Pavel Shumyatsky_Applications of Lie Ring Methods to Group Theory

8/11/2019 Pavel Shumyatsky_Applications of Lie Ring Methods to Group Theory

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Applications of Lie ring methods to group theory

Pavel Shumyatsky1

Department of MathematicsUniversity of Brasilia

70910-900 Brasilia - DF, Brazile-mail: [email protected]

Abstract. The aim of the paper is to illustrate how a Lie-theoretic result ofZelmanov enables one to treat various problems of group theory.

1. Introduction

Lie rings were associated with p-groups in the 30s in the context of the RestrictedBurnside Problem and since then Lie ring methods proved to be an important and veryeective tool of group theory. In the last 10 years the sphere of use of Lie rings wasamplied considerably, mainly due to Zelmanov's outstanding contribution.

It has been known for some time that the following assertions are equivalent.

1.1. Letm and n be positive integers. Then there exists a numberB(m; n) dependingonly onm andnsuch that the order of anym-generator nite group of exponentn is atmost B(m; n).

1.2. The class of locally nilpotent groups of exponentn is a variety.

1.3. The class of locally nite groups of exponentn is a variety.

1.4. Any residually nite group of exponentn is locally nite.

The Restricted Burnside Problem is exactly the question whether the rst of the aboveassertions is true. In 1956 P. Hall and G. Higman reduced the problem to the case ofprime-power exponent [11]. This shows that 1.1 is equivalent to 1.2. In [22] A. I. Kostrikinsolved the Restricted Burnside Problem armatively forn = p, a prime. His proof relieson a profound study of Engel Lie algebras of positive characteristic. The reduction toLie algebras was made earlier by W. Magnus [27] and independently by I. N. Sanov[31]. Finally, in 1989, E. Zelmanov gave a complete solution of the Restricted BurnsideProblem [52], [53]. In [54] Zelmanov deduces the positive solution of the RestrictedBurnside Problem from the following theorem.

1Research supported by FAPDF and CNPq

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Theorem 1.5. LetL be a Lie algebra generated bya1; : : : ; am. Suppose thatL satises

a polynomial identity and each commutator in a1; : : : ; am is ad-nilpotent. Then L isnilpotent.

The above theorem proved to be a clue for some other deep results on pronite andresidually nite groups (see [54], [55]). In the present paper we will discuss some recentlydiscovered group-theoretic corollaries of this theorem. Each of the principal results de-scribed in Sections 3-5 can be viewed as a development around the Restricted BurnsideProblem in the sense of at least one of the assertions 1.1-1.4. Thus, Theorem 1.5 is notonly one of the main tools in the solution of the Restricted Burnside Problem but also animportant factor extending our understanding of the problem itself. The present paper isan attempt to make this subject accessible to a broader audience.

In the next section we describe the construction that associates a Lie algebraL(G) toany groupG and establish basic facts reecting relationship betweenGandL(G). Thiswill be a bridge linking the above theorem of Zelmanov with group theory.

In Section 3 we consider residually nite groups in which all commutators[x1; x2; : : : ; xk] satisfy some restrictive condition. In particular we describe a proof ofthe following theorem.

Theorem 3.1. Letk be an integer,q = ps a prime-power, andGa residually nite groupsuch that [x1; x2; : : : ; xk]

q = 1 for all x1; x2; : : : ; xk 2 G. Thenk(G) is locally nite.

This extends the positive solution of the Restricted Burnside Problem for groups of

prime-power exponent (takek = 1). Another result considered in Section 3 deals withresidually nite groups in which the commutators [x1; x2; : : : ; xk] are Engel.

Theorem 3.2. Let k; n be positive integers, andG a residually nite group such that[x1; x2; : : : ; xk] isn-Engel for anyx1; x2; : : : ; xk 2G. Thenk(G) is locally nilpotent.

Fork = 1 this is a well-known result of J. Wilson [49].Section 4 is devoted to some questions on exponent of a nite group with automor-

phisms. The main result of the section is the following theorem.

Theorem 4.3 (Khukhro, Shumyatsky [21]). Letq be a prime, n an integer. Supposethat a non-cyclic groupA of orderq2 acts on a nite groupG of coprime order in such a

manner that the exponents of the centralizersCG(a) of non-trivial elementsa 2 A dividen. Then the exponent ofG is bounded in terms ofnand q .

Note that the exponent of the centralizer of a single automorphisma of a nite groupGhas no impact over the exponent ofG. Indeed, any abelian group of odd order admitsa xed-point-free automorphism of order two. Hence, we cannot bound the exponent ofGsolely in terms of the exponent ofCG(a). In view of this the following theorem seemsto be interesting.

Theorem 4.4. Letn be a positive integer, G a nite group of odd order admitting aninvolutory automorphismasuch thatCG(a) is of exponent dividingn. Suppose that for

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anyx2 G the element [x; a] = x1xa has order dividingn. Then the exponent ofG is

bounded in terms ofn.

In Section 5 we obtain some sucient conditions for a periodic residually nite groupto be locally nite.

Theorem 5.3 (Shalev [36]). LetG be a periodic residually nite group having a nite2-subgroupA such that the centralizerCG(A) is nite. ThenG is locally nite.

Theorem 5.4. Letq be a prime, G a residually nite group in which each 2-generatorsubgroup is nite. Suppose that G has a niteq-subgroupA such that the centralizerCG(A) is nite. ThenG is locally nite.

This paper certainly is not a comprehensive survey on Lie methods in group theory.Some important areas are not even mentioned here. For the reader willing to learnmore on this subject we recommend the survey [35] and the textbooks [18, ChapterVIII], [20], [47]. Throughout the paper p stands for a xed prime. We use the term\fa ;b ;c ; : : :g-bounded" to mean \bounded from above by some function ofa;b ;c; : : :".

I am grateful to A. Mann for a number of suggestions simplifying proofs in Section 3.

2. Associating a Lie algebra to a group

LetLbe a Lie algebra over a eldk. Letk; nbe positive integers and letx1; x2; : : : ; xk;x; y be elements ofL. We dene inductively

[x1] =x1; [x1; x2; : : : ; xk] = [[x1; x2; : : : ; xk1]; xk]

and[x; 0y] =x; [x; ny] = [[x; n1y]; y]:

An element a 2 L is called ad-nilpotent if there exists a positive integer n such that[x; na] = 0 for all x 2 L. Ifn is the least integer with the above property then we say

that a is ad-nilpotent of index n. Let X L be any subset of L. By a commutatorin elements of X we mean any element of L that can be obtained as a Lie product ofelements ofXwith some system of brackets. Denote byFthe free Lie algebra over koncountably many free generatorsx1; x2; : : :. Letf=f(x1; x2; : : : ; xn) be a non-zero elementofF. The algebraL is said to satisfy the identityf 0 iff(a1; a2; : : : ; an) = 0 for anya1; a2; : : : ; an 2L. In this case we say that L is PI. The next theorem is straightforwardfrom Theorem 1.5 quoted in the introduction.

