennikov and Stöhr in  call CT commutation transitivity. Thus, we have chosen to avoid the terminology “commutation transitivity” altogether.
CT and NZCT are the 0th and 1st terms of the following hierarchy of universal sentences CT() introduced in :
Remark 26. If , then CT() implies CT().
Remark 27. The interpretation of CT() in a group is that the centralizers of an element outside of the th term of the upper central series be abelian.
It was observed in  that, for an integer , the free nilpotent groups satisfy CT().
Definition 28. Let be a group. The set of all universal sentences of true in is the universal theory of . One says and are universally equivalent and writes provided that .
Analogous to the definition of a universal sentence, we define an existential sentence of to be one of the form , where is a tuple of distinct variables and is a formula of containing no quantifiers and involving (at most) the variables in . The meanings of and are apparent. Since an existential sentence is logically equivalent to the negation of a universal sentence and vice versa one concludes that if and only if . The following, for which we provide a proof, is standard fare.
Lemma 29. Let be a group and a subgroup of . Then if and only if, to every finite system (call it of equations and inequations in a finite tuple of unknowns, it is the case that has a solution in whenever it has a solution in .
Proof. Suppose first that . Then . So,
is true in ; it must also be true in .
Now suppose that has a solution in whenever it has a solution in . It will suffice to show . Since , every existential sentence of true in must also be true in . Suppose that is true in . We may assume that is written in disjunctive normal form; namely, Then is logically equivalent to Since a disjunction is true if and only if at least one of the disjuncts is true, for at least one fixed , is true in , then the system has a solution in and hence also a solution in . Working backwards, so to speak, is true in .
Corollary 30. Let be a group and a subgroup of . If discriminates , then .
Proof. Suppose the system in the tuple has a solution . Since discriminates , there is a homomorphism which does not annihilate any for . But then is a solution to the system in .
Proposition 31. Let be a nontrivial variety of groups and let be an infinite cardinal. Then the free group of countably infinite rank in discriminates . In particular, for all infinite cardinals .
Remark 32. Before giving a proof of Proposition 31, we remark that the universal equivalence of free algebras of infinite rank is also a very special consequence of a general theorem of Vaught, which is recorded as Theorem 4 of Section 3.8 of Grätzer  and proven therein.
Proof of Proposition 31. Let freely generate and view as the subgroup generated (necessarily freely) by the initial segment . Let . Let be finitely many nontrivial elements of . For each , fix a word on representing . Each can involve only finitely many generators from and only finitely many generators from . Let be the set of generators from involved in and the set of generators from involved in so that and are finite. Say and . We may assume that , , where and , , where . We may view each as a word on . Since every map on into any group in extends to a homomorphism, we may map into (and onto which we may identify with ) by fixing , annihilating , and mapping to for all . The image of is .
Suppose that, for some fixed , , we would have . Since is free on in we would have, for any and any , that . That is, is a law in . But that contradicts the fact that in . The contradiction shows that all of the images are nontrivial in , . Hence, discriminates .
Given a nontrivial variety , every universal sentence true in must also be true in for all (where we view to be free on the empty set). The can conceivably satisfy more universal sentences of than , but they must satisfy at least those true in and hence true in for all cardinals . It therefore makes sense to call the universal theory of the free groups in for that is the set of universal sentences of that must be true in every group free in . In some cases, we can do better than rank . Let us say that is finitely discriminated if there is a positive integer such that is -discriminated. In that event, discriminates and since, for any finite , we may view a map as a map , such also discriminate . It follows that for all . Therefore, if we let when is not finitely discriminated and otherwise, then is precisely the set of universal sentences of true in every group free in . In particular, since quasi-identities are instances of universal sentences, if is the set of all quasi-identities of true in every free in , then is precisely the set of all quasi-identities of true in .
Definition 33. Let be a nontrivial variety of groups with the above notation being the universal theory of the free groups in .
Proposition 34. Let be a nontrivial variety of groups. Suppose that for some integer the free groups in satisfy the universal sentence CT. Let be --freely separated. Then satisfies CT().
Proof. Suppose not. Then there are such that and and , but . Since is --freely separated, there is a group free in and a homomorphism such that simultaneously and , but that contradicts the fact that satisfies CT() since in . The contradiction shows that satisfies CT().
Remark 35. We may as well take in Proposition 34 to be the least integer such that the free groups in satisfy CT() as CT() implies CT() whenever .
