Positive laws on word values in residually-p groups

Journal of Algebra 425 (2015) 524–545

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Journal of Algebra www.elsevier.com/locate/jalgebra

Positive laws on word values in residually-p groups ✩ Gustavo A. Fernández-Alcober a,∗ , Pavel Shumyatsky b a

Department of Mathematics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain b Department of Mathematics, University of Brasília, 70910 Brasília DF, Brazil

a r t i c l e

i n f o

Article history: Received 29 May 2014 Available online 30 December 2014 Communicated by E.I. Khukhro Keywords: Positive laws Word-values Commutators

a b s t r a c t We address the following problem: if the set of all values of a word w in a group G satisfies a positive law, does it follow that the whole verbal subgroup w(G) also satisfies a positive law? In the realm of finitely generated residually-p groups, we obtain a positive answer for the simple commutator words for all but finitely many primes p, depending on the positive law. Furthermore, if we assume that the set of all powers of the values of w satisfies a positive law, then the conclusion holds for all primes. We extend these results to any outer commutator word, in the case that the verbal subgroup w(G) is finitely generated. © 2014 Elsevier Inc. All rights reserved.

1. Introduction Let F denote the free group on countably many symbols x0, x1 , x2 , . . . . The elements of F are called group words in the xi ’s, and a positive word is a word which does not ✩ Supported by the Brazilian and Spanish Governments, under the project with the following references: CAPES/DGU 304/13; PHB2012-0217-PC. The first author is also supported by the Spanish Government, grant MTM2011-28229-C02-02, and by the Basque Government, grant IT753-13. * Corresponding author. E-mail addresses: [email protected] (G.A. Fernández-Alcober), [email protected] (P. Shumyatsky).

http://dx.doi.org/10.1016/j.jalgebra.2014.12.005 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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involve any inverses of the xi ’s. Let T be a subset of a group G. If α and β are two different positive words, we say that T satisfies the positive law α ≡ β if every substitution xi → ti with ti ∈ T gives the same value for α and β. Then the degree of the positive law is the maximum of the lengths of α and β. Groups satisfying a positive law have received attention in recent times. Trivial examples are groups of finite exponent and abelian groups. Clearly, if G/Z(G) satisfies the law α ≡ β then the law αβ ≡ βα holds in G. Thus all nilpotent groups satisfy a positive law in two symbols, which depends on the class c of the group. This is the so-called Malcev law Mc (x, y). Hence all nilpotent-by-(finite exponent) groups satisfy a positive law. Remarkably, the converse to this result holds for the large class of locally graded groups (i.e. groups in which every non-trivial finitely generated subgroup contains a proper subgroup of finite index), and in particular for residually finite groups. More precisely, if a locally graded group G satisfies a positive law of degree n then G is nilpotent-by-(locally finite of finite exponent), with both the class and the exponent bounded in terms of n (see the papers [3] by Burns, Macedońska and Medvedev, [4] by Burns and Medvedev, and [2] by Bajorska and Macedońska). A different kind of question about positive laws is the following: if the set of all commutators in a group G satisfies a positive law, does a positive law hold in the derived subgroup G ? Recall that a group G is called residually-p if for every x ∈ G there exists a normal subgroup N such that the index |G : N | is a p-power and x ∈ / N . Riley and Shumyatsky [11] have obtained a positive answer to the previous question for finitely generated residually-p groups under the stronger condition that the set of all products tuk , with t and u commutators and k ≥ 0, satisfies a positive law. If this is the case, the degree of the positive law satisfied by G is bounded in terms of the prime p, the number d of generators of G, and the degree n of the original positive law. It is also indicated in [11] that the same conclusion is true if we substitute simple commutators of length k for commutators and the subgroup γk (G) for G . In the present paper we get some improvements and generalizations of these results. We begin by showing that it is usually enough to require the positive law to hold on the set of simple commutators of length k, or in the worst situation, on the set of powers of these commutators. In the following theorems, and all throughout the paper, if S is a set of parameters, we use the expression ‘S-bounded’ to mean ‘bounded from above in terms of the parameters in S’. Theorem A. For every positive integer n there exists a finite set P (n) of primes such that, if p ∈ / P (n) and G is a d-generator residually-p group in which the set of all simple commutators of length k satisfies a positive law of degree n, then γk (G) satisfies a positive law of {n, p, d, k}-bounded degree. Theorem B. Let G be a d-generator residually-p group. If the set of all powers of simple commutators of length k in G satisfies a positive law of degree n, then γk (G) satisfies a positive law of {n, p, d, k}-bounded degree.

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We obtain both Theorems A and B as corollaries to Theorem C below, which deals with the following more general question: if w is a word and the set of all values of w in a residually-p group G satisfies a positive law, does the verbal subgroup w(G) also satisfy a positive law? Here w(G) is the subgroup generated by the subset Gw consisting of all values of w in G (also referred to as w-values in G). We say that a word w is commutator-closed in a group G if the set Gw is commutator-closed, that is, closed under taking commutators of its elements. On the other hand, w is of finite width in G if there exists an integer m such that every element of w(G) can be expressed as a product of at most m elements which are either w-values or inverses of w-values. The smallest such m is then called the width of w in G. We have the following result. Theorem C. Let G be a residually-p group, and let w be a word which is commutatorclosed and has finite width m in the pro-p completion of G. Suppose one of the following two conditions holds: / P (n). (i) The set of all w-values in G satisfies a positive law of degree n, and p ∈ (ii) The set of all powers of w-values in G satisfies a positive law of degree n. Then w(G) satisfies a positive law of {n, p, m, w}-bounded degree. Now Theorems A and B readily follow, since the simple commutator of length k has {d, k}-bounded width in a d-generator nilpotent group. The non-trivial words w that have {d, p, w}-bounded width in all d-generator finite p-groups for every d (equivalently, the words which have finite width in all finitely generated pro-p groups) have been determined by Jaikin-Zapirain in [9, Theorem 1.1]. These words, now known as J(p)-words, are characterized by the condition w ∈ / (F  )p F  . It is not clear, however, whether there are any J(p)-words which are commutator-closed, other than the simple commutators γk . The good news is that Theorem C may be applied to any group G and any commutator p of G, so it can be used closed word w that has finite width in the pro-p completion G in situations where w is not a J(p)-word. For example, since all words have finite width in a p-adic analytic pro-p group [9, Theorem 1.3], we have the following result. Theorem D. Let G be a p-adic analytic pro-p group, and let w be any word which is commutator-closed in G. Suppose one of these two conditions holds: / P (n). (i) The set of all w-values in G satisfies a positive law of degree n, and p ∈ (ii) The set of all powers of w-values in G satisfies a positive law. Then w(G) also satisfies a positive law. More precisely, w(G) is nilpotent-by-finite. Assuming the hypotheses of Theorem D, it is natural to ask whether w(G) has a characteristic subgroup whose class and index are {n, p, r, w}-bounded, where r stands

