Profinite groups and the fixed points of coprime automorphisms
aa r X i v : . [ m a t h . G R ] M a r Profinite groups and the fixed pointsof coprime automorphisms
Cristina Acciarri and Pavel Shumyatsky
Abstract.
The main result of the paper is the following theorem.Let q be a prime and A an elementary abelian group of order q .Suppose that A acts coprimely on a profinite group G and assumethat C G ( a ) is locally nilpotent for each a ∈ A . Then the group G is locally nilpotent.
1. Introduction
Let A be a finite group acting on a finite group G . Many well-known results show that the structure of the centralizer C G ( A ) (thefixed-point subgroup) of A has influence over the structure of G . Theinfluence is especially strong if ( | A | , | G | ) = 1, that is, the action of A on G is coprime. Let A denote the set of non-identity elements of A.The following theorem was proved in [ ]. Theorem . Let q be a prime and A an elementary abelian q -group of order at least q . Suppose that A acts coprimely on a finitegroup G and assume that C G ( a ) is nilpotent for each a ∈ A . Then G is nilpotent. There are well-known examples that show that the above theoremfails if the order of A is q . Indeed, let p and r be odd primes and H and K the groups of order p and r respectively. Denote by A = h a , a i the noncyclic group of order four with generators a , a and by Y thesemidirect product of K by A such that a acts on K trivially and a takes every element of K to its inverse. Let B be the base group ofthe wreath product H ≀ Y and note that [ B, a ] is normal in H ≀ Y . Set Mathematics Subject Classification.
Key words and phrases.
Profinite groups, Automorphisms, Centralizers.This work was supported by the Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq), Brazil. G = [ B, a ] K . The group G is naturally acted on by A and C G ( A ) = 1.Therefore C G ( a ) is abelian for each a ∈ A . But, of course, G is notnilpotent.In [ ] the situation of Theorem 1.1 was studied in greater detailand the following result was obtained. Theorem . Let q be a prime and A an elementary abelian q -group of order at least q . Suppose that A acts coprimely on a finitegroup G and assume that C G ( a ) is nilpotent of class at most c for each a ∈ A . Then G is nilpotent and the class of G is bounded by a functiondepending only on q and c . Of course, the above results have a bearing on profinite groups. Byan automorphism of a profinite group we always mean a continuousautomorphism. A group A of automorphisms of a profinite group G is coprime if A has finite order while G is an inverse limit of finitegroups whose orders are relatively prime to the order of A . Using theroutine inverse limit argument it is easy to deduce from Theorem 1.1and Theorem 1.2 that if G is a profinite group admitting a coprimegroup of automorphisms A of order q such that C G ( a ) is pronilpotentfor all a ∈ A , then G is pronilpotent; and if C G ( a ) is nilpotent forall a ∈ A , then G is nilpotent. Yet, certain results on fixed points inprofinite groups cannot be deduced from corresponding results on finitegroups. The purpose of the present paper is to establish the followingtheorem. Theorem . Let q be a prime and A an elementary abelian q -group of order at least q . Suppose that A acts coprimely on a profinitegroup G and assume that C G ( a ) is locally nilpotent for each a ∈ A .Then G is locally nilpotent. Recall that a group is locally nilpotent if every finitely generatedsubgroup is nilpotent. Though Theorem 1.3 looks similar to Theorems1.1 and 1.2, in fact it cannot be deduced directly from those results.Moreover, the proof of Theorem 1.3 is very much different from those ofTheorems 1.1 and 1.2. In particular, unlike the other results, Theorem1.3 relies heavily on the Lie-theoretical techniques created by Zelmanovin his solution of the restricted Burnside problem [
14, 15 ]. The generalscheme of the proof of Theorem 1.3 is similar to that of the result in[ ].
2. Preparatory work
Throughout the paper we use without special references the well-known properties of coprime actions:
ROFINITE GROUPS AND COPRIME AUTOMORPHISMS 3
Lemma . If a group A acts coprimely on a finite group G , then C G/N ( A ) = C G ( A ) N/N for any A -invariant normal subgroup N . Lemma . If A is a noncyclic abelian group acting coprimely ona finite group G , then G is generated by the subgroups C G ( B ) , where A/B is cyclic.