It was shown in  that the universal theory of the groups free in the variety of all groups is axiomatizable by the quasi-identities true in the free groups together with CT. It was shown in [7, 11] that the universal theory of the free metabelian groups is axiomatizable by the quasi-identities true in the free metabelian groups together with CT. With those facts in mind, we assert the next proposition.
Proposition 36. Let be a nontrivial variety of groups. Suppose that there is an integer (which we may assume minimal) such that the universal theory of the free groups in is axiomatizable by a set of quasi-identities together with CT(). If has a finite presentation relative to , is -freely separated, and satisfies CT(), then is -freely discriminated.
Proof. Let be a finite presentation of relative to . Suppose that is not -freely discriminated. Then there exist finitely many nontrivial elements (with ) of such that every homomorphism must annihilate at least one of . Let , . It follows that the following universal sentence of is true in every : Since is -freely separated, it is isomorphic to a subgroup of a direct product of groups free in . Quasi-identities are preserved in direct products and subgroups. Hence, is a model of . But also satisfies CT() and so, by hypothesis, is a model of the universal theory of the free groups in . In particular, must satisfy . But then at least one of , or must be trivial in -contradicting the choice of . The contradiction shows that is -freely discriminated.
Philip Hall showed (recorded as Observation (2) on page 22 of ) that every finitely generated metabelian group has a finite presentation relative to the variety of all metabelian groups. Putting that observation together with the fact that the universal theory of the free metabelian groups is axiomatizable by and Propositions 34 and 36, we immediately deduce the following metabelian analogue of Benjamin Baumslag’s classic theorem.
Theorem 37. Let be a finitely generated metabelian group which is -freely separated. Then is -freely discriminated if and only if it is CT.
With the same proof as Corollary 8, we immediately deduce the following.
Corollary 38. Let be a finitely generated metabelian group. Then is -freely discriminated if and only if it is --freely separated.
Lyndon showed in  that every finitely generated nilpotent group is finitely presented. Anticipating our main result, proven in the next section, that the relatively free groups satisfy NZCT = CT(), we may ask whether the universal theory of the free groups in is axiomatizable by . If it is, then we have the analogue of Theorem 37 for the varieties . Furthermore, following a suggestion of A. G. Myasnikov that the universal theory of the free groups in may be axiomatizable by for a suitable (which we may assume minimal), we may ask whether or not this is so. If true, then we also have the analogue of Theorem 37 for the varieties .
3. The Main Result
3.1. Philip Hall’s Basic Commutators
The general reference for this section as well as the justification for all subsequent commutator calculations will be in Chapter 5 of . We begin by fixing an integer . Let be a rank free group and suppose that such that is freely generated by the set . is residually nilpotent. That is, It follows that, for every element , there is a positive integer such that . The least such positive integer is called the weight of and is denoted by .
For each positive integer , the quotient group is a finitely generated free abelian group. Suppose that we order a basis for each such quotient. Say (here is the number of basis elements of weight ; see Theorem 40 below) is a basis for ordered according to their subscripts. Then every element of is uniquely represented modulo in the form where are fixed preimages in of the modulo . Furthermore, if we order the union of these preimages over all by asserting that those of smaller weight precede those of larger weight, we have for each fixed integer that every element of is uniquely represented modulo in the form Equivalently, every element of the nilpotent free group is uniquely represented in the form .
Starting with as before, we extend to an infinite sequence of distinct terms: according to a construction which guarantees that the terms of any fixed weight in the sequence not only are pairwise incongruent modulo but also project to a basis for modulo .
Although it is now generally agreed that Philip Hall never himself actually constructed such a sequence (Marshall Hall, Jr. did), the sequence constructed according to the recipe to follow is named in his honor. The terms of such a sequence are called (Hall) basic commutators.
Remark 39. The terminology “basic commutators” is standard. However, technically, it is actually the sequence rather than the individual commutators which is basic.
Let us observe, for example, that So the representation of a commutator need not be unique. In order to finesse questions about the uniqueness of representations of basic commutators in terms of prior ones, we construct formal basic commutators in an abstract setting. With apologies to our topologist friends who wish to reserve the word “groupoid” for a concept in category theory, we follow the usage of Baumslag in  and let groupoid be a set provided with a binary operation. Let be the class of all groupoids and let be free of rank relative to . Suppose that such that freely generates . Thus, every element of is uniquely a nonassociative word on . If the length of such a nonassociative word is , then we say has weight and write .
Starting with , we extend to a basic sequence of distinct terms according to the recipe below.(1) are the formal basic commutators of weight and are ordered according to their subscripts.(2)If , then the formal basic commutators of weight less than precede those of weight while the formal basic commutators of weight are ordered arbitrarily among themselves.(3)If , then is a formal basic commutator of weight provided that the following conditions are satisfied:(i);(ii) in the ordering of formal basic commutators;(iii)if , then in the ordering of formal basic commutators.