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for the rank of G. This is an open problem, which is linked to the difficult question as whether the width of w is {p, r, w}-bounded over all p-adic analytic pro-p groups of rank r. Unfortunately, if we only assume that the set of simple commutators satisfies a positive law, Theorem A does not provide any information in the case where the prime p lies in the set P (n). Is it possible to get rid of this restriction? The set P (n) arises in Section 2 when we prove general results for groups generated by a commutator-closed normal subset satisfying a positive law, which are obtained under the condition that p ∈ / P (n). In Section 4 we provide counterexamples showing that it is not possible to cover all primes in this general setting. This does not mean that the restriction p ∈ / P (n) is necessary for simple commutators, since in that case the set of generators is not only closed for commutators of its elements, but also closed when making commutators with an arbitrary element of the ambient group. In any case, if we want to improve Theorem A, we need extra tools to complement the results of Section 2. In our next theorem we see that, if we impose the condition of finite generation not on the whole group G, but on the subgroup γk (G), then we can conclude that γk (G) satisfies a positive law for all primes without exception. This result combines the methods of Section 2 with ingredients from the theory of powerful p-groups, and is valid for a much larger family of words besides simple commutators, namely for all outer commutator words. Theorem E. Let w be an outer commutator word, and let G be a residually-p group in which the set of all w-values in G satisfies a positive law of degree n. If w(G) is finitely generated, and d(w(G)) = d, then w(G) satisfies a positive law of {n, p, d, w}-bounded degree. More precisely, w(G) contains a nilpotent characteristic subgroup of {n, p, d, w}-bounded class and index. Recall that an outer commutator word is a word which is formed by nesting commutators, always using different variables. A particular case is that of the derived words δk , which are defined recursively by the rules δ1 (x1 , x2 ) = [x1 , x2 ] and   δk (x1 , . . . , x2k ) = δk−1 (x1 , . . . , x2k−1 ), δk−1 (x2k−1 +1 , . . . , x2k ) . Observe that the corresponding verbal subgroup is the kth derived subgroup G(k) . Other examples of outer commutator words are [x1 , x2 , x3 , [x4 , x5 ], x6 ] or [x1 , [x2 , x3 ], [x4 , x5 , x6 ]]; on the other hand, the Engel words [x1 , x2 , . n. ., x2 ] are not outer commutators if n ≥ 2. 2. Positive laws on commutator-closed normal sets of generators In this section we study finitely generated residually-p groups G in which a ‘large’ set T of generators, in the sense that T is a normal commutator-closed subset, satisfies a positive law. Our goal is to show that, if we assume furthermore that G satisfies a law (by which we always mean a non-trivial law), then in most cases G satisfies a positive

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law. This will be proved in Theorem 2.10 below, where this statement will be made more precise. To begin with, we see that we may always assume that the positive law satisfied by T has a specific form. Lemma 2.1. Let G be a group, and let T be a subset of G. If T satisfies a positive law α ≡ β of degree n ≥ 2, then T also satisfies a positive law α∗ ≡ β ∗ , where α∗ and β ∗ are both of length n, only involve the variables x0 , . . . , xn−1 , and have different initial symbols. Proof. We may assume that α is of length n, and by renaming variables we may replace α and β with positive words α and β  , in such a way that α only involves the variables x0 , . . . , xn−1 . Let us first see how to obtain two different positive words α ˜ and β˜ such that the law α ˜ ≡ β˜ holds on T , both α ˜ and β˜ are of length n, and they only involve x0 , . . . , xn−1 . We  can take α ˜ equal to α . If β  only involves x0 , . . . , xn−1 and is of length n, we similarly put  ˜ , where m is the length of β  . β = β . Otherwise, we define β˜ as follows. Put β  = β  xn−m n  Observe that β is of length n and contains a variable which is not among x0 , . . . , xn−1 . For every i = 0, . . . , n − 1, let βi be the word which is obtained from β  by substituting xi for every variable which is distinct from x0 , . . . , xn−1 . These n words are different for different values of i, and then, since n ≥ 2, one of them is different from α ˜ . Let β˜ be that ˜ word. This way β has been defined in every case, and is different from α ˜. Also, one can readily check that the law α ˜ ≡ β˜ holds on T . ˜ Hence all requirements in the statement of the theorem are fulfilled by α ˜ and β, except possibly the one about the initial symbols. Let α ˜ 1 be the largest common prefix ˜ and write α of α ˜ and β, ˜=α ˜1 α ˜ 2 and β˜ = α ˜ 1 β˜2 . Then the proof is complete by putting α∗ = α ˜2α ˜ 1 and β ∗ = β˜2 α ˜1 . 2 Thus in the following we may always assume that a positive law of degree n ≥ 2 satisfied by a subset T is of the form xi1 . . . xin ≡ xj1 . . . xjn ,

where 0 ≤ ir , jr ≤ n − 1, and i1 = 1, j1 = 0.

(1)

However we cannot suppose that T satisfies a law in two variables, as is usually done when a positive law holds in a group: observe that the trick of substituting xy i for xi cannot be used in this setting since T need not be closed for products. Next we introduce, for every positive integer n, a finite set P (n) of primes which is going to play a fundamental role throughout the paper. Definition 2.2. Let n ≥ 2 be an integer. (i) Let α ≡ β be a positive law of degree n, written in the form (1), and consider the polynomial

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g(X) = X n−1 + (i2 − j2 )X n−2 + · · · + (in − jn ). Then we define P (α, β) as the set of prime divisors of the non-zero coefficient of smallest degree in the expansion of g(X) in powers of X − 1. (ii) We let P (n) be the union of the sets P (α, β), as α ≡ β runs over all positive laws of degree n of the form (1). Observe that P (n) is finite, since there are only finitely many positive laws of degree n for every n ≥ 1. Our first theorem analyzes the action of a normal subset T satisfying a positive law on the abelian normal sections of G in the case where the ambient group G is a finite p-group. We need the following lemma, which is well-known and easy to prove. Lemma 2.3. Let G = t, A be a group, where A is an abelian normal subgroup of G. Then γi (G) = [A,i−1 t] = {[a,i−1 t] | a ∈ A} for every i ≥ 2. We say that a subset X of a group G is power-closed if xi ∈ X for all x ∈ X and all i ≥ 1 (so X is closed under powers with positive exponent). If N is a normal subgroup of G, we say that X is power-closed modulo N if the image of X in G/N is power-closed, in other words, if xi ∈ XN for all x ∈ X and all i ≥ 1. Theorem 2.4. Let G be a finite p-group and suppose that T ⊆ G is a normal subset which satisfies a positive law of degree n. Then: (i) If A is an elementary abelian normal section of G, then [A,n t] = 1 for every t ∈ T . (ii) If p ∈ / P (n) and A is an abelian normal section of G, then [A,n t] = 1 for every t ∈ T. (iii) For every r ≥ 1, there exist {n, r}-bounded positive integers k and m with the following property: if A = K/L is an abelian normal section of G and T r is power-closed modulo K, then [A,m trk ] = 1 for every t ∈ T . Proof. All throughout the proof, we may assume without loss of generality that the normal sections in the statement of the theorem are in fact normal subgroups of G; thus we have A = K in (iii). On the other hand, observe that if n = 1 then T must necessarily consist of no more than one element. Since T is a normal subset of G, it follows that T ⊆ Z(G), and then the theorem trivially holds. Hence we may assume that n ≥ 2. By Lemma 2.1, T satisfies a positive law of the form xi1 . . . xin ≡ xj1 . . . xjn , where ir , jr ∈ {0, . . . , n − 1} for r = 1, . . . , n, and i1 = 1, j1 = 0. Now we consider an arbitrary i normal subgroup A of G, and we apply this law to the elements ti = ta = tai(1−t) , where t ∈ T and a ∈ A are arbitrary. Since ti tj = tai(1−t) taj(1−t) = t2 a(1−t)(it+j) ,

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we get n−1

tn a(1−t)(i1 t

+i2 tn−2 +···+in )

n−1

= tn a(1−t)(j1 t

+j2 tn−2 +···+jn )

.

