On verbal subgroups in finite and profinite groups
aa r X i v : . [ m a t h . G R ] M a r On verbal subgroups in finite and profinite groups
Cristina Acciarri and Pavel Shumyatsky
Abstract.
Let w be a multilinear commutator word. In the presentpaper we describe recent results that show that if G is a profinite groupin which all w -values are contained in a union of finitely (or in somecases countably) many subgroups with a prescribed property, then theverbal subgroup w ( G ) has the same property as well. In particular, weshow this in the case where the subgroups are periodic or of finite rank. Introduction
Let w be a group-word, and let G be a group. The verbal subgroup w ( G ) of G determined by w is the subgroup generated by the set of allvalues w ( g , . . . , g n ), where g , . . . , g n are elements of G . Most of the wordsconsidered in this paper are multilinear commutators , also known under thename of outer commutator words . These are words that have a form of amultilinear Lie monomial, i.e., they are constructed by nesting commutatorsusing always different indeterminates. For example the word[[ x , x ] , [ y , y , y ] , z ]is a multilinear commutator while the Engel word[ x, y, y, y ]is not.An important family of multilinear commutators are the lower centralwords γ k , given by γ = x , γ k = [ γ k − , x k ] = [ x , . . . , x k ] , for k ≥ γ k ( G ) are the terms of the lower centralseries of G . Another distinguished sequence of multilinear commutators arethe derived words δ k , on 2 k variables, which are defined recursively by δ = x , δ k = [ δ k − ( x , . . . , x k − ) , δ k − ( x k − +1 , . . . , x k )] , for k ≥ Mathematics Subject Classification.
Primary 20E18; Secondary 20F14.
Key words and phrases.
Profinite groups, verbal subgroups, coverings, multilinearcommutators.This research was supported by CNPq-Brazil.
The verbal subgroup that corresponds to the word δ k is the familiar k thderived subgroup of G usually denoted by G ( k ) .A word w is concise in a class of groups C if whenever the set of all w -values in G is finite, it always follows that w ( G ) is finite, for every group G ∈ C . P. Hall asked whether every word is concise, but it was later provedby Ivanov [ ] that this problem has a negative solution (see also [ ], p.439). On the other hand, many relevant words are known to be concise.For instance, Turner-Smith [ ] showed that the lower central words γ k andthe derived words δ k are concise. In 1974 Wilson [ ] extended the resultto all multilinear commutator words. New proofs of the results on γ k and δ k were obtained in [ ] and for arbitrary multilinear commutators in [ ].Turner-Smith also proved that every word is concise in the class formed bythe residually finite groups all of whose quotients are again residually finite[ ]. In 1967 Merzljakov [ ] answered P. Hall’s question in the affirmativefor the class of linear groups. It is an open question, due to Segal [ ],whether in the class of residually finite groups every word is concise.There are several natural ways to look at Hall’s question from a differentangle. The circle of problems arising in this context can be characterized asfollows.Given a word w and a group G , assume that certain restrictions areimposed on the set of all w -values in G . How does this influence the structureof the verbal subgroup w ( G )?In the present paper we describe some recent results related to the abovequestion. We concentrate on results that show that if G is a profinite groupin which all w -values are contained in a union of finitely (or in some casescountably) many subgroups with a prescribed property, then the verbalsubgroup w ( G ) has the same property as well. In particularly we show thisin the case where the subgroups are periodic or of finite rank.Throughout the paper we use the expression “( a, b, . . . )-bounded” tomean “bounded from above by a function depending only on the parameters a , b , . . . ”.