Notice that different orderings can lead to different basic sequences. If we wish to pin down a unique basic sequence given , we can, for example, insist on the lexicographical ordering of terms of each fixed weight in the sequence. That is, and , each of weight , will be ordered according to provided either or and .
Now may be viewed as a groupoid under the derived operation of commutation. That is, we define a new binary operation on by . The assignment , , extends to a groupoid homomorphism from into . The images of the formal basic commutators (given a fixed basic sequence) are the basic commutators in (with respect to a fixed basic sequence).
For example, if has rank , then (if we insist on the lexicographical ordering) the basic commutators of weights through are the following. Weight 1: Weight 2: Weight 3: Weight 4:
These project modulo to a basic sequence in the free nilpotent group . A result of Magnus recorded as Theorem 36.32 of  asserts that if is a nonabelian free group, then the left-normed basic commutators of weight project modulo to a basic sequence in the relatively free group . Thus, in our concrete example, if we omit the non-left-normed terms, namely, , , and , then the remaining terms are basic modulo and so project to a basic sequence in the relatively free group .
3.2. Deducing the Main Result
If the , then is abelian, so it certainly satisfies NZCT. So suppose that each of and is at least . Then is -discriminated (see ). Thus, every universal sentence of true in must also be true in every group free in . NZCT is a universal sentence of ; hence, it suffices to show that satisfies NZCT. Let be an absolutely free group of rank . Let be free of rank in . Then and is isomorphic to .
Our strategy will be to work in modulo . We must show that, modulo , the centralizer of an element which projects to a noncentral element modulo is abelian. It will be useful to know what subgroups of free nilpotent groups look like. Our principal tool will be the following theorem of Moran whose proof may be found in .
Theorem 40 (Moran’s subgroup theorem). Let and let be a subgroup of . Then , where , if and if . The are constructed by taking as generators individual preimages in of basis elements of the free abelian group . The group is free abelian of rank given by the Witt formula, , where is the Möbius function defined for positive integers by the rules and if with the ’s being distinct primes, if any , and .
Remark 41. Of course, if is an absolutely free group of finite rank , and , then, for all , , and so, for all , , is isomorphic to . In the application at hand will be .
Lemma 42. The free groups in are torsion free.
Proof. Let be a positive integer and suppose that in and assume that . Let be the least integer such that . Let map to in . Then and so, by Theorem 36.32 of , is uniquely the product of powers of weight left-normed basic commutators in . But are central in . Thus, in again by the uniqueness of the representation. That contradicts .
Definition 43. A group is power commutative provided for ordered pairs of positive integers and all such that , , and it is the case that .
Definition 44 (Baumslag ). A group is a -group provided for every integer it is the case that whenever in .
Lemma 45. Every -group is power commutative.
Proof. Suppose that , , and in the -group . Then , so and commutes with . Thus, , so and consequently and commute.
By a result of A. I. Mal’cev recorded as Lemma 2.1 in , it follows that torsion-free nilpotent groups are -groups. We therefore conclude.
Corollary 46. The free groups in are power commutative.
The next result is Lemma 34.51 of .
Lemma 47. If is a metabelian group and , then for every permutation of .
Theorem 48 (the main theorem). The free groups in satisfy NZCT.
Proof. If , then is abelian, so it certainly satisfies NZCT. We assume that each of and is at least . Since is -discriminated, every universal sentence of true in is true in every group free in . Thus it suffices to show that the universal sentence NZCT is true in .
Let and let where , .
Since is metabelian commutators which commute in , it suffices to determine what can commute with an element . Since is power commutative, we may take not to be a proper power in . Now for integers let be the left-normed basic commutator and let be its image in . Note that, when , the weight commutators and are central in and , respectively. In determining commutativity, we may ignore central factors, so we can assume that contains no factors .
Thus where and is not a proper power in . Let be the preimage of in . Note that cannot be a proper power in ; otherwise, would be a proper power in . Let be the centralizer in of . We wish to show that is abelian.
Let be the preimage of in and note that . Our strategy will be to use Moran’s subgroup theorem to show that is commutative modulo . We can take for generators of preimages in of a basis for So suppose that and . (Note: .) Now , so implies . So by collecting and to the front in , we see that and and are scalar multiples of a common nonzero ordered pair with GCD(. There is a least positive integer such that lies in . Then is infinite cyclic generated by