Theorem 2.1. Let L be a Lie algebra over a eld k generated by a1; a2; : : : ; am. As-sume that L satises an identity f 0 and that each commutator in the generators

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a1; a2; : : : ; am is ad-nilpotent of index at most n. Then L is nilpotent of ff;n;m; kg-

bounded class.Proof. Consider the freem-generated Lie k-algebraFmon free generatorsf1; : : : ; f m, andlet T be the ideal ofFm generated by all values off on the elements ofFm and by allelements of the form [g; nc], where g 2 Fm and c is an arbitrary commutator in the fi.Then F=T satises the hypothesis of Theorem 1.5 and hence is nilpotent of some classu = u(m;n;f; k). Clearly, L is the image ofF=Tunder the homomorphism induced bythe mappingfi! ai. HenceL is nilpotent of class at mostu. 2

An important criterion for a Lie algebra to be PI is the following

Theorem 2.2(Bahturin-Linchenko-Zaicev). LetLbe a Lie algebra over a eldk. Assume

that a nite group A acts on L by automorphisms in such a manner that CL(A), thesubalgebra formed by xed elements, is PI. Assume further that the characteristic of k iseither 0 or prime to the order ofA. ThenL is PI.

This theorem was rst proved by Yu.A. Bahturin and M. V. Zaicev for solvable groupsA[2] and later extended by V. Linchenko to the general case [25].

Corollary 2.3. Let Fbe the free Lie algebra of countable rank over k. Denote by F

the set of non-zero elements ofF. For any nite groupAthere exists a mapping

: F !F

such that ifL andA are as in Theorem 2.2, and ifCL(A) satises an identityf0, thenLsatises the identity(f) 0.

Proof. Assume that the assertion is false. Then there existsf 2 F such that for anyg 2 F we can choose a Lie algebra Lg which does not satisfy the identity g 0 andadmits an action byA withCLg(A) satisfyingf0. Consider the direct sumM=Lg,where the summation is taken over allg 2 F. It is easy to see that Malso admits anaction by A with CM(A) satisfying f 0. ObviouslyMcannot be PI, a contradictionagainst 2.2. 2

We now turn to groups. LetG be any group. Forx; y 2 G we use [x; y] to denotethe group-commutatorx1y1xy. The long commutators [x1; x2; : : : ; xk] and [x; ny] aredened as in Lie algebras:

[x1] =x1; [x1; x2; : : : ; xk] = [[x1; x2; : : : ; xk1]; xk]

and[x; 0y] =x; [x; ny] = [[x; n1y]; y]:

Ifx1; x2; : : : ; xk belong to a setA then we say that [x1; x2; : : : ; xk] is a simple commutatorof weightk in elements ofA.

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The following commutator identities hold in any group and can be checked manually.

Lemma 2.4.

[x; y]1 = [y; x]

[xy;z] = [x; z][x;z;y][y; z]

[x;yz] = [x; z][x; y][x;y;z]

[x; y1; z]y[y; z1; x]z[z; x1; y]x = 1:

Let us also record the Collection Formula (see [18, p. 240]). For any integern and

any subgroupHof a groupG we denote byHn the subgroup ofG generated by then-thpowers of elements fromHand usen(G) for the n-th term of the lower central series ofG.

Lemma 2.5. Letx; y be elements of a group G. Then

(xy)pn

xpn

ypn

mod 2(G)pn

Y1rn

pr(G)pnr :

A series of subgroups

G= G1 G2 : : : ()is called an N-series if it satises [Gi; Gj] Gi+j for all i; j. Obviously anyN-series iscentral, i.e. Gi=Gi+1Z(G=Gi+1) for any i. An N-series is called Np-series ifG

pi Gpi

for alli.Generalizing constructions discovered by Magnus [27] and Zassenhaus [51], Lazard

noticed in [23] that a Lie ringL(G) can be associated to anyN-series () of a group G.He also discovered some very useful special properties ofL(G) in case () is anNp-series.Let us briey describe the construction.

Given anN-series (), letL(G) be the direct sum of the abelian groupsLi =Gi=Gi+1,written additively. Commutation inG induces a binary operation [; ] in L. For homoge-

neous elementsxGi+12L

i ; yGj+12 L

j the operation is dened by

[xGi+1; yGj+1] = [x; y]Gi+j+12 Li+j

and extended to arbitrary elements ofL(G) by linearity. It is easy to check that theoperation is well-dened and thatL(G) with the operations + and [; ] is a Lie ring.

The above procedure can be performed for eachN-series ofG. If all quotientsGi=Gi+1of anN-series () have exponentp thenL(G) can be viewed as a Lie algebra over Fp, theeld withp elements. This is always the case if () is anNp-series. We are now concernedwith the relationship betweenG and L(G). For any x 2 Gin Gi+1 let x denote theelementxGi+1ofL

(G).

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Lemma 2.6(Lazard, [23]). If () is anNp-series then (ad x)p =ad (xp) for anyx 2 G.

Consequently, ifx is of nite ordert thenx

is ad-nilpotent of index at mostt.

Let F r denote the free group on free generatorsx1; x2; : : :, and choose a non-trivialelementw =w(x1; x2; : : : ; xs)2 F r. We say that a groupG satises the identityw 1ifw(g1; g2; : : : ; gs) = 1 for any g1; g2; : : : ; gs2 G.

The following proposition can be deduced from the proof of Theorem 1 in the paperof J. Wilson and E. Zelmanov [50]

Proposition 2.7. Let G be a group satisfying an identityw 1. Then there exists anon-zero Lie polynomialfover Fpdepending only onp andw such that for anyNp-series() ofG the algebraL(G) satises the identityf0.

In fact J. Wilson and E. Zelmanov describe in [50] an eective algorithm allowing towrite fexplicitely for any p and w but we do not require this. Let us just record animportant special case of the above proposition.

Proposition 2.8 (Higman, [16]). Letn be a p-power, G a group such thatxn = 1 foranyx 2 G. Then for anyNp-series () the algebraL(G) satises the identity

X2Sn1

[x0; x(1); x(2); : : : ; x(n1)] = 0: (2:9)

In general a group G has many Np-series so that there are many ways to associateto G a Lie algebra as described above. We will introduce now anNp-series which is ofparticular importance for applications of Lie-theoretic results to group theory.

To simplify the notation we write i for i(G). Set Di = Di(G) =Q

jpki

pk

j . The

subgroupsDi form a series G= D1 D2 : : : in the groupG.

Proposition 2.10([18, p. 250]). The series fDig is anNp-series.

We will call fDig the p-dimension central series ofG. It is also known as the Lazard

series or the Jennings-Lazard-Zassenhaus series.In conformity with the earlier described procedure we can associate to G a Lie al-gebra DL(G) = Li over Fp corresponding to the p-dimension central series. HereLi= Di=Di+1. This algebra plays a crucial role in all results considered in the paper.

Let Lp(G) = hL1i be the subalgebra of DL(G) generated by L1. If G is nitelygenerated then nilpotency of Lp(G) has strong impact over the structure of G. Thefollowing proposition is implicit in [55].