1. On groups in which w -values admit a finite covering A covering of a group G is a family { S i } i ∈ I of subsets of G such that G = S i ∈ I S i . If { H i } i ∈ I is a covering of G by subgroups, it is natural to askwhat information about G can be deduced from properties of the subgroups H i . In the case where the covering is finite actually quite a lot about thestructure of G can be said. The first result in this direction is due to Baer[ ], who proved that G admits a finite covering by abelian subgroups ifand only if it is central-by-finite. The nontrivial part of this assertion isan immediate consequence of a subsequent result of B.H. Neumann [ ]: if { S i } is a finite covering of G by cosets of subgroups, then G is also coveredby the cosets S i corresponding to subgroups of finite index in G . In other N VERBAL SUBGROUPS IN FINITE AND PROFINITE GROUPS 3 words, we can get rid of the cosets of subgroups of infinite index withoutlosing the covering property.If the set of all w -values in a group G can be covered by finitely manysubgroups, one could hope to get some information about the structure ofthe verbal subgroup w ( G ).In this direction we mention the following result that was obtained in[ ]. Let w be either the lower central word γ k or the derived word δ k .Suppose that G is a group in which all w -values are contained in a union offinitely many Chernikov subgroups, then w ( G ) is Chernikov. Remind thata group is Chernikov if and only if it is a finite extension of a direct sum offinitely many Pr¨ufer groups C p ∞ .Another result of this nature was established in [ ]: If G is a groupin which all commutators are contained in a union of finitely many cyclicsubgroups, then G ′ is either cyclic or finite. Later G. Cutulo and C. Nicoterashowed that if G is a group in which all γ k -values are contained in a unionof finitely many cyclic subgroups, then γ k ( G ) is finite-by-cyclic. They alsoshowed that γ k ( G ) can be neither cyclic nor finite [ ].In our recent paper [ ] we dealt with profinite groups in which all w -values are contained in a union of finitely many subgroups with certainprescribed properties. A profinite group is a topological group that is iso-morphic to an inverse limit of finite groups. The textbooks [ ] and [ ]provide a good introduction to the theory of profinite groups. In the con-text of profinite groups all the usual concepts of group theory are interpretedtopologically. In particular, by a verbal subgroup of a profinite group cor-responding to the word w we mean the closed subgroup generated by all w -values. Theorem . Let w be a multilinear commutator word and G a profi-nite group that has finitely many periodic subgroups G , G , . . . , G s whoseunion contains all w -values in G . Then w ( G ) is locally finite. A group is periodic (torsion) if every element of the group has finite orderand a group is called locally finite if each of its finitely generated subgroupsis finite. Periodic profinite groups have received a good deal of attention inthe past. In particular, using Wilson’s reduction theorem [ ], Zelmanov hasbeen able to prove local finiteness of periodic compact groups [ ]. EarlierHerfort showed that there exist only finitely many primes dividing the ordersof elements of a periodic profinite group [ ]. It is a long-standing problemwhether any periodic profinite group has finite exponent. Remind that agroup G has exponent e if x e = 1 for all x ∈ G and e is the least positiveinteger with that property.In [ ] we also considered the situation where all w -values are containedin a union of finitely many subgroups of finite rank. A group G is said to beof finite rank r if every finitely generated subgroup of G can be generatedby r elements. Very often the term “special rank” is used with the samemeaning. However in the present paper there is no danger of confusing CRISTINA ACCIARRI AND PAVEL SHUMYATSKY different notions so we will just use the term “rank”. We have the followingresult.
Theorem . Let w be a multilinear commutator and G a profinitegroup that has finitely many subgroups G , G , . . . , G s whose union containsall w -values in G . If each of the subgroups G , G , . . . , G s is of finite rank,then w ( G ) has finite rank as well. It follows from the proof that if under the hypotheses of Theorem 1.1the subgroups G , G , . . . , G s have finite exponent, then w ( G ) has finiteexponent as well. It is natural to address the question whether the ex-ponent (rank) of w ( G ) is bounded in terms of the exponents (ranks) of G , G , . . . , G s and s .We have been able to answer the question in the affirmative only in theparticular case where w = γ k . The case of the exponent was treated usingthe Lie-theoretic techniques that Zelmanov created in his solution of therestricted Burnside problem [
31, 32 ]. Theorem . Let e, k, s be positive integers and G a profinite group thathas subgroups G , G , . . . , G s whose union contains all γ k -values in G . Sup-pose that each of the subgroups G , G , . . . , G s has finite exponent dividing e . Then γ k ( G ) has finite ( e, k, s ) -bounded exponent. The situation where the subgroups G , G , . . . , G s are of finite rank doesnot require the use of Zelmanov’s Lie-theoretic techniques. Instead an im-portant role in the proof of the next theorem is played by the Lubotzky-Mann theory of powerful p -groups [ ]. Theorem . Let k, r, s be positive integers and G a profinite groupthat has subgroups G , G , . . . , G s whose union contains all γ k -values in G .Suppose that each of the subgroups G , G , . . . , G s has finite rank at most r .Then γ k ( G ) has finite ( k, r, s ) -bounded rank. The proofs of the above theorems will be sketched out in Section 4 below.In the next section we will discuss one specific tool used in the proofs ofTheorems 1.3 and 1.4 –namely, a version of the Focal Subgroup Theorem.