Hence af (t) = 1, where   f (X) = (1 − X) X n−1 + (i2 − j2 )X n−2 + · · · + (in − jn ) is a polynomial of degree n with integer coefficients, which depends only on the positive law we are working with, and not on the choice of t and a. Observe that f (X) = (1 − X)g(X), where g(X) is as in part (i) of Definition 2.2. We are going to use this polynomial in the proof of the three parts of the theorem. (i) Now let A be elementary abelian. We consider every element t ∈ T as an endomorphism of A, viewed as an Fp -vector space, and we write m(X) for the minimal polynomial of t. Then, since af (t) = 1 for all a ∈ A, it follows that m(X) divides f (X). On the other hand, if the nilpotency class of G is c, then we have [A,c t] = 1, and so m(X) divides (X − 1)c . Since deg f (X) = n, we deduce that m(X) divides (X − 1)n . This proves that [A,n t] = 1. (ii) Let us write f in powers of X − 1, f (X) =

n 

λi (X − 1)i ,

i=0

and let k be the first index for which λk = 0 (note that k ≥ 1). Then the condition af (t) = 1 reads [a,k t]λk [a,k+1 t]λk+1 . . . [a,n t]λn = 1, and consequently [a,k t]λk ∈ [A,k+1 t].

(2)

Now, by the definition of the set P (n), the fact that p does not lie in P (n) implies that p does not divide λk . It follows from (2) that [a,k t] ∈ [A,k+1 t] for all a ∈ A. By Lemma 2.3, we have γk+1 (H) = γk+2 (H) in the group H = t, A . Since H is nilpotent, we have γk+1 (H) = 1, and since k ≤ n, it follows in particular that [A,n t] = 1. (iii) Let us consider the quotient ring R = Z[X]/(f (X)), where (f (X)) denotes the ideal generated by f (X) in Z[X]. Since the leading coefficient of f (X) is −1, it follows that R is a finitely generated Z-module, actually free of rank n. By Proposition 5.1 of [1], the image of X r in R is integral over Z, i.e. there exists a monic polynomial g(X) ∈ Z[X], depending on r, such that g(X r ) ∈ (f (X)). Furthermore, the proof of this fact shows that g(X) can be taken of degree n. Since f (t) annihilates A, we obtain that g(tr ) annihilates A for every t ∈ T .

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Since T r is power-closed modulo A, for every t ∈ T and every i ≥ 1, there exists u ∈ T such that tri ≡ ur (mod A). Consequently the endomorphism g(tri ) annihilates A for all i ≥ 1. If we define I to be the ideal of Z[X] generated by all polynomials g(X i ) then h(tr ) annihilates A for all h(X) ∈ I. By Lemma 3.3 of [14], we have qX  (X k − 1) ∈ I for some positive integers q, k and  determined by g(X), and consequently by the positive law holding on T and by the integer r. Since there are only finitely many positive laws of degree n, we can further assume that q, k and  depend only on n and r. If ps is s the highest power of p dividing q, we deduce that [Ap , trk ] = 1 for all t ∈ T . Now, by i i+1 part (i), we have [Ap ,n t] ≤ Ap for every i, and then Lemma 2.3 implies that also i i+1 s p rk p rk [A ,n t ] ≤ A . Hence [A,sn t ] ≤ Ap , and we conclude that [A,sn+ trk ] = 1, as desired. 2 Now we return to our original setting, where G is a finitely generated residually-p group having a ‘large’ set of generators which satisfies a positive law, and G is known to satisfy some law. As a first application of the theorem above, we can bound the derived length of G and the rank of all its finite p-quotients. For this purpose, we need the following well-known result of Zelmanov [18, page 36]. In the remainder, when we speak about commutators formed with elements of a certain subset S in a Lie algebra, we also include commutators of length 1, that is, the elements of S themselves. Theorem 2.5 (Zelmanov). Let L be a Lie algebra satisfying a polynomial identity. If L can be generated by a finite set S such that every commutator of elements of S is ad-nilpotent, then L is nilpotent. Theorem 2.6. Let G be a d-generator residually-p group which satisfies a certain law v ≡ 1. Suppose further that G can be generated by a commutator-closed normal subset  p of G is a soluble satisfying a positive law of degree n. Then the pro-p completion G p-adic analytic group, and furthermore:  p is {n, p, d, v}-bounded. Hence the ranks of all finite p-quotients of G (i) The rank of G are {n, p, d, v}-bounded.  p is {n, p, d, v}-bounded. Hence, G is soluble of {n, p, (ii) The derived length of G d, v}-bounded derived length. Proof. Consider the free pro-p group F on a set X of cardinality d and let Y be the set of all commutators (of any length and shape) formed with conjugates of elements of X. In other words, Y is the smallest normal subset of F which contains X and is commutator-closed. Let N be the (normal) subgroup of F generated by all v-values on F and by all values of αβ −1 on Y , where α ≡ β is the positive law of degree n in the statement of the theorem. Now we let M be the topological closure of N , and put K = F/M . Then K is a d-generator pro-p group with a commutator-closed normal set of generators satisfying the positive law α ≡ β, and the law v ≡ 1 holds in K.