The above results both easily extend to the case of coprime auto-morphisms of profinite groups (see for example [ , Lemma 3.2]). Let x, y be elements of a group, or a Lie algebra. We define inductively[ x, y ] = x and [ x, n y ] = [[ x, n − y ] , y ] for n ≥ . Let L be a Lie algebra. An element a ∈ L is called ad-nilpotentif there exists a positive integer n such that [ x, n a ] = 0 for all x ∈ L .Let X ⊆ L be any subset of L . By a commutator in elements of X we mean any element of L that can be obtained as a Lie product ofelements of X with some system of brackets. The next theorem is dueto Zelmanov (see [ ] or [ ]). Theorem . Let L be a Lie algebra generated by finitely manyelements a , a , . . . , a m such that each commutator in these generatorsis ad-nilpotent. If L satisfies a polynomial identity, then L is nilpotent. An important criterion for a Lie algebra to satisfy a polynomialidentity is the following theorem.
Theorem . Assume that a finitegroup A acts on a Lie algebra L by automorphisms in such a mannerthat C L ( A ) , the subalgebra formed by fixed elements, satisfies a polyno-mial identity. Assume further that the characteristic of the ground fieldis either 0 or prime to the order of A . Then L satisfies a polynomialidentity. The above theorem was first proved by Bahturin and Zaicev in thecase where A is soluble [ ] and later extended by Linchenko to thegeneral case [ ]. In the present paper we only require the case where A is abelian.Let G be a (profinite) group. A series of subgroups G = G ≥ G ≥ . . . ( ∗ )is called an N -series if it satisfies [ G i , G j ] ≤ G i + j for all i, j ≥ N -series is central,i.e. G i /G i +1 ≤ Z ( G/G i +1 ) for any i . Let p be a prime. An N -seriesis called N p -series if G pi ≤ G pi for all i . Given an N -series ( ∗ ), let CRISTINA ACCIARRI AND PAVEL SHUMYATSKY L ∗ ( G ) be the direct sum of the abelian groups L ∗ i = G i /G i +1 , writtenadditively. Commutation in G induces a binary operation [ , ] in L . Forhomogeneous elements xG i +1 ∈ L ∗ i , yG j +1 ∈ L ∗ j the operation is definedby [ xG i +1 , yG j +1 ] = [ x, y ] G i + j +1 ∈ L ∗ i + j and extended to arbitrary elements of L ∗ ( G ) by linearity. It is easyto check that the operation is well-defined and that L ∗ ( G ) with theoperations + and [ , ] is a Lie ring. If all quotients G i /G i +1 of an N -series ( ∗ ) have prime exponent p then L ∗ ( G ) can be viewed as a Liealgebra over F p , the field with p elements. In the important case wherethe series ( ∗ ) is the p -dimension central series (also known under thename of Zassenhaus-Jennings-Lazard series) of G we write L p ( G ) forthe subalgebra generated by the first homogeneous component G /G in the associated Lie algebra over the field with p elements. Observethat the p -dimension central series is an N p -series (see [ , p. 250] fordetails).Any automorphism of G in the natural way induces an automor-phism of L ∗ ( G ). If G is profinite and α is a coprime automorphismof G , then the subring (subalgebra) of fixed points of α in L ∗ ( G ) isisomorphic with the Lie ring associated to the group C G ( α ) via theseries formed by intersections of C G ( α ) with the terms of the series ( ∗ )(see [ ] for more details).Let w = w ( x , x , . . . , x k ) be a group-word. Let H be a subgroupof a group G and g , g , . . . , g k ∈ G . We say that the law w ≡ g H, g H, . . . , g k H if w ( g h , g h , . . . , g k h k ) = 1for all h , h , . . . , h k ∈ H . Wilson and Zelmanov showed in [ ] that ifa profinite group G has an open subgroup H and elements g , g , . . . , g k such that the law w ≡ g H, g H, . . . , g k H ,then L p ( G ) satisfies a polynomial identity for each prime p . Moreprecisely, the proof in [ ] shows that whenever a profinite group G has an open subgroup H and elements g , g , . . . , g k such that the law w ≡ g H, g H, . . . , g k H , the Lie algebra L ∗ ( G ) satisfies a multilinear polynomial identity for any prime p andany N p -series ( ∗ ) in G . Lemma . For any locally nilpotent profinite group G there exista positive integer n , elements g , g ∈ G and an open subgroup H ≤ G such that the law [ x, n y ] ≡ is satisfied on the cosets g H, g H . Proof.