2. Around the focal subgroup theorem
The Focal Subgroup Theorem [ , Theorem 7.3.4] says that if P is aSylow subgroup of a finite group G , then P ∩ G ′ is generated by elements ofthe form [ x, y ] ∈ P , where x ∈ P and y ∈ G . In particular, it follows thatSylow subgroups of G ′ are generated by commutators. Thus, the followingquestion arises.Let w be a commutator word, G a finite group and P a Sylow p -subgroupof w ( G ). Is it true that P can be generated by w -values lying in P ?The above question was considered in [ ] where the following result wasproved. N VERBAL SUBGROUPS IN FINITE AND PROFINITE GROUPS 5
Theorem . Let w be a multilinear commutator, G a finite group and P a Sylow p -subgroup of w ( G ) . Then P is generated by powers of w -values. The theorem depends on the classification of finite simple groups. It isused in the proof of Theorems 1.3 and 1.4.Another obvious corollary of the Focal Subgroup Theorem is that if G isa finite group in which every nilpotent subgroup generated by commutatorshas exponent dividing e , then the exponent of G ′ divides e as well. It is easyto deduce from Theorem 2.1 that if every nilpotent subgroup generated by γ k -commutators has exponent dividing e , then γ k ( G ) has ( k, e )-boundedexponent. The latest result in this direction is the following theorem [ ]. Theorem . Let e be a positive integer and w a multilinear commu-tator. Suppose that G is a finite group in which any nilpotent subgroupgenerated by w -values has exponent dividing e . Then the exponent of thecorresponding verbal subgroup w ( G ) is bounded in terms of e and w only. The proof of this result uses a number of deep tools. In particular, ituses the classification of finite simple groups and Zelmanov’s solution ofthe restricted Burnside problem. It is interesting that the reduction fromthe general case to the case where G is soluble is somewhat reminiscent ofthe Hall-Higman’s reduction [ ] for the restricted Burnside problem. In thesame time in the case of Theorem 2.2 the reduction uses the solution of therestricted Burnside problem.As a by-product of the proof of Theorem 2.2, it is shown that if G is afinite soluble group in which any nilpotent subgroup generated by w -valueshas exponent dividing e , then the Fitting height of G is bounded in termsof e and w only.
3. About proofs
In the present section we will describe ideas behind the proof of Theorem1.1. Details can be found in [ ]. As the reader will see the arguments arepretty universal and can be used to obtain other results of this kind. Inparticular, in the next section we use the same scheme of reasoning to deducenew results on covering of w -values in profinite groups. Idea of the proof of Theorem 1.1.