Proposition 2.11. LetG be generated by elementsa1; a2; : : : ; am, and assume that Lp(G)is nilpotent of class at most c. Let 1; 2; : : : ; s be the list of all simple commutators in

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a1; a2; : : : ; am of weight c. Then for any non-negative integer i the group G can be

written as a product G= h1ih2i : : : hsiDi+1

of the cyclic subgroups generated by1; 2; : : : ; s andDi+1.

Proof. We start with the following remark. For any positive integeri the subgroupDiisgenerated byDi+1 and elements of the form [b1; : : : ; bj]

pk , wherejpk i and b1; : : : ; bj 2fa1; : : : ; amg. This can be shown using for example formulae 2.4 and 2.5.

The proposition will be proved by induction oni, the casei = 0 being trivial. Assumethati 1 and

G=h1ih2i : : : hsiDi:

Then any elementx 2 G can be written in the form

x=11 22 : : :

ss y; (2:12)

wherey 2 Di. Without any loss of generality we can assume thatDi+1= 1.By the remark made in the beginning of the proof we can write

y= (pk1

1 )1

(pk2

2 )2

: : : (pkt

t )t

; (2:13)

where eachn is of the form [b1; : : : ; bj], with j pkn i andb1; : : : ; bj 2 fa1; : : : ; amg.

Let ~al denotealD2 2 Lp(G); l = 1; : : : ; m. By the hypothesisLp(G) is nilpotent ofclass c, that is [~b1; : : : ; ~bc+1] = 0 for any b1; : : : ; bc+1 2 fa1; : : : ; amg. This implies that

[b1; : : : ; bc+1] 2 Dc+2 for any b1; : : : ; bc+1 2 fa1; : : : ; amg and c+1 Dc+2. Then, byProposition 2.10, for anyd c+ 1 we haved Dd+1.

Now, ifn is of the form [b1; : : : ; bj] with j c + 1 then

pkn

n 2pkn

j Dpkn

j+1 D(j+1)pkn Di+1= 1:

Hence we can assume that eachn is of the form [b1; : : : ; bj] withj c, in which casenbelongs to the list1; 2; : : : ; s.

It remains to remark that by 2.10pkn

n 2Z(G). Comparing now (2.12) and (2.13) weobtain that

x2 h1ih2i : : : hsi;

as required. 2

The following corollary is now immediate.

Corollary 2.14. Assume the hypotheses of 2.6 and suppose that eachj is of order atmost K. Then Di is of index at mostK

s for anyi. In particular, the seriesDi becomesstationary after nitely many steps.

A groupG is said to be residually-p if for any non-trivial elementx2 G there exists anormal subgroupNG such thatx 62NandG=Nis a nitep-group. M. Lazard proved

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in [24] that ifG is a nitely generated pro-p group such thatLp(G) is nilpotent thenG

is p-adic analytic. Since any nitely generated residually-p group can be embedded in anitely generated pro-p group (see for example [4]), we obtain the following

Proposition 2.15. If G is a nitely generated residually-p group such that Lp(G) isnilpotent thenG has a faithful linear representation over the eld ofp-adic numbers.

3. Groups with commutators of bounded order

This section is devoted to groups in which all commutators [x1; x2; : : : ; xk] satisfy somerestrictive condition. In particular we describe a proof of the following theorem.

Theorem 3.1. Letk be an integer, q = ps a prime-power, G a residually nite groupsuch that [x1; x2; : : : ; xk]

q = 1 for all x1; x2; : : : ; xk 2 G. Thenk(G) is locally nite.

This generalizes the positive solution of the Restricted Burnside Problem for groupsof prime-power exponent (takek = 1). The theorem is no longer true if the assumptionthat G is residually nite is dropped. Using the technique developed by A. Ol'shanskii[29] it is possible (for suciently big values ofq) to construct a groupG that satises theidentity [x; y]q = 1 and has the derived groupG0 non-periodic.

Another result considered in this section deals with residually nite groups in whichthe commutators [x1; x2; : : : ; xk] are Engel. An element x of a group G is called (left)

n-Engel if [g; nx] = 1 for any g 2 G. A groupG is called n-Engel if all elements ofG aren-Engel. It is a long-standing problem whether anyn-Engel group is locally nilpotent. In[49] J. Wilson proved that this is true ifG is residually nite. His proof relies on a resultof A. Shalev [32] which uses the positive solution of the Restricted Burnside Problem[52],[53]. We will prove

Theorem 3.2. Let k; n be positive integers, G a residually nite group such that[x1; x2; : : : ; xk] is n-Engel for any x1; x2; : : : ; xk 2 G. Then k(G) is locally nilpotent.

We will consider rst nite groups satisfying the hypotheses of 3.1 and 3.2. To prepare

the use of Theorem 1.5 we will show that ifG is a nite group satisfying the hypothesisof Theorem 3.1 (respectively, 3.2), thenk(G) is ap-group (respectively, is nilpotent).

In the proof of the next lemma we follow advice of A. Mann and R. Solomon.

Lemma 3.3. Let G be a nite group in which all commutators [x1; x2; : : : ; xk] are p-elements. Thenk(G) is a p-group.

Proof. Assume that the result is false and letGbe a counterexample of minimal possibleorder. ObviouslyG has no non-trivial normalp-subgroups. Letr be a prime divisor ofjGjdistinct fromp and suppose thatG contains anr-subgroupR such that the quotient

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NG(R)=CG(R) has an element z of order prime to r. Then z can be viewed as a non-

trivial automorphism ofRand therefore [R ; z ; : : : ; z| {z }k1

]6= 1, [7, Theorem 5.3.6]. On the other

hand, by the hypothesis, [R ; z ; : : : ; z| {z }k1

] must containp-elements, a contradiction. Therefore

NG(R)=CG(R) is an r-group for any r-subgroup R of G. Hence G possesses a normalr-complement K [7, 7.4.5]. The induction on jGj shows that k(K) is a p-group. SinceG has no non-trivial normalp-subgroups, it follows thatk(K) = 1 and K is nilpotent.Again becauseGhas no non-trivial normalp-subgroups, we conclude thatKis ap0-group.But then so isG, a contradiction. 2

Lemma 3.4. If for any elementsy; x1; x2; : : : ; xkof a nite groupGthere exists an integer

nsuch that [y; n[x1; x2; : : : ; xk]] = 1 then k(G) is nilpotent.Proof. Engel elements of any nite group generate a nilpotent subgroup [17, III, 6.14].SinceG is nite andk(G) =h[x1; x2; : : : ; xk]; x1; x2; : : : ; xk2Gi, the result follows. 2

The next lemma is a particular case of Lemma 2.1 in the paper of J. Wilson [49].

Lemma 3.5. Let G be a nitely generated residually nite-nilpotent group. For eachprimep let Jp be the intersection of all normal subgroups ofG of nitep-power index. IfG=Jp is nilpotent for eachp thenG is nilpotent.