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Now any pro-p group with these properties is a quotient of K. In particular, the pro-p  p of G is a quotient of K. If we prove that K is a soluble pro-p group completion G of finite rank, then the values of the rank and derived length of K are bounds for the  p . Since K only depends on p, d, v, and the positive law corresponding invariants of G  p are {n, p, d, v}-bounded. of degree n, it follows that the rank and derived length of G  p is p-adic analytic by Corollary 8.34 of [5]. Also, being a pro-p group of finite rank, G  p , and so is soluble of {n, p, d, v}-bounded Now since G is residually-p, it embeds in G derived length. On the other hand, since the finite p-quotients of G are the same as the  p by (a) and (d) of [12, Proposition 3.2.2], the second assertion finite p-quotients of G of (i) also follows. Thus we only need to prove that K is a soluble pro-p group of finite rank. For this purpose, it suffices to prove that the Lie algebra Lp (K) associated to the dimension subgroup series {Di } of K is nilpotent. Indeed, it then follows from Interlude A in [5] that K has finite rank and that K is linear over a field of characteristic 0. By the Tits Alternative [15], either K is soluble-by-finite or contains a non-abelian free subgroup. Since K is a pro-p group and it satisfies the law v ≡ 1, we deduce that K is soluble. For any k ∈ K, let k denote its image in Lp (K), and use a similar notation for subsets  of K. Write also Li = Di /Di+1 , so that Lp (K) = i Li . According to Lemma 11.11 and Proposition 11.15 of [5], in order to see that Lp (K) is nilpotent, it suffices to prove that the Lie algebra L generated by the first homogeneous component L1 is nilpotent. Since the law v ≡ 1 holds in K, it follows from Theorem 1 of [17] that the Lie algebra Lp (K) satisfies a polynomial identity, and hence also L does. Let S and T be the images of X and Y in K, respectively. Note that both S and T generate the Lie algebra L. Since |S| = d is finite and T contains all the commutators of elements of S, according to Theorem 2.5 we only need to check that all elements of T are ad-nilpotent. Let then t be an arbitrary element of T , and let i be such that t ∈ Di \ Di+1 . Then we have [Dj ,2n t] ≤ [Dj+ni ,n t] for all j. On the other hand, T satisfies a positive law of degree n, and the section Dj+ni /Dj+2ni+1 is elementary abelian in the finite p-group K/Dj+2ni+1 . Thus it follows from part (i) of Theorem 2.4 that [Dj+ni ,n t] ≤ Dj+2ni+1 . Hence [Dj ,2n t] ≤ Dj+2ni+1 for all j, and consequently t is ad-nilpotent of index at most 2n. This completes the proof. 2 We still require a couple of lemmas before proceeding to the proof of our main result in this section. Lemma 2.7. Let G = t1 , . . . , td , A be a group, where A is an abelian subgroup of G containing G . Then the following hold: (i) If [A,n ti ] = 1 for all i = 1, . . . , d, then G is nilpotent of class at most dn. (ii) If [A,n tki ] = 1 for all i = 1, . . . , d, then G has a normal subgroup of index at most kd which is nilpotent of class at most dn, and G has {d, k, n}-bounded rank.

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Proof. (i) Since A is abelian and normal in G, and since [A,n ti ] = 1, Lemma 2.3 implies that each of the normal subgroups Ni = ti , A is nilpotent of class ≤ n for i = 1, . . . , d. Since G = N1 . . . Nd , it follows that G is nilpotent of class at most dn. (ii) Let K = tk1 , . . . , tkd , A = Gk A. By part (i), K is nilpotent of class at most dn. On the other hand, we have |G : K| = |G/A : (G/A)k | ≤ kd , since G/A is an abelian d-generator group. Then both K and G/K have {d, k, n}-bounded rank, and so does G. 2 Remark 2.8. Assume that a group G has a nilpotent subgroup of class c and finite index e. Then its core in G is nilpotent of class c and index at most e!. Then, by Lemma 3.1 of [11], G contains a characteristic nilpotent subgroup of class at most e! · c and index at most e!. As a consequence, in results such as (ii) in the last theorem, where we claim the existence of a (normal) subgroup in a group G with bounded class and index, it also happens that G contains a characteristic subgroup of bounded class and index. This remark will be freely used in the remainder of the paper, most of the times without further mention. The following lemma shows how to combine Theorem 2.4 with Theorem 2.6. Lemma 2.9. Let G be a finite p-group of rank r and derived length , and let T be a commutator-closed normal set of generators of G. Then: (i) If [A,n t] = 1 for every abelian characteristic section A of G and for every t ∈ T , then the nilpotency class of G is {n, r, }-bounded. (ii) If [A,n tk ] = 1 for every abelian characteristic section A of G and for every t ∈ T , then G has a characteristic subgroup whose nilpotency class and index are {n, k, r, }-bounded. Proof. Clearly, every quotient of G over a characteristic subgroup satisfies the hypotheses of the theorem. On the other hand, since T is a normal set of generators of G, the derived subgroup G can be generated by the subset of T consisting of all elements of the form [t, u] with t, u ∈ T . This subset is commutator-closed and normal in G . Also, since it is contained in T , its elements have the desired kind of action on abelian characteristic sections. Thus, G satisfies the hypotheses of the theorem, and consequently the same is true for every term of the derived series of G. Therefore both (i) and (ii) can be proved by induction on the derived length of G. (i) By a well-known theorem of P. Hall, if N is normal in G and both G/N  and N are nilpotent then also G is nilpotent. Also, the class of G is bounded in terms of the classes of G/N  and N [10, Theorem 3.26]. Hence, it suffices to deal with the case when G is metabelian. Then, by the hypothesis, [G ,n t] = 1 for every t ∈ T . Since G is an r-generator finite p-group, by the Burnside Basis Theorem we can find elements t1 , . . . , tr in T such that G = t1 , . . . , tr , and then the result follows from (i) of Lemma 2.7.

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(ii) We follow the ideas of the proof of Proposition 3.3 in [11]. In the rest of the proof, when we say ‘bounded’ we mean ‘{n, k, r, }-bounded’. Arguing by induction on , we may assume that G has a characteristic subgroup N of bounded class and index. As above, let t1 , . . . , tr be generators of G taken from T . Put Hi = tki , G and Ji = tki , N for all i = 1, . . . , r. Then Hi is normal in G and |Hi : Ji | ≤ |G : N | is bounded. Since [N,n tki ] ≤ N  , it follows that the nilpotency class of Ji /N  is at most n, and consequently also the class of Ji is bounded, by Hall’s result. By Remark 2.8, Hi has a characteristic subgroup Ki whose class and index in Hi are bounded. It follows that Ki is normal in G and then K = K1 . . . Kr has bounded class. Finally, if H = H1 . . . Hr = tk1 , . . . , tkr G then |G : K| = |G : H||H : K| ≤ kr

r

|Hi : Ki |

i=1

is bounded, and we are done. 2 Now we can easily derive the desired result. Theorem 2.10. Let G be a d-generator residually-p group which satisfies a certain law v ≡ 1. Suppose that G can be generated by a commutator-closed normal subset T satisfying a positive law of degree n. Then: (i) If p ∈ / P (n), then G is nilpotent of {n, p, d, v}-bounded class. (ii) If T is power-closed, then G contains a nilpotent characteristic subgroup of {n, p, d, v}-bounded class and index. In both cases, G satisfies a positive law of {n, p, d, v}-bounded degree. Proof. It suffices to prove the theorem when G is a finite p-group. Recall that, by Theorem 2.6, G has {n, p, d, v}-bounded rank and derived length. Now the result follows by combining Theorem 2.4 and Lemma 2.9. 2 3. Positive laws on word values In this section we apply the general results of Section 2 to the situation in which the set of all w-values in a residually-p group G satisfies a positive law, and see that Theorems A, B, C, and D in the introduction follow immediately. The key step is to prove Theorem C. Proof of Theorem C. Let Q be a finite p-quotient of G. Since w has finite width m  p , it has width at most m in Q. Thus if x and y are two arbitrary elements of in G w(Q), they belong to a subgroup R which can be generated by at most 2m w-values. Let α ≡ β be the positive law satisfied by Gw , and consider the word v which is obtained by