Without explicit references weuse Zelmanov’s theorem that a periodic profinite group is locally finite [ ].It is a general property of multilinear commutators that for every w thereexists k such that every δ k -commutator is also a w -value [ , Lemma 4.1].For each integer i = 1 , . . . , s we set S i = n ( x , . . . , x k ) ∈ G × · · · × G | {z } k | δ k ( x , . . . , x k ) ∈ G i o . The sets S i are closed in G × · · · × G | {z } k and cover the group G × · · · × G | {z } k . ByBaire’s Category Theorem [ , p. 200] at least one of these sets contains a CRISTINA ACCIARRI AND PAVEL SHUMYATSKY non-empty interior. Hence, there exist an open subgroup H of G , elements a , . . . , a k in G and an integer j such that δ k ( a h , . . . , a k h k ) ∈ G j for all h , . . . , h k ∈ H. Thus, all commutators of the form δ k ( a h , . . . , a k h k ) belong to the givenperiodic subgroup G j .Without loss of generality we can assume that the subgroup H is nor-mal. In this case H normalizes the set of all commutators of the form δ k ( a h , . . . , a k h k ), where h , . . . , h k ∈ H . Let K be the subgroup of G generated by all commutators of the form δ k ( a h , . . . , a k h k ), where h , . . . , h k ∈ H . Note that K ≤ G j . Since the subgroup G j is locally finite,so is K . Let D = K ∩ H . Then D is a normal locally finite subgroup of H and the normalizer of D in G has finite index. Therefore there are onlyfinitely many conjugates of D in G . Let D = D , D , . . . , D r be all theseconjugates. All of them are normal in H and so their product D D . . . D r is locally finite. By passing to the quotient G/D D . . . D r we may assumethat D = 1. Since D = K ∩ H and H has finite index in G , it follows that K is finite. On the other hand, the normalizer of K has finite index in G andso, by Dicman’s Lemma [ , 14.5.7] the normal closure, say L , of K in G isalso finite. We can pass to the quotient group G/L and assume that K = 1.In that case we have δ k ( a h , . . . , a k h k ) = 1 for all h , . . . , h k ∈ H .This is a so-called coset identity , i.e., the cosets a H, . . . , a k H satisfythe law δ k ≡
1. Note that coset identities often play important role inthe results on profinite groups. For example, the problem whether everycompact periodic group has finite exponent can be reduced to the followingquestion: Let n be a positive integer. Suppose that a periodic profinitegroup G has an element a and an open subgroup H such that ( ah ) n = 1 forevery h ∈ H . Does it follow that H has finite exponent? So far the positiveanswer to the above question is known only in the case where n is prime[ ].Our case is much easier, due to the fact that the word δ k is multilinear.We prove that the condition δ k ( a h , . . . , a k h k ) = 1 for all h , . . . , h k ∈ H implies that H is soluble with derived length at most k . Thus, the group G is soluble-by-finite and we can use induction on the minimal derived length d of a normal open subgroup of G .So assume that H has derived length d and let N = H ( d − . Let M be the subgroup of N generated by all w -values contained in N . Then M = M M . . . M s , where M i = M ∩ G i for i = 1 , . . . , s . It follows that M is locally finite and has finite exponent. Thus we can pass to the quotient G/M and assume that there are no nontrivial w -values contained in N .It is a property of multilinear commutators that a normal subgroupcontaining no w -values must centralize w ( G ). Thus, [ w ( G ) , N ] = 1. Recallthat we are using induction on d and N = H ( d − . Since [ w ( G ) , N ] = 1,we conclude that w ( G ) /Z ( w ( G )) is locally finite. For abstract groups wehave a locally finite version of Schur’s Theorem: if K/Z ( K ) is locally finite, N VERBAL SUBGROUPS IN FINITE AND PROFINITE GROUPS 7 then K ′ is locally finite. However it is unclear whether this can be used onprofinite groups. Fortunately, we have the additional information that w ( G )is soluble-by-finite.Since w ( G ) /Z ( w ( G )) is locally finite and soluble-by-finite, it easily fol-lows that w ( G ) /Z ( w ( G )) has finite exponent. By a profinite version ofMann’s theorem [ ] we conclude that the derived group of w ( G ) has finiteexponent. Therefore we can pass to the quotient G/w ( G ) ′ and assume that w ( G ) is abelian. But now it is clear that w ( G ) is the product of subgroups w ( G ) ∩ G i for i = 1 , . . . , s , and so w ( G ) is locally finite. (cid:3) Theorem 1.2 can be proved following, by and large, similar arguments.One essential difference is that in place of Mann’s theorem we apply a result,due to S. Franciosi, F. de Giovanni, and L.A. Kurdachenko [ , Theorem 2.5]which says that if K is a soluble-by-finite group such that K/Z ( K ) has finiterank, then K ′ has finite rank bounded in terms of the derived length of thesoluble radical of K , its index in K and the rank of K/Z ( K ).In the proofs of Theorems 1.3 and 1.4 techniques of quite a differentnature are required. Let us focus our attention on Theorem 1.3. Let e, k, s be positive integers and G a profinite group that has subgroups G , G , . . . , G s whose union contains all γ k -values in G . Suppose that eachof the subgroups G , G , . . . , G s has finite exponent dividing e . Then γ k ( G ) has finite ( e, k, s ) -bounded exponent. Note that without loss of generality G can be assumed finite. The fol-lowing elementary lemma plays a crucial role in the proof. Lemma . Let G be a nilpotent group and suppose that G is generatedby a commutator-closed subset X which is contained in a union of finitelymany subgroups G , G , . . . , G s . Then G can be written as the product G = G G . . . G s . Proof.