Proof. Let R be the intersection of all normal subgroups N of G such that G=N istorsion-free and nilpotent. Since a torsion-free nilpotent group is residually-p for any

prime p, it follows that Jp R for anyp. We therefore conclude thatG=R is nilpotent.Then so isG=[R ; G ; : : : ; G| {z }

k

] for any positive integerk . SetR0= R,Rk+1= [R ; G ; : : : ; G| {z }k

].

SinceG is residually nilpotent, the intersection\iRi is trivial. By the choice ofR thequotient R=R1 must be periodic. This is a subgroup of a nitely generated nilpotentgroupG=R1and thereforeR=R1 is nitely generated too. HenceR=R1is nite, and letbe the set of primes dividing the order ofR=R1. ThenR=Rk is a -group for anyk. (Itis a well-known general fact: IfR is any normal subgroup of a nilpotent groupG, and ifR=[R; G] is a -group, thenR is a -group.) It follows that\p2Jp \iRi= 1. Sinceis nite,G embeds into the direct product of nitely many nilpotent groupsG=Jp; p2 .

HenceG is nilpotent. 2

Proof of Theorem 3.1. By Lemma 3.3 k(G) is residually-p. To prove thatk(G) islocally nite let us take arbitrarily a nite subsetSofk(G) and show thatSgenerates anite subgroup. Sincek(G) is generated by elements of the form [x1; x2; : : : ; xk], it followsthatSis contained in some subgroupHgenerated by nitely many elementsa1; a2; : : : ; ameach of which can be written in the form [x1; x2; : : : ; xk]. Now it is important to observethat if is an arbitrary commutator ina1; a2; : : : ; amwith some system of brackets thenalso can be written in the form = [x1; x2; : : : ; xk] for suitably chosenx1; x2; : : : ; xk 2 G.We want to show thatH is nite. LetL = Lp(H) be the Lie algebra associated with the

p-dimension central series

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H=D1 D2 : : :

ofH (see Section 2). ThenL is generated by ~ai = aiD2; i = 1; 2; : : : m. Let b be anyLie-commutator in ~a1; ~a2; : : : ; ~am and c the group-commutator in a1; a2; : : : ; am havingthe same system of brackets as b. We have already observed thatc= [x1; x2; : : : ; xk] forsome x1; x2; : : : ; xk 2 G, so we conclude thatcq = 1. Suppose thatc 2 Djn Dj+1. It isimmediate from the denition ofL that eitherb = 0 orb = cDj+1. This implies (Lemma2.6) thatb is ad-nilpotent of index at mostq . Further, the groupHsatises the identity[y1; y2; : : : ; yk]

q = 1. Therefore, by 2.7, L satises some non-trivial polynomial identity.By Theorem 1.5 we conclude thatLis nilpotent. Corollary 2.14 now shows that the indexofDi in H is bounded from above by some number that depends only onq, m and thenilpotency class ofL. SinceHis residually-p, it follows that the intersection of allDi is

trivial and we derive thatHis nite. 2

The proof of 3.2 is similar to that of 3.1.

Proof of Theorem 3.2. By Lemma 3.4 k(G) is residually nite-nilpotent. To provethatk(G) is locally nilpotent let us take arbitrarily a nite subsetSofk(G) and showthatS generates a nilpotent subgroup. Sincek(G) is generated by elements of the form[x1; x2; : : : ; xk], it follows that S is contained in some subgroupH generated by nitelymany elementsa1; a2; : : : ; am each of which can be written in the form [x1; x2; : : : ; xk]. Asin the proof of 3.1 we observe that if is an arbitrary commutator ina1; a2; : : : ; am withsome system of brackets then also can be written in the form = [x1; x2; : : : ; xk] forsuitably chosen x1; x2; : : : ; xk 2 G. We want to show that H is nilpotent. Lemma 3.5allows us to assume thatHis residually-p for some primep. Let L= Lp(H) be the Liealgebra associated with thep-dimension central series

H=D1 D2 : : :

ofH. ThenL is generated by ~ai= aiD2; i= 1; 2; : : : m. Letb be any Lie-commutator in~a1; ~a2; : : : ; ~am and c be the group-commutator ina1; a2; : : : ; amhaving the same system ofbrackets asb. We have already observed thatc = [x1; x2; : : : ; xk] for somex1; x2; : : : ; xk2G, so we conclude thatc isn-Engel. This implies thatb is ad-nilpotent of index at mostn. Further, the group H satises the identity [y; n[y1; y2; : : : ; yk]] = 1. Therefore, by2.7, L satises some non-trivial polynomial identity. Theorem 1.5 now implies thatL is

nilpotent. Hence, by Proposition 2.15,Hhas a faithful linear representation over the eldofp-adic numbers. ClearlyHcannot have a free subgroup of rank two, and so by Tits'Alternative [46] Hhas a solvable subgroup of nite index. Lemma 3.4 shows that eachnite quotient ofHis solvable, so that we conclude thatHis solvable too. It is now easyto deduce from a result of K. Gruenberg [9] thatHis nilpotent, as required. 2

We conclude this section by listing some open problems related to the results describedhere.

Problem 1. Does there exist a group G satisfying the hypothesis of Theorem 3.1 andhavingk(G) of unbounded exponent?

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The answer to this problem is likely to be \yes" but nding an example does not seem

to be easy. Using results of B. Hartley [12] one can show that such an example cannot benitely generated ( see [42] for detail in the casek = 2).

Problem 2. Let k; n be positive integers, G a residually nite group such that[x1; x2; : : : ; xk]

n = 1 for any x1; x2; : : : ; xk 2G. Is thenk(G) necessarily locally nite?

4. Bounding the exponent of a nite group with automorphisms

Letq be a prime, and letA be a non-cyclic group of orderq2 acting on a nite groupG. It is well-known (see [7, Theorem 6.2.4, Theorem 5.3.16 ]) that ifG is any group oforder prime toq then

G= CG(a); a2 A#

; (4:1)whereA# =A n f1g. If, moreover, G is a p-group then we even have

G=Y

a2A#

CG(a): (4:2)

Since A normalizes some Sylow p-subgroup of G for any p dividing jGj ([7, Theorem6.2.4]), it is immediate that ifjCG(a)j n for anya2 A# thenjGj nq+1 (we use thatAhas exactlyq +1 cyclic subgroups). How profound is the connection between the structureofGand that ofCG(a); a2 A#? It is known that ifCG(a) is nilpotent for eacha2 A#

then Gis metanilpotent [48]. In some situations this result holds even ifG is allowed to

be innite periodic [38]. In this section we will prove the following theorem.

Theorem 4.3 (Khukhro, Shumyatsky [21]). Suppose that A is a non-cyclic group oforder q2 acting on a nite groupG of coprime order, and let n be such an integer thatthe exponents of the centralizersCG(a) of non-trivial elementsa 2 A# dividen. Thenthe exponent ofG is bounded in terms ofn and q.

Note that the exponent of the centralizer of a single automorphisma of a nite groupGhas no impact over the exponent ofG. Indeed, any abelian group of odd order admitsa xed-point-free automorphism of order two. Hence, we cannot bound the exponent ofGsolely in terms of the exponent ofCG(a). In view of this the following theorem seems

to be interesting.