Since any finitely generated subgroup of G is nilpotent, forevery pair of elements g , g there exists a positive number j such that ROFINITE GROUPS AND COPRIME AUTOMORPHISMS 5 [ g , j g ] = 1. For each integer i we set S i = { ( x, y ) ∈ G × G : [ x, i y ] = 1 } . Since the sets S i are closed in G × G and have union G × G , by Bairecategory theorem [ , p. 200] at least one of these sets has a non-emptyinterior. Therefore we can find an open subgroup H in G , elements g , g ∈ G and an integer n with the required property. (cid:3) The following proposition is now straightforward.
Proposition . Assume that a finite group A acts coprimely ona profinite group G in such a manner that C G ( A ) is locally nilpotent.Then for each prime p the Lie algebra L p ( G ) satisfies a multilinearpolynomial identity. Proof.
Let L = L p ( G ). In view of Theorem 2.4 it is sufficientto show that C L ( A ) satisfies a polynomial identity. We know that C L ( A ) is isomorphic with the Lie algebra associated with the centralseries of C G ( A ) obtained by intersecting C G ( A ) with the p -dimensioncentral series of G . Since C G ( A ) is locally nilpotent, Lemma 2.5 applies.Thus, the Wilson-Zelmanov result [ , Theorem 1] tells us that C L ( A )satisfies a polynomial identity. (cid:3) We will also require the following lemma that essentially is due toWilson and Zelmanov (cf [ , Lemma in Section 3]. Lemma . Let G be a profinite group and g ∈ G an elementsuch that for any x ∈ G there exists a positive n with the propertythat [ x, n g ] = 1 . Let L ∗ ( G ) be the Lie algebra associated with G usingan N p -series ( ∗ ) for some prime p . Then the image of g in L ∗ ( G ) isad-nilpotent. Finally, we quote a useful lemma from [ ]. Lemma . Let L be a Lie algebra and H a subalgebra of L gen-erated by m elements h , . . . , h m such that all commutators in the gen-erators h i are ad-nilpotent in L . If H is nilpotent, then we have [ L, H, . . . , H | {z } d ] = 0 for some number d .
3. Proof
As usual, for a profinite group G we denote by π ( G ) the set ofprime divisors of the orders of finite continuous homomorphic imagesof G . We say that G is a π -group if π ( G ) ⊆ π and G is a π ′ -groupif π ( G ) ∩ π = ∅ . If m is an integer, we denote by π ( m ) the set ofprime divisors of m . If π is a set of primes, we denote by O π ( G ) the CRISTINA ACCIARRI AND PAVEL SHUMYATSKY maximal normal π -subgroup of G and by O π ′ ( G ) the maximal normal π ′ -subgroup.We are ready to embark on the proof of Theorem 1.3. Proof of Theorem 1.3.
Recall that q is a prime and A an ele-mentary abelian group of order q acting coprimely on a profinite group G in such a manner that C G ( a ) is locally nilpotent for all a ∈ A . Wewish to show that G is locally nilpotent. In view of Ward’s Theorem 1.1the group G is pronilpotent and therefore G is the Cartesian productof its Sylow subgroups.Choose a ∈ A . By Lemma 2.5 C G ( a ) contains an open subgroup H and elements u, v such that for some n the law [ x, n y ] ≡ uH, vH . Let [ C G ( a ) : H ] = m and let π = π ( m ).Denote O π ′ ( C G ( a )) by T . Since T is isomorphic to the image of H in C G ( a ) /O π ( C G ( a )), it is easy to see that T satisfies the law [ x, n y ] ≡ T is n -Engel. By the result of Burns and Medvedev [ ] thesubgroup T has a nilpotent normal subgroup U such that T /U hasfinite exponent, say e . Set π = π ( e ). Of course, the sets π and π depend on the choice of a ∈ A so strictly speaking they should bedenoted by π ( a ) and π ( a ). For each such choice let π a = π ( a ) ∪ π ( a ).We repeat this argument for every a ∈ A . Set π = ∪ a ∈ A π a and K = O π ′ ( G ). Since all sets π ( a ) and π ( a ) are finite, so is π . The choiceof the set π guarantees that C K ( a ) is nilpotent for every a ∈ A . Thus,by Theorem 1.2, the subgroup K is nilpotent. Let p , p , . . . , p r be thefinitely many primes in π and let P , P , . . . , P r be the correspondingSylow subgroups of G . Then G = P × P × · · · × P r × K and thereforeit is sufficient to show that each subgroup P i is locally nilpotent. Thus,from now on without loss of