Let K be the last nontrivial term of the lower central series of G . Then K is generated by elements of X and so K = K K . . . K s , where K i = K ∩ G i for i = 1 , . . . , s . Arguing by induction on the nilpotency classof G assume G = G G . . . G s K K . . . K s . Since all K i are central, we can move them to the left in the above productand the lemma follows. (cid:3) Idea of the proof of Theorem 1.3.
Let P be a Sylow p -subgroupof γ k ( G ). It is sufficient to show that the exponent of P is ( e, k, s )-bounded.By Theorem 2.1 we know that P is generated by powers of γ k -values –elements of order dividing e . Therefore it is sufficient to show that theexponent of γ k ( P ) is ( e, k, s )-bounded. Let Q = γ k ( P ). Then, by Lemma3.1, Q is the product Q Q . . . Q s , where Q i = G i ∩ Q for i = 1 , . . . , s . CRISTINA ACCIARRI AND PAVEL SHUMYATSKY
Using Zelmanov’s Lie-theoretic techniques one can show that Q has ( e, k, s )-bounded exponent. Hence P has ( e, k, s )-bounded exponent. This holds forevery Sylow subgroup of γ k ( G ) and so the result follows. (cid:3)
4. On countable coverings of w -values in profinite groups The last section of the paper deals with the situation where w -values arecovered by countably many subgroups with prescribed properties. Theorem . Let w be a multilinear commutator word and G a profi-nite group having countably many soluble subgroups whose union containsall w -values. Then w ( G ) is soluble-by-finite. Proof.
Let G , G , . . . be the countably many soluble subgroups whoseunion contains all w -values. Since w is a multilinear commutator there existsan integer k such that every δ k -commutator is also a w -value [ , Lemma4.1]. For each integer i ≥ S i = n ( x , . . . , x k ) ∈ G × · · · × G | {z } k | δ k ( x , . . . , x k ) ∈ G i o . The sets S i are closed in G × · · · × G | {z } k and cover the group G × · · · × G | {z } k . ByBaire’s Category Theorem at least one of these sets contains a non-emptyinterior. Hence, there exist an open subgroup H of G , elements a , . . . , a k in G and an integer j such that δ k ( a h , . . . , a k h k ) ∈ G j for all h , . . . , h k ∈ H. Without loss of generality we can assume that the subgroup H is nor-mal. In this case H normalizes the set of all commutators of the form δ k ( a h , . . . , a k h k ), where h , . . . , h k ∈ H . Let K be the subgroup of G generated by all commutators of the form δ k ( a h , . . . , a k h k ), where h , . . . , h k ∈ H . Note that K ≤ G j . Since the subgroup G j is soluble, sois K . Let D = K ∩ H . Then D is a normal soluble subgroup of H and thenormalizer of D in G has finite index. Therefore there are only finitely manyconjugates of D in G . Let D = D , D , . . . , D r be all these conjugates. Allof them are normal in H and so their product D D . . . D r is soluble. Bypassing to the quotient G/D D . . . D r we may assume that D = 1. Since D = K ∩ H and H has finite index in G , it follows that K is finite.On the other hand, the normalizer of K has finite index in G and so thenormal closure, say L , of K in G is finite. We can pass to the quotient group G/L and assume that K = 1. In that case we have δ k ( a h , . . . , a k h k ) = 1for all h , . . . , h k ∈ H . It follows that the group H is soluble-by-finite-by-soluble, so in particular it is soluble-by-finite. Since H has finite index in G we conclude that G is soluble-by-finite, as desired. (cid:3) Our attempts to treat the groups in which w values are contained in aunion of countably many subgroups that are periodic or of finite rank so farwere successful only in the case where w = [ x, y ]. N VERBAL SUBGROUPS IN FINITE AND PROFINITE GROUPS 9
Theorem . Let G be a profinite group having countably many periodicsubgroups whose union contains all commutators [ x, y ] of G . Then G ′ islocally finite. Theorem . Let G be a profinite group having countably many sub-groups of finite rank whose union contains all commutators [ x, y ] of G . Then G ′ is of finite rank. We will now describe in detail the proof of Theorem 4.2. Theorem 4.3can be proved in a somewhat similar way.