Theorem 4.4. Letn be a positive integer, G a nite group of odd order admitting aninvolutory automorphismasuch thatCG(a) is of exponent dividingn. Suppose that foranyx2 G the element [x; a] = x1xa has order dividingn. Then the exponent ofG isbounded in terms ofn.

Apart from the technique described in Section 2 the proof of Theorems 4.3 and 4.4involves using powerfulp-groups. These were introduced by A. Lubotzky and A. Mannin [26]: a nitep-group G is powerful if and only ifGp [G; G] forp 6= 2 (orG4 [G; G]forp = 2).

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Powerful p-groups have many nice linear properties, of which we need the following:

if a powerful p-group G is generated by elements of exponentpe

, then the exponent ofG is pe too (see [4, Lemma 2.2.5]). Combining this with 4.1, we conclude that if G isa powerful p-group satisfying the hypothesis of Theorem 4.3, then the exponent of Gdividesn. Thus, it is sucient to reduce the proof of Theorem 4.3 to the case of powerful

p-groups. A. Shalev was the rst to discover relevance of powerfulp-groups to problemson automorphisms of nite groups [33], [34]. However our situation is quite dierent fromthat considered in the papers of A. Shalev. The reason powerfulp-groups emerge in thecontext of the Restricted Burnside Problem is the following lemma.

Lemma 4.5. Suppose thatP is a d-generator nitep-group such that the Lie algebraLp(P) is nilpotent of classc. ThenPhas a powerful characteristic subgroup offp; c; dg-

bounded index.Proof. Let1; : : : ; sbe all simple commutators of weight cin the generators ofP; heresis a fd; cg-bounded number. Since Pis a nitep-group, the nilpotency ofLp(P) of classcimplies that every elementg2 Pcan be written in the form g= k11 : : :

kss (see Proposition

2.11). Hence jP=Ppm

j psm for any positive integer m. Let V be the intersectionof the kernels of all homomorphisms of P into GLs(Fp). Put W = V if p 6= 2 (orW =V2 ifp = 2). The exponent of the Sylowp-subgroup ofGLs(Fp) is afp; sg-boundednumber. Then Pp

a

W for some fp; sg-bounded number a, which is also fp; c; dg-bounded, sinces is fc; dg-bounded. There is afp; c; dg-bounded numberu a such thatjPp

u

=Ppu+1

j ps, for otherwise the inequalityjP=Ppm

j psm would be violated for some

m. ThenPpu

Ppa

W, andPpu

iss-generator since jPpu

=(Ppu

)j jPpu

=Ppu+1

j ps

.Now, by [4, Proposition 2.12], Pp

u

is a powerful subgroup. The index ofPpu

is at mostpus and hence isfp; c; dg-bounded. 2

We will now quote a well-known lemma which is of fundamental importance wheneverone uses Lie algebra methods to study nite groups with automorphisms of coprime order.

Lemma 4.6([7, 6.2.2 (iv)]). Let A be a nite group acting on a nite groupG. Assumethat (jAj; jGj) = 1 and thatN is a normalA-invariant subgroup ofG. ThenCG=N(A) =CG(A)N=N.

LetHbe a subgroup of a groupG. Set Hj =Dj\ H, where Dj is thej-th term ofthep-dimension central series ofG. Write

L(G; H) =HjDj+1=Dj+1 andLp(G; H) =Lp(G) \ L(G; H):

Observation 4.7. L(G; H) is a subalgebra ofDL(G). It is isomorphic to the Lie algebraassociated with theNp-seriesfHjgofH.

Let now a nite groupA act on a group G. ObviouslyA induces an automorphismgroup of every quotient Dj=Dj+1. This action extends to the direct sum Dj=Dj+1.

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Thus,A can be viewed as a group acting onLp(G) by Lie algebra automorphisms. The

following remark is immediate from Lemma 4.6.

Observation 4.8. IfG is nite and (jGj; jAj) = 1 then

Lp(G; CG(A)) =CLp(G)(A):

The next lemma will be helpful in the proof of Theorem 4.3.

Lemma 4.9. Suppose that Lis a Lie algebra,Ha subalgebra ofL generated by relementsh1; : : : ; hr such that all commutators in thehi are ad-nilpotent inL of indext. IfH is

nilpotent of classu, then for somefr;t;ug-bounded numberv we have [L ; H ; : : : ; H | {z }v

] = 0.

Proof. We apply to a suciently long (but offr;t;ug-bounded length) commutator

[l; hi1; hi2 ; : : :]

a collecting process whose aim is to rearrange thehi(and emerging by the Jacobi identitycommutators in the hi) in an ordered string after l, where all occurrences of a given element(hior a commutator in thehi) would form an unbroken segment. SinceHis nilpotent, thisprocess terminates at a linear combination of commutators with suciently long segments

of equal elements. All these commutators are equal to 0 because all commutators in thehi are ad-nilpotent by the hypothesis. 2

Proof of Theorem 4.3. As noted above, for every prime p dividing jGj there is anA-invariant Sylowp-subgroupP ofG. SinceP =

CP(a); a2 A#

, such a primep must

be a divisor of n. If Theorem 4.3 is valid in the case whereG is a nite p-group, theexponents of all Sylow subgroups ofG are bounded in terms ofq and n, which impliesthat the exponent ofG is bounded. Thus, we may assumeG to be a nitep-group (for aprimep 6=q). We may also assumeG to be generated byq2 elements, since every elementg2 G is contained in theA-invariant subgrouphga; a2 Ai.

From now on, in addition to the hypothesis of Theorem 4.3,G is a niteq2-generator

p-group andn is a power ofp. LetA1; A2; : : : ; Aq+1be the distinct cyclic subgroups ofA.SetDj = Dj(G), L = Lp(G), Lj =L \ Dj=Dj+1, so thatL =Lj. Let Lij =CLj(Ai).Then, by 4.2, for anyj we have

Lj =X

1iq+1

Lij:

By Lemma 4.6 for any l 2 Lij there exists x 2 Dj\ CG(Ai) such thatl = xDj+1. By thehypothesisx is of order at most n, whencel is ad-nilpotent of index at mostn (Lemma2.6). Thus,

any element inLij is ad-nilpotent of index at mostn: (4:10)

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Since G is generated by q2 elements, the Fp-space L1 is spanned by q2 elements. In

particular,L is generated by at mostq2

ad-nilpotent elements, each fromLi1 for somei.But we cannot claim that every Lie commutator in these generators is again in someLijand hence is ad-nilpotent too.

To overcome this diculty, we extend the ground eld ofL by a primitiveqth root ofunity !, forming L = L Fp[!]. The idea is to replaceL byL and to prove that L isnilpotent offq; ng-bounded class, which will, of course, imply the same nilpotency resultforL. Before that we translate the properties ofL into the language ofL.

It is natural to identifyLwith the Fp-subalgebraL1 ofL. We note that if an elementx2 L is ad-nilpotent of indexm, say, then the \same" elementx 1 is ad-nilpotent inLof the same indexm.