generality we assume that G is a pro- p group for some prime p . Since every finite subset of G is contained ina finitely generated A -invariant subgroup, we can further assume that G is finitely generated.Let A , A , . . . , A s be the distinct maximal subgroups of A . Wedenote by D j = D j ( G ) the terms of the p -dimension central series of G . Set L = L p ( G ) and L j = L ∩ ( D j /D j +1 ), so that L = ⊕ L j . Thegroup A naturally acts on L . Since each subgroup A i is noncyclic, byLemma 2.2 we have L = P a ∈ A i C L ( a ) for every i ≤ s .Let L ij = C L j ( A i ). Again by Lemma 2.2, for any j we have L j = X ≤ i ≤ s L ij . In view of Lemma 2.1 for any l ∈ L ij there exists x ∈ D j ∩ C G ( A i ) suchthat l = xD j +1 . Therefore, by Lemma 2.7, the element l is ad-nilpotent ROFINITE GROUPS AND COPRIME AUTOMORPHISMS 7 in C L ( a ) for every a ∈ A i . Since L = P a ∈ A i C L ( a ), we conclude thatany element l in L ij is ad-nilpotent in L. ( ∗∗ )Let ω be a primitive q th root of unity and L = L ⊗ F p [ ω ]. We canview L both as a Lie algebra over F p and that over F p [ ω ]. It is naturalto identify L with the F p -subalgebra L ⊗ L . We note that if anelement x ∈ L is ad-nilpotent of index r , say, then the “same” element x ⊗ L of the same index r . Put L j = L j ⊗ F p [ ω ]; then L = (cid:10) L (cid:11) , since L = h L i , and L is the direct sum of the homogeneouscomponents L j .The group A acts naturally on L , and we have L ij = C L j ( A i ), where L ij = L ij ⊗ F p [ ω ]. Let us show thatany element y ∈ L ij is ad-nilpotent in L . ( ∗ ∗ ∗ )Since L ij = L ij ⊗ F p [ ω ], we can write y = x + ωx + ω x + · · · + ω q − x q − for some x , x , x , . . . , x q − ∈ L ij , so that each of the summands ω t x t is ad-nilpotent by (**). Set J = h x , ωx , . . . , ω q − x q − i . This is thesubalgebra generated by x , ωx , . . . , ω q − x q − . Note that J ⊆ C L ( A i ).A commutator of weight k in the elements ω t x t has the form ω s x forsome x that belongs to L im , where m = kj . By (**) the element x isad-nilpotent and so such a commutator must be ad-nilpotent.Proposition 2.6 tells us that the Lie algebra L satisfies a multilinearpolynomial identity. The multilinear identity is also satisfied in L andso it is satisfied in J . Hence by Theorem 2.3 J is nilpotent. Lemma2.8 now says that [ L, J, . . . , J | {z } d ] = 0 for some d . This establishes (***).Since A is abelian and the ground field is now a splitting field for A , every L j decomposes in the direct sum of common eigenspaces for A . In particular, L is spanned by finitely many common eigenvectorsfor A . Hence L is generated by finitely many common eigenvectorsfor A from L . Every common eigenspace is contained in the central-izer C L ( A i ) for some i ≤ s , since A is of order q . We also note thatany commutator in common eigenvectors is again a common eigenvec-tor. Thus, if l , . . . , l r ∈ L are common eigenvectors for A generating L then any commutator in these generators belongs to some L ij andtherefore, by (***), is ad-nilpotent.As we have seen, L satisfies a polynomial identity. It follows fromTheorem 2.3 that L is nilpotent. We now deduce that L is nilpotentas well. CRISTINA ACCIARRI AND PAVEL SHUMYATSKY
According to Lazard [ ] the nilpotency of L is equivalent to G being p -adic analytic. The Lubotzky-Mann theory [ ] now tells us that G isof finite rank, that is, all closed subgroups of G are finitely generated.In particular, we conclude that C G ( a ) is finitely generated for every a ∈ A . It follows that the centralizers C G ( a ) are nilpotent. Theorem1.2 now tells us that G is nilpotent. The proof is complete. (cid:3) References [1] Y. A. Bahturin, M. V. Zaicev, Identities of graded algebras,
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ROFINITE GROUPS AND COPRIME AUTOMORPHISMS 9
Cristina Acciarri: Department of Mathematics, University of Brasilia,Brasilia-DF, 70910-900 Brazil
E-mail address : [email protected] Pavel Shumyatsky: Department of Mathematics, University of Brasilia,Brasilia-DF, 70910-900 Brazil
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