ON PROFINITE GROUPS WITH ENGEL-LIKE CONDITIONS 5 1 RAIMUNDO BASTOS AND PAVEL SHUMYATSKY 0 2 n Abstract. LetG be aprofinitegroupinwhichforeveryelement a x∈Gthereexistsanaturalnumberq =q(x)suchthatxq isEngel. J We show that G is locally virtually nilpotent. Further, let p be a 2 primeandGafinitelygeneratedprofinitegroupinwhichforevery 2 γ -value x ∈ G there exists a natural p-power q = q(x) such that k xq is Engel. We show that γ (G) is locally virtually nilpotent. ] k R G . h at 1. Introduction m The positive solution of the Restricted Burnside Problem [14, 15] [ had led to many remarkable results on profinite groups. In particular, 1 usingWilson’sreductiontheorem[11],Zelmanovhasbeenabletoprove v 0 local finiteness of profinite periodic groups [16]. Recall that a group is 7 periodic if all of its elements have finite order. The group G is said to 6 5 have a certain property locally if any finitely generated subgroup of 0 G possesses that property. Another result that was deduced following . 1 the solution of the Restricted Burnside Problem is that any profinite 0 Engel group is locally nilpotent [12]. For elements x,y of G we define 5 1 [x,1y] = [x,y] and [x,i+1y] = [[x,iy],y] for i ≥ 1. The group G is called : an Engel group if for all x,y ∈ G there is an integer n = n(x,y) such v i that [x, y] = 1. An element y ∈ G is called Engel if for anyx ∈ G there X n is an integer n = n(x) such that [x, y] = 1; and y is called n-Engel if r n a for any x ∈ G we have [x, y] = 1. n In the present article we consider profinite groups in which a power of any element is Engel. Our result can be viewed as a common gener- alization of both of the above results. Theorem 1.1. Let G be a profinite group in which for every element x ∈ G there exists a natural number q = q(x) such that xq is Engel. Then G is locally virtually nilpotent. 2010 Mathematics Subject Classification. 20E18;20F45. Key words and phrases. Profinite groups; Engel elements. ThefirstauthorwaspartiallysupportedbyCAPESandCNPq-Brazil;thesecond author was supported by FAPDF and CNPq-Brazil. 1 2 RAIMUNDOBASTOS ANDPAVEL SHUMYATSKY Werecall thataprofinitegrouppossesses acertainpropertyvirtually if it has an open subgroup with that property. Given an integer k ≥ 1, the word γ = γ (x ,...,x ) is defined k k 1 k inductively by the formulae γ1 = x1, and γk = [γk−1,xk] = [x1,...,xk] for k ≥ 2. The subgroup of a group G generated by all values of the word γ is k denoted by γ (G). Of course, this is the familiar k-th term of the lower k central series of G. When G is a profinite group, γ (G) denotes the k closed subgroup generated by all values of the word γ . It was shown k in [8] that if G is a finitely generated profinite group in which all γ - k values are Engel, then γ (G) is locally nilpotent. In the present paper k we establish the following related result. Theorem 1.2. Let p be a prime and G a finitely generated profinite group in which for every γ -value x ∈ G there exists a natural p-power k q = q(x) such that xq is Engel. Then γ (G) is locally virtually nilpotent. k We do not know whether the hypothesis that G is finitely generated is really necessary in Theorem 1.2. The proof that we present here uses this assumption in a very essential way. Another natural question arising in the context of Theorem 1.2 is whether the theorem remains valid with q allowed to be an arbitrary natural number rather than p- power. This is related to the conjecture that if G is a finitely generated profinite group in which every γ -value has finite order, then γ (G) is k k locally finite (cf. [8]). Using the results obtained in [8] one can easily show that if G is a finitely generated profinite group in which every γ -value has finite p-power order, then indeed γ (G) is locally finite. k k This explains why Theorem 1.2 is proved only in the case where q is a p-power. 