PutLj =Lj Fp[!]; thenL=

L1

, since L= hL1i, and L is the direct sum of the

homogeneous componentsLj. Since the Fp-spaceL1 is spanned byq2 elements, so is theFp[!]-spaceL1.

The groupAacts naturally onL, and we haveLij =CLj(Ai), whereLij=Lij Fp[!].Let us show that

any elementy 2 Lij is ad-nilpotent offq; ng-bounded index. (4:11)

SinceLij =Lij Fp[!], we can write

y= x0+ !x1+ !2x2+ : : :+ !

q2xq2

for somexs2 Lij, so that each of the summands!sxs is ad-nilpotent of indexn by 4.10.

SetH=hx0; !x1; : : : ; !q2xq2i. Note thatHCL(Ai), since!sxs2CL(Ai) for all s.A commutator of weightk in the!sxs has the form!

txfor somex 2 Lim, wherem =kj.By 4.10 such anx is ad-nilpotent of indexn and hence so is! tx.

Further, combining Observations 4.7 and 4.8 with Proposition 2.8, we conclude thatCL(Ai) satises the polynomial identity 2.9. This identity is polylinear and so it is alsosatised byCL(Ai)Fp[!] =CL(Ai). We have already observed thatHCL(Ai), whencethe identity 2.9 is satised inH. Hence by Theorem 2.1His nilpotent offq; ng-boundedclass. Lemma 4.9 now says that [L ; H ; : : : ; H | {z }

v

] = 0 for some fq; ng-bounded numberv.

This establishes 4.11.SinceA is abelian, and the ground eld is now a splitting eld forA, every Lj de-

composes in the direct sum of common eigenspaces forA. In particular,L1 is spannedby common eigenvectors for A, and it requires at most q2 of them to span L1. HenceL is generated by q2 common eigenvectors forA from L1. Every common eigenspace iscontained in the centralizerCL(Ai) for some 1 i q+ 1, since A is non-cyclic. Notethat any commutator in common eigenvectors is again a common eigenvector. The mainadvantage of extending the ground eld now becomes clear: ifl1; : : : ; lq2 2L1are commoneigenvectors forA generatingL then any commutator in these generators belongs to someLij and therefore, by 4.11, is ad-nilpotent offq; ng-bounded index.

We already know that the identity 2.9 is satised inCL(Ai) Fp[!] = CL(Ai). So,iff denotes the Lie polynomial in 2.9 then, by 2.3, L satises some identity (f) 0

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which depends only onn and q . Theorem 2.1 now shows thatL (henceL) is nilpotent of

fq; ng-bounded class.By Lemma 4.5 G contains a characteristic powerful subgroupG1 offq; ng-boundedindex. Combining the hypothesis with 4.1 we see thatG1 is generated by elements oforder dividingn. It follows that the exponent ofG1 dividesn [4, 2.2.5]. Therefore theexponent ofG is fq; ng-bounded, as required. 2

In the proof of Theorem 4.4 we use the following well-known fact.

Lemma 4.12. LetG be a nite group of odd order with an involutory automorphisma.Then any elementx 2 G can be written uniquely in the formx = gh, where ga = g1

andh 2 CG(a). Moreoverxa =x1 if and only ifx = [y; a] =y1ya for somey 2 G.

Proof of Theorem 4.4. As in Theorem 4.3 we remark thatGpossesses ana-invariantSylow p-subgroup for any prime p dividing jGj. It is therefore sucient to bound theexponent of a-invariant p-subgroups of G. So, without any loss of generality we mayassume that G is a p-group and n is a p-power. Take an arbitrary element x 2 G.According to 4.12 we can writex = gh, wherega =g1 andh 2 CG(a). So to prove thatthe order ofx is n-bounded it is sucient to prove that the order ofhg; hi is n-bounded.Thus, we can assume thatG =hg; hi.

LetfDjg be thep-dimension central series ofG, Lj =Dj=Dj+1, L= DL(G) the Liealgebra corresponding to the seriesfDjg. We can naturally viewa as an automorphismofL. Set

L+ =CL(a) andL =fl2 L; la =lg:

Then one has:[L+; L+] L+; [L+; L] L; [L; L] L+: (4:13)

For a xedj let us denote for a momentLj byM. SinceM isa-invariant, 4.12 shows thatM=M+ M, whereM+ =L+ \ M andM =L \ M. It is easy to check (using 4.6) that for anym 2 M+ there existsd 2 CDj(a) such thatdDj+1= m. By the hypothesisdis of order dividingn. Therefore, by Lemma 2.6, m is ad-nilpotent of index at mostn.Similarly one concludes that any element inM is ad-nilpotent of index at mostn.

SinceG= hg; hi, it follows that the subalgebraLp(G) ofL is generated by ~g =gD2and~h = hD2. The assumption thatg

a =g1 andha =h implies that ~g 2 L, ~h2 L+.By 4.13 we conclude that any commutator in ~g; ~h lies either in Lj\L+ or in Lj \L

for some j. Combining this with the observations made in the preceding paragraph, wearrive at the conclusion that any commutator in ~g; ~h is ad-nilpotent of index at mostn.

Arguing like in the proof of Theorem 4.3, remark that the identity 2.9 is satised inCL(a). Therefore, by Theorem 2.3, L satises a certain polynomial identity dependingonly onn. Theorem 2.1 now shows thatLp(G) is ofn-bounded nilpotency class.

By Lemma 4.5 we derive that G contains a characteristic powerful subgroup G1 ofn-bounded index. Lemma 4.12 implies thatG1is generated by elements of order dividingn. Therefore the exponent ofG1 dividesn. The theorem follows. 2

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It is not clear whether the above theorem can be extended to the case of automorphism

of any order prime to that ofG.Problem 3. Let n be a positive integer, G a nite group admitting an automorphisma, of order prime to jGj, such that CG(a) is of exponent dividingn. Suppose that foranyx 2 G the element [x; a] = x1xa has order dividingn. Is then the exponent ofGbounded in terms ofn and jaj?

5. On centralizers in periodic residually nite groups

In this section we nd some sucient conditions for a periodic residually nite groupto be locally nite. As is attested by the groups constructed in [1], [6], [8], [10], [44], ingeneral a periodic residually nite group need not be locally nite. Our theme will be thefollowing.

Given a periodic residually nite groupG acted on by a nite groupA, under whatconditions onA and CG(A) does it follow thatG is locally nite? Since any subgroup ofGacts on G by inner automorphisms, this problem includes problems on centralizers ofnite subgroups ofG.

In 1972 V. Shunkov proved that if a periodic groupG admits an involutory automor-phisma with nite centralizerCG(a) thenG contains a solvable subgroup of nite index[43]. This was strengthened later by B. Hartley and Th. Meixner who showed thatGhas a nilpotent subgroup of index depending only on jCG(a)j and of nilpotency class at

most two [15]. Locally nite groupsGhaving an automorphismaof arbitrary prime orderp with nite centralizerCG(a) have been studied intensively in seventies and eities (seefor example [13]). Khukhro showed that these groups have a nilpotent subgroup of niteindex depending only onjCG(a)jand onp, and nilpotency class depending only onp [19].In general, a very interesting direction in locally nite group theory is to classify in somesense locally nite groupsG having a nite subgroupA such thatCG(A) possesses someprescribed property, as for example the property to be a linear group [14].