Proof of Theorem 4.2.
Let G , G , . . . be the countably many peri-odic subgroups whose union contains all commutators [ x, y ] of G . For eachinteger i ≥ S i = n ( x, y ) ∈ G × G | [ x, y ] ∈ G i o . The sets S i are closed in G × G and cover the whole of G × G . By Baire’sCategory Theorem at least one of these sets contains a non-empty interior.Hence, there exist an open subgroup H of G , elements a, b in G and aninteger i such that [ ah , bh ] ∈ G i for all h , h ∈ H. Without loss of generality we can assume that the subgroup H is normal. Inthis case H normalizes the set of all commutators [ ah , bh ], where h , h ∈ H . Let K be the subgroup of G generated by all commutators of the form[ ah , bh ], where h , h ∈ H . Note that K ≤ G i . Since the subgroup G i islocally finite, so is K . Let K = K ∩ H . Then K is a normal locally finitesubgroup of H and the normalizer of K in G has finite index. Thereforethere are only finitely many conjugates of K in G . Let K , K , . . . , K r be theconjugates. All of them are normal in H and so their product K K . . . K r is locally finite. By passing to the quotient G/K K . . . K r we may assumethat K = 1. Since K = K ∩ H and H has finite index in G , it followsthat K is finite. On the other hand, the normalizer of K has finite index in G and so the normal closure, say L , of K in G is also finite. We can passto the quotient group G/L and assume that K = 1. In that case we have[ ah , bh ] = 1 for all h , h ∈ H . As we have seen in the proof of Theorem1.1 this implies that the subgroup H is abelian.Since H is open, we can choose finitely many elements a , a , . . . , a s ∈ G such that G = h H, a , a , . . . , a s i . Then [ H, G ] = Q ≤ i ≤ s [ H, a i ] . Because H is abelian, every element of the subgroup [ H, a i ] is a commutator. Againby Baire’s Category Theorem [ H, a i ] contains an open subgroup M and anelement c such that cM is contained in G j , for some index j . So M islocally finite and it follows that [ H, a i ] is locally finite as well. Since we canrepeat the same argument for every i = 1 , . . . , s , we conclude that [ H, G ]is locally finite. We now pass to the quotient G/ [ H, G ] and assume that[
H, G ] = 1. Then H ≤ Z ( G ) and by Schur’s Theorem G ′ is finite. The proofis complete. (cid:3) References [1] C. Acciarri, G.A. Fern´andez-Alcober, P. Shumyatsky,
A focal subgroup theorem forouter commutator words . J. Group Theory, DOI: 10.1515/jgt.2011.113, to appear.[2] C. Acciarri, P. Shumyatsky,
On profinite groups in which commutators are covered byfinitely many subgroups , submitted. arXiv:1112.5879.[3] S. Brazil, A. Krasilnikov, P. Shumyatsky,
Groups with bounded verbal conjugacyclasses , J. Group Theory, N. , 2006, pp.127–137.[4] G. Cutolo, C. Nicotera, Verbal sets and cyclic coverings , J. Algebra, N. , 2010,pp.1616–1624.[5] G.A. Fern´andez-Alcober, M. Morigi,
Outer commutator words are uniformly concise ,J. London Math. Soc., N. , 2010, pp.581–595.[6] G.A. Fern´andez-Alcober, P. Shumyatsky, On groups in which commutators are cov-ered by finitely many cyclic subgroups , J. Algebra, N. , 2008, pp.4844–4851.[7] S. Franciosi, F. de Giovanni, L.A. Kurdachenko,