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the composition of w and αβ −1 . More precisely, if w is a word in k different variables, then v is the image of αβ −1 under the endomorphism of the free group F sending each xi to w(xik , xik+1 , . . . , xik+k−1 ). Then it is clear that the law v ≡ 1 holds in G, and consequently also in R. It follows from Theorem 2.10 that R has a nilpotent characteristic subgroup S of class at most c such that the quotient group R/S has exponent at most e, with c and e depending only on n, p, m and w, not on R. Thus x and y satisfy the law Mc (xe , y e ). Hence the positive law Mc (xe , y e ) holds over all of w(Q), for every p-quotient Q of G. Since G is residually-p, we conclude that the same law holds on w(G), and we are done. 2 Proof of Theorem D. According to a result of Jaikin-Zapirain [9, Theorem 1.3], if G is a compact p-adic analytic group then all words have finite width in G. By Proposition 4.1.2 of [13], w(G) is closed in G, and consequently w(G) is a pro-p group. On the other hand, Theorem C applies and w(G) satisfies a positive law. By the result of Burns, Macedońska and Medvedev that we mentioned in the introduction, it follows that w(G) has a nilpotent normal subgroup K such that the quotient w(G)/K is of finite exponent. By replacing K with its topological closure, we may assume that K is closed in G. Since G is a p-adic analytic group, G is of finite rank, and consequently w(G) is (topologically) finitely generated. Thus w(G)/K is a finitely generated pro-p group of finite exponent. It then follows from Zelmanov’s positive solution to the Restricted Burnside Problem that w(G)/K is finite, and we are done. 2 Proof of Theorem A and Theorem B. By Corollary 1.2.8 of [13], the word γk has width at most dk−1 in any d-generator nilpotent group. This applies in particular to all finite  p over its p-quotients of G or, what is the same, to all quotients of the pro-p completion G k−1 p . open normal subgroups. By Proposition 4.1.2 of [13], γk has width at most d in G Thus both Theorem A and Theorem B follow from Theorem C. 2 Remark 3.1. In the proof of Theorem C, if p ∈ / P (n) then Theorem 2.10 yields the stronger conclusion that x, y is nilpotent of bounded class. Thus w(G) is a bounded Engel group. Since w(G) is residually-p, the verbal subgroup w(G) is locally nilpotent by Theorem 2 of [16]. As a consequence, we can also guarantee local nilpotency in Theorem A and in part (i) of Theorem D. Next we study the case when the assumption of finite generation is set on the verbal subgroup w(G), and w is an outer commutator word. Again, we need to develop general tools to study finite p-groups with some normal set of generators which satisfies a positive law, where in this case the set need not be commutator-closed. To begin with, we prove the following useful result about powerful finite p-groups. Lemma 3.2. Let G be a powerful finite p-group and let N be a normal subgroup of G of index e and nilpotency class c. Then the nilpotency class of G is {c, e}-bounded.

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Proof. By [10, Corollary 11.6], all terms of the lower central series of G are powerfully embedded in G. Then, by applying [10, Lemma 11.12], we obtain 

Ge

c+1

   c+1 , G, . c. ., G = γc+1 (G)e = γc+1 Ge ≤ γc+1 (N ) = 1.

On the other hand, if we write e = pm then, since G is powerful, we have m(c+1)

γm(c+1)+1 (G) ≤ Gp

= Ge

c+1

.

Thus γ(m+1)(c+1) (G) = 1, which proves the result. 2 Theorem 3.3. Let G be a d-generator finite p-group and let 1 = N0 ≤ N1 ≤ · · · ≤ N = G be a normal abelian series of G. Suppose that every Ni can be generated by a normal subset Ti which is power-closed modulo Ni−1 , and that Ti satisfies a positive law of degree n. Then the following hold: (i) The rank of G is {d, , n}-bounded. (ii) G has a characteristic subgroup whose index and nilpotency class are {d, , n}-bounded, and so G satisfies a positive law of {d, , n}-bounded degree. Proof. We use ‘bounded’ for ‘{d, , n}-bounded’ all throughout the proof. (i) We argue by induction on . Since T is a normal subset of G which is power-closed modulo N−1 and satisfies a positive law of degree n, we may apply (iii) of Theorem 2.4. Thus there exist n-bounded positive integers k and m such that 

  N−1 ,m tk ≤ N−1

for every t ∈ T .

(3)

By the Burnside Basis Theorem we can find T ⊆ T such that |T | = d and G = T . Then,  by applying part (ii) of Lemma 2.7 to G = G/N−1 , it follows that G has a nilpotent subgroup whose index and nilpotency class are bounded. Also, G has bounded rank. Now d(N−1 ) coincides with d(N−1 ), and so is bounded. By applying the induction hypothesis to N−1 , it follows that this group has bounded rank. As a consequence, the rank of G is bounded, as desired. (ii) Again, we use induction on . Since G has bounded rank, Proposition 2.12 and Theorem 2.13 of [5] yield the existence of a normal subgroup P of G of bounded index, say r, and such that every subgroup of P which is normal in G is powerful. By the induction hypothesis applied to N−1 , there exists a normal subgroup of N−1 of bounded index and bounded nilpotency class. Then the same holds for P ∩ N−1 , which is on the other hand normal in G, and so powerful. By applying Lemma 3.2, it follows that P ∩N−1 is nilpotent of bounded class. Let us put R = Tr (P ∩ N−1 ), which is a subgroup of P . One readily checks that Tr is power-closed modulo P ∩ N−1 , and so we may apply (iii) of Theorem 2.4 to the abelian section (P ∩ N−1 )/(P ∩ N−1 ) . Since d(R) is bounded, we can use Lemma 2.7

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as in the proof of (i), to obtain that R/(P ∩ N−1 ) has a nilpotent subgroup whose index and nilpotency class are bounded. But again R is powerful, so R/(P ∩ N−1 ) itself has bounded class. It follows that R is of bounded class, by P. Hall’s criterion of nilpotency. By Dedekind’s Law, we have