An immediate corollary of the theorem of Shunkov is that G is locally nite. Thispart of the theorem, and in fact most dicult part, has no analogue for periodic groupsadmitting an automorphism of odd order. According to G. Deryabina and A. Ol'shanskii,for any positive integern which has at least one odd divisor there exists an innite group

Ghaving a non-central element of ordern such that all proper subgroups ofG are nite[3]. Therefore the result of Shunkov cannot be extended to periodic groups having anautomorphism whose order is not a 2-power. Moreover, Obraztsov and Miller constructedfor any (not necessarily distinct) odd primespandq a nitely generated innite residuallynite periodicp-group admitting a xed-point-free automorphism of orderq [28].

On the other hand, in the recent years new means to treat the problem in the residuallynite case have been found. N. R. Rocco and the author proved in [30] that if a periodicresidually nite groupG admits an automorphisma of order 2s withCG(a) nite thenGis locally nite. Somewhat later ([39], [40]) the author proved local niteness of a periodicresidually nite groupGin the following cases:

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1)G admits a 4-groupA of automorphisms withCG(A) nite; or

2) G has no elements of order two and admits an involutory automorphism a suchthatCG(a) is abelian.Then A. Shalev obtained in [36] some very general results on local niteness of periodic

residually nite groups acted on by a nite 2-group (see 5.2 and 5.3 bellow). These resultswere extended in [41] to the case when the acting group is not necessarily of 2-power order.

IfA is any nite group, letq(A) denote the maximal prime divisor ofjAj. One of theresults described in this section is the following theorem.

Theorem 5.1. Let Gbe a residually nite group acted on by a nite solvable groupAwithq = q(A). Assume thatG has nojAj-torsion andCG(A) is either solvable or of niteexponent. If anyq 1 elements ofG generate a nite solvable subgroup thenG is locally

nite.

We should mention that for any integerd 2 there exist innited-generator residuallynite groups in which all (d 1)-generator subgroups are nite. The correspondingexamples are provided by Golod's groups [6].

If under the hypothesis of 5.1A is a 2-group then the condition imposed onG is thatGis merely periodic. This important special case is due to A. Shalev [36].

Theorem 5.2 (Shalev). Let G be a periodic residually nite group with no 2-torsionacted on by a nite 2-group A. Suppose the centralizer CG(A) is solvable, or of niteexponent. ThenG is locally nite.

Other results to be described in this section are as follows.

Theorem 5.3 (Shalev). Let G be a periodic residually nite group having a nite 2-subgroupAsuch that the centralizerCG(A) is nite. ThenG is locally nite.

Theorem 5.4. Letq be a prime, G a residually nite group in which each 2-generatorsubgroup is nite. Suppose that G has a niteq-subgroupA such that the centralizerCG(A) is nite. ThenG is locally nite.

Recall that the Fitting height of a nite solvable groupGis dened as the least number

h=h(G) such thatG possesses a normal series

G= G1 G2: : : Gh+1= 1;

all of whose quotientsGi=Gi+1are nilpotent. Thus,Gis nilpotent if and only ifh(G) = 1.In this section we will use some deep results on Fitting height of nite solvable groups.

Theorem 5.5 (Thompson, [45]). Let G and A be nite solvable groups such that(jGj; jAj) = 1. Assume that A acts on G in such a manner that h(CG(A)) = h. As-sume further that the order ofAis a product ofk not necessarily distinct primes. Thenh(G) isfh; kg-bounded.

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The following lemma is a corollary of the famous theorem of P. Hall and G. Higman

onp-length of ap-solvable nite group of given exponent [11].

Lemma 5.6(Shalev, [36]). The Fitting height of a nite solvable group of exponentn isn-bounded.

Lemma 5.7. Leth be a positive integer, G a residually nite-solvable group such thath(Q) h for any nite quotientQ ofG. ThenG possesses a normal series

G= G1 G2: : : Gh+1= 1;

all of whose quotients are residually nite-nilpotent.

Proof. Let us use the induction on h, the case h = 1 being obvious. Assume thath 2, and let Hbe the intersection of all nite index normal subgroupsN ofG suchthath(G=N) h 1. By the induction hypothesisG possesses a normal series

G=G1 G2 : : : Gh= H;

all of whose quotients are residually nite-nilpotent. Therefore, it suces to show thatHis residually nilpotent. Letx be any non-trivial element ofH. SinceG is residually nite,there exists a normal subgroupNof nite index inG such thatx 62N. By hypothesisGpossesses a normal series

G= N1N2 : : : Nh+1= N;

all of whose quotients are nilpotent. We note thath(G=N) = h for Ndoes not containH. Thereforeh(G=Nh) =h 1 and soHNh. Since the quotientNh=N is nilpotent, soisH=H\ N. Thus for any non-trivial elementx2 Hwe can nd a normal inG subgroupNsuch thatx 62N andH=N\ His nite and nilpotent. This means thatHis residuallynite-nilpotent, as required. 2

LetG be a periodic group acted on by a nite groupA. Suppose thatGhas no non-trivial elements of order dividing that ofA, and letNbe anA-invariant normal subgroupofG. Lemma 4.6 says that ifG is nite then CG=N(A) = CG(A)N=N. We saw in the

previous section that this fact is of fundamental importance for using Lie ring methods inthe study of nite groupsG having automorphisms of coprime order. The applicabilityof Lie ring methods to the study of innite periodic groupsG acted on by a nite groupAdepends ultimately on how successful one is in extending Lemma 4.6 to the case whenG is allowed to be innite periodic. Since in generalCG=N(A)6=CG(A)N=N, one has toimpose additional conditions onG and A.

Given a positive integern, a groupG is said to ben-nite if anyn-generator subgroupofGis nite. Thus, a group is 1-nite if and only if it is periodic. It was proved in [37]that the equalityCG=N(A) = CG(A)N=Nholds wheneverA is a 2-group. More generally,we have.

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Lemma 5.8. LetA be a nite solvable group with q = q(A) acting on a (q 1)-nite

group G with no jAj-torsion. Let N be a normal A-invariant subgroup of G. ThenCG=N(A) =CG(A)N=N.

Proof. It suces to show that anyA-invariant cosetxNcontains an element fromCG(A).Suppose rst thatAis of orderq and leta be a generator ofA. Set

x0= x1xa,x1= x

a0, : : :, xq1= x

aq2.

Then all x0; x1; : : : ; xq1 lie in N and x0x1 : : : xq1 = 1. It follows that the subgroupF =hx0; x1; : : : ; xq1iis generated by at most q 1 elements. HenceFis nite. We notethatF isA-invariant and consider the natural split extensionF A.

Since G has no jAj-torsion, it is easy to see that A and B = hax10 i are Sylow q-

subgroups of F A. Hence there exists an elementy 2 F such that B = Ay. Thereforexy1 2NG(A) = CG(A) andx 2 CG(A)N, as required.