The Schur property and groups withuniform conjugacy classes , J. Algebra, N. , 1995, pp.823–847.[8] D. Gorenstein,
Finite Groups , Chelsea Publishing Company, New York, 1980.[9] P. Hall, G. Higman,
The p -length of a p -soluble group and reduction theorems forBurnside’s problem , Proc. London Math. Soc., N.(3) , 1956, pp.1–42.[10] W. Herfort, Compact torsion groups and finite exponent , Arch. Math., N. , 1979,pp.404–410.[11] S. V. Ivanov, P. Hall’s conjecture on the finiteness of verbal subgroups , Izv. Vyssh.Ucheb. Zaved., N. , 1989, pp.60–70.[12] J.L. Kelley,
General Topology , van Nostrand, Toronto - New York - London, 1955.[13] E.I. Khukhro,
A comment on periodic compact groups , Sib. Math. J., N. (3), 1989,pp.493–496.[14] A. Lubotzky, A. Mann, Powerful p -groups. I: finite groups , J. Algebra, N. , 1987,pp.484–505.[15] A. Mann, The exponents of central factor and commutator groups , J. Group Theory,N. , 2007, pp.435–436.[16] Ju.I. Merzlyakov, Verbal and marginal subgroups of linear groups , Dokl. Akad. NaukSSSR, N. , 1967, pp.1008–1011.[17] B.H. Neumann,
Groups covered by finitely many cosets , Publ. Math. Debrecen, N. ,1954, pp.227–242.[18] B.H. Neumann, Groups covered by permutable subsets , J. London Math. Soc., N. ,1954, pp.236–248.[19] A. Yu. Ol’shanskii, Geometry of Defining Relations in Groups , Mathematics and itsapplications v. (Soviet Series), Kluwer Academic Publishers, Dordrecht, 1991.[20] L. Ribes, P. Zalesskii, Profinite Groups , 2nd Edition, Springer Verlag, Berlin – NewYork, 2010.[21] D.J.S. Robinson,
A Course in the Theory of Groups , 2nd Edition., Springer-Verlag,1995.[22] J.R. Rog´erio, P. Shumyatsky,
A finiteness condition for verbal subgroups , J. GroupTheory, N. , 2007, pp.811–815.[23] D. Segal, Words: notes on verbal width in groups , LMS Lecture Notes , CambridgeUniversity Press, Cambridge, 2009.[24] P. Shumyatsky,
Verbal subgroups in residually finite groups , Quart. J. Math., N. ,2000, pp.523–528.[25] P. Shumyatsky, On the exponent of a verbal subgroup in a finite group , J. Aust. Math.Soc., to appear.[26] R.F. Turner-Smith,
Finiteness conditions for verbal subgroups , J. London Math. Soc.,N. , 1966, pp.166–176.[27] J. Wilson, On outer-commutator words , Can. J. Math., N. , 1974, pp.608–620. N VERBAL SUBGROUPS IN FINITE AND PROFINITE GROUPS 11 [28] J.S. Wilson,
On the structure of compact torsion groups , Monatsh. Math., N. , 1983,pp.57–66.[29] J.S. Wilson, Profinite Groups , Clarendon Press, Oxford, 1998.[30] E. Zelmanov,
On periodic compact groups , Israel J. Math., N. , 1992, pp.83–95.[31] E. Zelmanov, Lie methods in the theory of nilpotent groups , in Groups ’93 Galaway/St Andrews, Cambridge University Press, Cambridge, 1995, pp.567–585.[32] E. Zelmanov,
Nil rings and periodic groups , The Korean Mathematical Society Lec-ture Notes in Mathematics, Seoul, 1992.(C. Acciarri)
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil
E-mail address : [email protected] (P. Shumyatsky) Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900 Brazil
E-mail address ::