 P ∩ Tr N−1 = Tr (P ∩ N−1 ) = R. Consequently |P : R| ≤ |G : Tr N−1 | ≤ rd , since G/N−1 is an abelian d-generator group. Since P is powerful, it follows that P has bounded class. This completes the proof of (ii), since |G : P | is bounded. 2 We can apply the previous result to the case where the set of values of an outer commutator word satisfies a positive law. We rely on the following result of Morigi and the first author [6, Theorem B]. Theorem 3.4. Let w be an outer commutator word, and let G be a soluble group. Then there exists a series of subgroups from 1 to w(G) such that: (i) All subgroups of the series are normal in G. (ii) Every section of the series is abelian and can be generated by values of w all of whose powers are also values of w in that section. Furthermore, the length of this series only depends on the word w and on the derived length of G. Now we are ready to prove Theorem E. Proof of Theorem E. Let us use ‘bounded’ for ‘{n, p, d, w}-bounded’ throughout the proof. If w(G) satisfies a positive law of bounded degree then, by the result of Burns, Macedońska and Medvedev mentioned in the introduction, it follows that w(G) has a nilpotent characteristic subgroup of bounded class and index. Thus it suffices to prove that w(G) satisfies a positive law of bounded degree. Observe that we may assume that G is a finite p-group. To begin with, suppose that the derived length of G is bounded. By Theorem 3.4, there exists a normal abelian series 1 = N0 ≤ N1 ≤ · · · ≤ N = w(G), where  is bounded, such that every quotient Ni /Ni−1 can be generated by the image of a certain subset Si of w-values all of whose powers are again w-values modulo Ni−1 . Thus for every s ∈ Si and every m ∈ N there exists g = g(s, m) ∈ Gw such that sm ≡ g (mod Ni−1 ). Define

 Si∗ = g(s, m) s ∈ Si , m ∈ N . Then, one readily checks that Si∗ is power-closed modulo Ni−1 . Let Ti be the union of i all conjugates in w(G) of the union j=1 Sj∗ . Then Ti is a normal subset of w(G) which

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generates Ni , and Ti is power-closed modulo Ni−1 . Also, since Ti is a subset of Gw , it satisfies a positive law. By applying Theorem 3.3 to w(G), we conclude that w(G) is of bounded rank and satisfies a positive law of bounded degree. Thus in order to complete the proof we only need to prove that the derived length of G is bounded. If w is a derived word then w is commutator-closed and, as in the proof of Theorem C, if v is the composition of the words w and αβ −1 , then w(G) satisfies the law v ≡ 1. Then it follows from (ii) of Theorem 2.6 that the derived length of w(G) is bounded, and so also the derived length of G is bounded. Hence the result is valid for derived words. Now let w be an arbitrary commutator word, say in q variables. Then every δq -value is also a w-value, and G(q) ≤ w(G). If G = G/G(q+1) then the derived length of G is bounded, and we are in the situation of the first paragraph of the proof. As a consequence, w(G) has bounded rank, and then d(G(q) ), which coincides with d(G(q) ), is bounded. On the other hand, Gδq satisfies the same positive law as Gw . Since the result is already known to hold for derived words, we deduce that G(q) has a nilpotent subgroup whose index and nilpotency class are bounded. Thus the derived length of G(q) is bounded, and so is the derived length of G. This completes the proof. 2 4. Examples In Theorem A there is a set P (n) of ‘bad primes’ for which we are not able to conclude that a positive law of degree n on simple commutators of length k implies a positive law in the whole of γk (G). However, we do not know of any examples for which the conclusion of Theorem A fails, so it could be the case that it holds for all primes. Our proof of Theorem A relies directly on the more general Theorem 2.10, and this is the reason why the set P (n) arises in Theorem A. The purpose of this final section is to show that we cannot get rid of P (n) in the context of Theorem 2.10, which therefore is not valid for all primes. More precisely, we are going to prove the following result. Theorem 4.1. For every prime p there exists a metabelian 2-generator residually-p group G such that: (i) G can be generated by a commutator-closed normal subset satisfying a positive law in two variables. (ii) G is not nilpotent-by-finite, and so G cannot satisfy a positive law. Note that, being metabelian, G satisfies the law δ2 (x, y, z, t) = 1, so every condition in Theorem 2.10 (i) is fulfilled, with the exception that p must lie outside P (n). However, the conclusion of this theorem does not hold for G. The prototype for our example is a semidirect product constructed by using the ring Zp of p-adic integers. More specifically, choose a non-identity element t in the multiplicative group 1 + p Zp if p is odd, and in 1 + 4 Z2 if p = 2. As a consequence, t is of infinite order. Let B be the multiplicative subgroup generated by t, and let A = Z[t, t−1 ] be the subring

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of Zp generated by B. Then B acts on the additive group of A by multiplication and we can consider the corresponding semidirect product G = B  A. Our first observation is the following. Lemma 4.2. The semidirect product G is a 2-generator residually-p group which is not nilpotent-by-finite. Proof. By definition, the elements of G are pairs of the form (b, a) with b ∈ B and a ∈ A, and the operation of G is given by (b1 , a1 ) · (b2 , a2 ) = (b1 b2 , a1 b2 + a2 ). The identity element of G is (1, 0), and G is clearly generated by (t, 0) and (1, 1). Note also the value of the commutator of an element of A with an element of B:     (1, a), (b, 0) = 1, a(b − 1) . For every k ≥ 0, consider the subgroup

  k   k Gk = (b, a) b ∈ B p , a ∈ A ∩ pk Zp = tp , 0 , (1, a) a ∈ A ∩ pk Zp . Now A ∩ pk Zp is B-invariant, and 

  k    k (1, a), tp , 0 = 1, a tp − 1

k

with tp ≡ 1 (mod pk+1 Zp ), since t ≡ 1 (mod p Zp ). Consequently Gk is normal in G. Since the subgroups Gk have trivial intersection, we only need to prove that Gk has p-power index in G to conclude that G is residually-p. We have



k

|G : Gk | = B : B p

A : A ∩ pk Zp = pk A : A ∩ pk Zp . Since







Zp : pk Zp ≥ A : A ∩ pk Zp ≥ Z : Z ∩ pk Zp = Z : pk Z , it follows that |A : A ∩ pk Zp | = pk and consequently |G : Gk | = p2k is a p-power. Finally, suppose that G has a normal subgroup N of finite index n which is nilpotent of class c. Then (tn , 0) and (1, n) belong to N and therefore 

    c  (1, n),c tn , 0 = 1, n tn − 1

must coincide with the identity element (1, 0). It follows that tn = 1, which is a contradiction since t is of infinite multiplicative order. 2