Now letAbe of non-prime order. LetDbe a non-trivial proper normal subgroup ofA.SinceD is of order less than that ofA, we can assume by induction thatxN\ CG(D)6=;.Therefore without any loss of generality we can assume x 2 CG(D). LetH = CG(D),A = A=CA(H), M = N \H. The coset xM is obviously A-invariant. Arguing by

induction onjAjand using that Ahas order less than that ofA we can assume thatxMcontains an elementz 2 CH(A). Now it remains to notice thatzN=xNandz 2 CG(A).2

Lemma 5.9. Letq be a prime,A a niteq-group acting on a residually nite groupG.

Suppose thatG is (q 1)-nite andCG(A) has noq-torsion. ThenG has noq-torsion.

Proof. Suppose that the lemma is false and assume rst thatA is of order q. Let abe a generator of A. Since G is residually nite, we can choose a normal A-invariantsubgroupNsuch thatG=Nis a nite group whose order is divisible byq. It follows thatAcentralizes some elementxNof orderq inG=N. Thenx1xa 2N. Set

x0= x1xa,x1= x

a0, : : :, xq1= x

aq2.

Arguing like in the previous lemma we observe thatF =hx0; x1; : : : ; xq1i is nite. IfqdividesjFj then obviouslyA must centralize some elements of order q in F. This yieldsa contradiction, so assume that jFj is prime to q. Thenhax01i is a Sylow q-subgroupofF A. Therefore it is a conjugate ofA. Hence there exists an elementy 2F such thaty1ay =x1ax. Then z =xy1 2 CG(A). Since the image ofz in G=N is of order q, itfollows that the order ofz is divided byq. ThereforeCG(A) contains an element of orderq, a contradiction.

Suppose now thatA is of order qn and use induction onn. Leta be an element ofprime order inZ(A). SetC= CG(a). IfCis q-torsion free then by the previous paragraphso is G. Assume that C has non-trivial q-elements. SinceA induces an automorphismgroup ofCwhose order is strictly less than that ofA, the induction hypothesis impliesthat some ofq-elements ofCmust lie in CG(A). 2

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Obviously the conclusions of Lemmas 5.8 and 5.9 remain true if we replace the as-

sumption thatG is (q 1)-nite by the assumption that the semidirect productGA is2-nite. Really, keeping notation like in Lemma 5.8 let us note that ifGA is 2-nite thenF is nite because F hx; ai. Leaving other parts of proofs unchanged we reach thefollowing results.

Lemma 5.80. LetAbe a nite solvable group acting on a groupG with nojAj-torsion.Suppose that GA is 2- nite. Let N be a normal A-invariant subgroup of G. ThenCG=N(A) =CG(A)N=N.

Lemma 5.90. Letq be a prime,A a niteq-group acting on a residually nite groupG.Suppose thatGA is 2-nite andCG(A) has noq-torsion. ThenG has noq-torsion.

Proposition 5.10. Letp be a prime andG a periodic residually nitep-group acted onby a nite solvable groupA whose order is prime top. Suppose thatq =q(A) andG is(q 1)-nite. IfCG(A) satises a non-trivial identity thenG is locally nite.

Proof. Since any nite set of elements of G is contained in a nitely generated A-invariant subgroup, we can assume thatG is nitely generated. LetL = Lp(G) be theLie algebra associated with thep-dimension central series ofG. We regardA as a groupacting onL. Just as in Section 4 (but using Lemma 5.8 in place of Lemma 4.6) we remarkthat CL(A) = Lp(G; CG(A)) and so, because CG(A) satises a non-trivial identity, byProposition 2.7CL(A) is PI. SinceG is periodic, it follows from Lemma 2.6 that L hasa nite set of generators in which every commutator is ad-nilpotent. Consequently, by

Theorem 1.5 combined with Theorem 2.2L is nilpotent.Let g1; g2; : : : ; gm be some generating set of G, let c be the nilpotency class of L

and write 1; 2; : : : ; s for the list of all simple commutators of weight at most c ing1; g2; : : : ; gm. ChooseK to be the maximum of orders ofi. Then, by 2.14,Dj(G) hasindex at most Ks for any j. SinceG is residually-p, it follows that the order ofG is atmost Ks. The proof is complete. 2

Proof of Theorems 5.1 and 5.2. Let us note that G is residually solvable. Indeed,ifq = 2 then G has no 2-torsion and soG is residually solvable by the Feit-ThompsonTheorem [5]. Ifq 3 then any two elements ofG generate a nite solvable subgroup.Since any simple nite group can be generated by two elements, it follows that G isresidually solvable.

Assume thatCG(A) is of nite exponent. LetNbe anyA-invariant normal subgroupof nite index inG and Q = G=N. ThenQ is solvable becauseG is residually so. ByLemma 5.8A acts on Q in such a way thatCQ(A) is of exponent at most that ofCG(A).By Lemma 5.6 the Fitting heighth(CQ(A)) ofCQ(A) is bounded in terms of the exponentofCQ(A). Theorem 5.5 now shows that the Fitting heighth(Q) ofQ is bounded in termsof the exponent ofCG(A) andjAj. Thus,h(Q) is bounded by a numberh which does notdepend onQ. Therefore, by Lemma 5.7,Gpossesses a normalA-invariant series of lengthat most h + 1 all of whose quotients are residually nilpotent. Induction onh shows that

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without any loss of generalityG can be assumed residually nilpotent. ThenG is a direct

product ofA-invariantp-subgroups and niteness ofG follows from Proposition 5.10.IfCG(A) is solvable andQ has the same meaning as above then h(CQ(A)) is at mostthe derived length ofCG(A) and we can repeat the argument. This completes the proof.2

Using Lemma 5.9 and 5.90 we can in some situations derive local niteness ofG evenwithout requiring thatG has nojAj-torsion.

Proof of Theorems 5.3 and 5.4. SinceG is residually nite, we can choose a normalsubgroupHof nite index inG such thatH\ CG(A) = 1. It suces to prove thatH islocally nite. Lemmas 5.9 and 5.90 imply thatHhas no jAj-torsion. Hence, by 5.8 (or

5.80

),A acts xed-point-freely on everyA-invariant nite quotientQ ofH. A well-knowncorollary of the classication of simple nite groups says that any nite group admitting axed-point-free automorphism group of coprime order is solvable. We now conclude thatQis necessarily solvable. HenceH is residually solvable. This places us in a position toapply Theorem 5.1 (or Theorem 5.2 in caseq=2) and derive thatG is locally nite. 2

Since there is no restriction on the identity satised byCG(A) in Proposition 5.10, itis not unreasonable to conjecture that the following problem can be answered positively.

Problem 4. Let A be a nite 2-group acting on a periodic residually nite group Gwhich has no 2-torsion. Assume thatCG(A) satises a non-trivial identity. Does it follow

thatG is locally nite?

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Pavel Shumyatsky_Applications of Lie Ring Methods to Group Theory