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The semidirect product G contains natural copies of A and B, which we still denote by these letters (but it is not wise to identify a with (1, a) and b with (b, 0), since A and B have elements in common). Then T = (t, 0)A ∪ A is a commutator-closed normal subset of G which generates the whole of G, and our goal now is to see how an appropriate choice of the element t = 1 in 1 + p Zp guarantees that T satisfies a certain positive law in two variables. Let α be a positive word in two variables x and y of length n. We associate to α three n−1 n−1 n−1 polynomials fα (X) = i=0 ai X i , gα (X) = i=0 bi X i and hα (X) = i=0 ci X i over the integers in the following way: if we label the letters in α from 0 to n − 1 beginning from the right, then ai = 1 if we find x at the i-th position and ai = 0 otherwise; on the other hand, bi is the number of x’s in α having exactly i occurrences of y to the right, and ci is defined symmetrically by interchanging the roles of x and y. For example, if α = xyxyyxxy then fα (X) = X 7 + X 5 + X 2 + X, gα (X) = X 4 + X 3 + 2X and hα (X) = X 3 + 2X 2 + 1. Now if L is a positive law in two variables x and y, given by α ≡ β, we associate to it the three polynomials fL (X) = fα (X) − fβ (X), gL (X) = gα (X) − gβ (X) and hL (X) = hα (X) − hβ (X). Thus the polynomials associated to the Malcev law M3 (x, y) are f (X) = X 7 − X 6 − X 5 + X 4 − X 3 + X 2 + X − 1, g(X) = X 4 − 2X 3 + 2X − 1 and h(X) = −g(X). Let us see another example which will be helpful later on. Example 4.3. Let m and n be two integers with n ≥ 1 and 1 ≤ m ≤ n. If L is the law given by (xy)n ≡ (yx)m (xy)n−m , then fL (X) = X 2(n−m) (X − 1)(X 2m−2 + X 2m−4 + · · · + X 2 + 1), gL (X) = X n−m (X m − 1) and hL (X) = −X n−m (X m − 1). The following result, whose proof is straightforward, will be useful when we need to calculate the polynomials associated to a positive law. If α is a positive word then we write (α) for the length of α, x (α) for the number of occurrences of x in α, and similarly y (α). We say that a positive law α ≡ β in two variables is balanced if x (α) = x (β) and y (α) = y (β). Lemma 4.4. Let α and β be two positive words, and let L be the positive law α ≡ β. Suppose that α = α1 α2 and β = β1 β2 are decomposed as a product of positive words, and let L1 and L2 be the positive laws α1 ≡ β1 and α2 ≡ β2 , respectively. Then: (i) fα (X) = X (α2 ) fα1 (X) + fα2 (X), and if α2 and β2 have the same length then fL (X) = X (α2 ) fL1 (X) + fL2 (X). (ii) gα (X) = X y (α2 ) gα1 (X) + gα2 (X), and if L2 is balanced then gL (X) = X y (α2 ) × gL1 (X) + gL2 (X). (iii) hα (X) = X x (α2 ) hα1 (X) + hα2 (X), and if L2 is balanced then hL (X) = X x (α2 ) × hL1 (X) + hL2 (X). In Example 4.3 above, we can write fL (X) = (X − 1)q(X 2 ) and gL (X) = −hL (X) = (X − 1)q(X) for a certain polynomial q(X). As an application of the previous lemma, we see that the same happens for a wide variety of positive laws.

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Lemma 4.5. Let α and β be positive words of the same length in two variables x and y, and suppose that both α and β can be written as words in xy and yx. If L denotes the positive law α ≡ β, then there exists a polynomial q(X) ∈ Z[X] such that fL (X) = (X − 1)q(X 2 ) and gL (X) = −hL (X) = (X − 1)q(X). Proof. We argue by induction on m, where m is the length of α and β as words in xy and yx. If m = 1 then the result holds trivially. Suppose the result true for m and let us check it for m +1. Write α = α1 α2 and β = β1 β2 , with α1 and β1 equal to xy or yx. Let L2 denote the law α2 ≡ β2 , and let us apply the previous lemma. If α1 = β1 then the three polynomials associated to L coincide with those corresponding to L2 and the result holds by the induction hypothesis. If α1 = xy and β1 = yx then fL (X) = X 2m (X −1) +fL2 (X), gL (X) = X m (X − 1) + gL2 (X) and hL (X) = −X m (X − 1) + hL2 (X), and again the result follows. The remaining case α1 = yx and β1 = xy is similar. 2 The reason why we have introduced the three polynomials fL (X), gL (X) and hL (X) is clear from the following lemma. Lemma 4.6. Let G = t, A be a group, where A is an abelian normal subgroup of G, and let L be a balanced positive law in two variables. Then the set T = tA ∪ A satisfies the law L if and only if the endomorphisms fL (t), gL (t) and hL (t) annihilate A. Proof. Let L be given by α ≡ β. We have to check that the following conditions hold for all u ∈ tA and a ∈ A: (i) α(ua, u) = β(ua, u), (ii) α(a, u) = β(a, u), (iii) α(u, a) = β(u, a). (Note that α(a, b) = β(a, b) is trivially satisfied for all a, b ∈ A, since L is balanced and A is abelian.) An easy calculation shows that α(ua, u) = u(α) afα (u) , α(a, u) = uy (α) agα (u) and α(u, a) = ux (α) ahα (u) . It follows that (i), (ii) and (iii) hold if and only if fL (u), gL (u) and hL (u) annihilate A for all u ∈ tA. Now, since A is abelian, the endomorphism F (u) coincides with F (t) for every polynomial F (X) over the integers, and the result follows. 2 The next step is to define the positive law that is going to be satisfied by the set of generators in our example. For this purpose, for every odd integer n ≥ 3 we define two positive words σn (x, y) and τn (x, y) as follows. Firstly, n

σn (x, y) = (xy)2

+2n−1 +1

.

(4)

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For the definition of τn (x, y) we need two auxiliary words, n−3

un (x, y) = (yx)4 (xy)3·2 1

=

−4

(yx)4 (xy)3·2

(yx)4 (xy)3·2

2i

−4

n−5

−4

. . . (yx)4 (xy)3·2

−4

. . . (yx)4 (xy)3·2−4

2

−4

,

i=(n−3)/2

and n−4

vn (x, y) = (yx)4 (xy)3·2 1

=

−4

(yx)4 (xy)3·2 2i−1

(yx)4 (xy)3·2

−4

n−6

.

i=(n−3)/2

Then τn (x, y) is defined recursively by means of the rule n−2

τn (x, y) = un (x, y)(yx)5 (xyyx)(xy)2 n−3

· (xy)2 (yx)2 (xy)2

−2

−3

vn (x, y)

τn−2 (x, y),

(5)

with the initial value τ3 (x, y) = (yx)5 (xyyx)(xy)(yx)2 (xy)2 (yx). We write L(n) to denote the positive law σn ≡ τn , of degree 2n+1 + 2n + 2. The following lemma gives the key for our construction to work. + X 2 −2 + · · · + X 8 + X 6 + 1, where Lemma 4.7. Let Fn (X) = X 2 + X 2 −2 + X 2 n ≥ 3 is odd. Then Fn divides all three polynomials fL(n) , gL(n) and hL(n) . n

n

n−2

n−2

Proof. Let + X 2 −1 + X 2 + X2  (n−1)/2  2i X 2 −1 + 1, = (X + 1) n−1

n−1

Gn (X) = X 2

n−3

n−3

−1

+ · · · + X4 + X3 + 1

i=1

so that Fn (X) = Gn (X 2 ). We claim that fL(n) (X) = (X − 1)Gn (X 2 )Gn (X 4 ) and gL(n) (X) = −hL(n) (X) = (X − 1)Gn (X)Gn (X 2 ). Since both σn and τn can be written as words in xy and yx, according to Lemma 4.5 it suffices to prove the factorization for gL(n) . We argue by induction on n. The case n = 3 can be verified directly. For general n, we apply repeatedly Lemma 4.4 in combination with Example 4.3. Let U (n) and V (n) denote the positive laws n−1

(xy)2

−4

= un (x, y)

n−2

and (xy)2

−2

= vn (x, y),

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respectively. According to the expressions (4) and (5) for σn and τn , we have  n n gU (n) (X) + X 2 X 5 − 1 + X 2 −2 (X − 1)   n−1 n−1 + X 2 +3 gV (n) (X) + X 2 −1 X 2 − 1 + gL(n−2) (X). n

gL(n) (X) = X 2

+5

(6)

Now a routine calculation shows that

gU (n)

  = (X − 1) X 3 + X 2 + X + 1

(n−3)/2 

 2i+2

X2

−8

i=1

   = (X − 1) X 2 + 1 Gn (X) − X 4 − X 3 − 1 /X 7

(7)

and

gV (n)

  = (X − 1) X 3 + X 2 + X + 1

(n−3)/2 

 X

22i+1 −6

i=1

    n n = (X − 1)(X + 1) Gn X 2 − X 2 − X 2 −2 − 1 /X 4 ,

(8)

whereas by the induction hypothesis,   gL(n−2) (X) = (X − 1)Gn−2 (X)Gn−2 X 2   n      n−1 = (X − 1) Gn (X) − (X + 1)X 2 −1 Gn X 2 − X 2 + 1 X 2 −2 .

(9)

By substituting the values of (7), (8) and (9) into (6), it follows without difficulty that gL(n) (X) = (X − 1)Gn (X)Gn (X 2 ), as desired. 2 We are now ready to proceed to the proof of Theorem 4.1. Proof of Theorem 4.1. Suppose first that p ≥ 3. As above, let Fp (X) = X 2 + X 2 −2 + p−2 p−2 X2 + X 2 −2 + · · · + X 8 + X 6 + 1. Since Fp (1) = p is congruent to 0 modulo p and p

 

(p−1)/2

Fp (1)

=

2

2i+1

+2

2i+1

−2 =

i=1

=

2

− 16 − (p − 1) ≡ 3

p+3





p

   22i+2 − 2

(p−1)/2

i=1

1 (mod p), 2 (mod p),

if p > 3, if p = 3,

it follows from Hensel’s Lemma (see Theorem 3.4.1 of [8]) that there exists t ∈ 1 + p Zp such that Fp (t) = 0. By the last lemma, also fL(n) (t) = gL(n) (t) = hL(n) (t) = 0. Consider now the semidirect product G = B  A described at the beginning of this section, where the element t of the construction is the one obtained in the last paragraph.

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As we know from Lemma 4.2, G is a metabelian 2-generator residually-p group which is not nilpotent-by-finite. On the other hand, since   (1, a)F ((t,0)) = 1, aF (t) for every a ∈ A and every polynomial F (X) ∈ Z[X], it follows that the endomorphisms fL(n) ((t, 0)), gL(n) ((t, 0)) and hL(n) ((t, 0)) annihilate A. By Lemma 4.6, the subset T = (t, 0)A ∪ A satisfies the positive law L(n). Thus G satisfies all properties in the statement of Theorem 4.1. If p = 2 then consider the law L(4) of degree 50 given by σ4 ≡ τ4 , where σ4 (x, y) = (xy)7 x2 y 2 (xy)10 x2 y 2 (xy)x2 (yx)y 2 and τ4 (x, y) = (yx)(xy)5 (yxxy)2 (xy)2 (yxxy)(xyyx)(yxxy)2 (yx)5 . It is easy to check that the polynomial F4 (X) = X 16 + X 4 + X + 1 is a factor of fL(4) , gL(4) and hL(4) . Since F4 (1) is congruent to 0 modulo 4 and F4 (1) ≡ 1 (mod 2), we can find a root t of F4 in 1 + 4 Z2 and we can argue as above. 2 Summarizing, Theorem 2.10 cannot be extended to cover all primes if the set T of generators satisfying a positive law is not power-closed, and the restriction p ∈ / P (n) is necessary in that case. This does not automatically imply that the condition p ∈ / P (n) is also necessary in the case of simple commutators satisfying a positive law, only that a new strategy, other than the one used in the current paper, should be developed if we want to try to get rid of that condition. Acknowledgments We thank A. Jaikin-Zapirain for helpful discussions. Also, we have made use of the computer algebra system GAP [7] in the search for positive laws with an appropriate common factor of the three polynomials associated to them, as in Lemma 4.7. More precisely, we have used GAP to find specific laws for the prime 3 and to check the validity of plausible generalizations of these laws to other small primes. References [1] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1969. [2] B. Bajorska, O. Macedońska, On positive law problems in the class of locally graded groups, Comm. Algebra 32 (2004) 1841–1846. [3] R.G. Burns, O. Macedońska, Y. Medvedev, Groups satisfying semigroup laws, and nilpotent-byBurnside varieties, J. Algebra 195 (2) (1997) 510–525. [4] R.G. Burns, Y. Medvedev, Groups laws implying virtual nilpotence, J. Aust. Math. Soc. 74 (2003) 295–312.

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[5] J.D. Dixon, M.P.F. du Sautoy, A. Mann, D. Segal, Analytic Pro-p Groups, 2nd ed., Cambridge Stud. Adv. Math., vol. 61, Cambridge University Press, 1999. [6] G.A. Fernández-Alcober, M. Morigi, Outer commutator words are uniformly concise, J. Lond. Math. Soc. 82 (2010) 581–595. [7] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4, http://www. gap-system.org, 2014. [8] F.Q. Gouvêa, p-Adic Numbers, Springer, 1997. [9] A. Jaikin-Zapirain, On the verbal width of finitely generated pro-p groups, Rev. Mat. Iberoamericana 168 (2008) 393–412. [10] E.I. Khukhro, p-Automorphisms of Finite p-Groups, London Math. Soc. Lecture Note Ser., vol. 246, Cambridge University Press, 1997. [11] D.M. Riley, P. Shumyatsky, Groups with a positive law on commutators, J. Pure Appl. Algebra 195 (3) (2005) 293–299. [12] L. Ribes, P. Zalesskii, Profinite Groups, second edition, Springer, 2010. [13] D. Segal, Words: Notes on Verbal Width in Groups, London Math. Soc. Lecture Note Ser., vol. 361, Cambridge University Press, 2009. [14] J.F. Semple, A. Shalev, Combinatorial conditions in residually finite groups, I, J. Algebra 157 (1) (1993) 43–50. [15] J. Tits, Free subgroups of linear groups, J. Algebra 20 (1972) 250–270. [16] J.S. Wilson, Two-generator conditions for residually finite groups, Bull. Lond. Math. Soc. 23 (1991) 239–248. [17] J.S. Wilson, E.I. Zelmanov, Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81 (1) (1992) 103–109. [18] E.I. Zelmanov, Nil Rings and Periodic Groups, KMS Lecture Notes in Math., Korean Mathematical Society, 1992.