2. Preliminary results ′ As usual, if π is a set of primes, we denote by π the set of all primes that do not belong to π. For a profinite group G we denote by π(G) the set of prime divisors of the orders of elements of G (understood as su- pernatural, or Steinitz, numbers). If a profinite group G has π(G) = π, then we say that G is a pro-π group. The maximal normal pro-π sub- group of G is denoted by O (G). Recall that Sylow theorems hold for π p-Sylow subgroups of a profinite group (see, for example, [13, Ch. 2]). When dealing with profinite groups we consider only continuous homo- morphisms and quotients by closed normal subgroups. If X is a subset of a profinite group G, the symbol hXi stands for the closed subgroup ON PROFINITE GROUPS WITH ENGEL-LIKE CONDITIONS 3 generated by X. The symbol hXGi stands for the normal closed sub- group generated by X. Throughout the paper a profinite pronilpotent group is just called pronilpotent. Of course, a pronilpotent group is a Cartesian product of its Sylow subgroups. Lemma 2.1. Let p be a prime and G a pro-p′ group admitting an auto- morphism a of finite p-power order. Then h[x,a]Gi = h[x, a]Gi for any k positive integer k and any x ∈ G. Proof. Fix k and x ∈ G. Set M = h[x,a]Gi and N = h[x, a]Gi. It is k obvious that M contains N and so we only need to show that M ≤ N. Wecan pass to the quotient G/N and without loss of generality assume that [x, a] = 1. Thus, we need to show that [x,a] = 1. Assume that k [x,a] 6= 1 and let j be the minimal integer such that [x, a] = 1. Set j y = [x,j−2a] with the assumption that y = x if j = 2. Thus, we ′ have [y,a] 6= 1 but [y,a,a] = 1. Since [y,a] is a nontrivial p-element commuting with a, the order of [y,a]a−1 cannot be a p-power. On the other hand, it is clear that [y,a]a−1 = a−y is a conjugate of a−1 and so the order of [y,a]a−1 must be a p-power. This is a contradiction. (cid:3) Lemma 2.2. Let G be a pronilpotent group and assume that b and c are elements of coprime orders in G. Set a = bc. Then a is an n-Engel element if and only if so are both b and c. The element a is Engel if and only if both b and c are Engel. Proof. Assume that a is n-Engel. Choose a Sylow p-subgroup P of G. It is clear that if p does not divide the order of b, then [P,b] = 1. Assume that p is a divisor of the order of b. In this case p is coprime with the order of c and so [P,c] = 1. Therefore [P,b] = [P,a] and [P, b] = [P, a] = 1. This happens for every choice of a Sylow subgroup n n P of G. Since G is pronilpotent, it follows that [G, b] = 1. By a n symmetric argument, [G, c] = 1. Thus, if a is n-Engel, then so are n both b and c. Now assume that both b and c are n-Engel. Let π be the set of 1 primes dividing the order of b and π the set of primes dividing the 2 order of c. Given x ∈ G write x = x x x , where x is a π -element, x 1 2 3 1 1 2 is a π -element and x whose order is not divisible by primes in π ∪π . 2 3 1 2 Since G is pronilpotent, the elements x ,x ,x areuniquely determined 1 2 3 by x. We have [x, a] = [x , b] = [x , c] = 1. n 1 n 2 n Thus, a is n-Engel. Further, using only obvious modifications of the above argument one can show that a is Engel if and only if both b and c are Engel. (cid:3) 4 RAIMUNDOBASTOS ANDPAVEL SHUMYATSKY Lemma 2.3. Let p be a prime and G a pro-p group generated by a set X. Suppose that G can be generated by m elements. Then there exist elements x ,...,x ∈ X such that G = hx ,...,x i. 1 m 1 m Proof. A simple inverse limit argument shows that without loss of gen- erality we can assume that G is a finite p-group. Now existence of the elements x ,...,x ∈ X is immediate from Bursnide Basis Theorem 1 m (cid:3) [7, 5.3.2]. Lemma 2.4. Letm,nbe positiveintegersand G anm-generated pronilpo- tent group. Suppose that G can be generated by n-Engel elements. Then G can be generated by m elements each of which is n-Engel. Proof. Since G is pronilpotent, G is the direct product P ×P ×..., 1 2 where the P are Sylow subgroups of G. Let X be a set of n-Engel i elements such that G is generated by X. For every element x ∈ X write x = x x ..., where x is the projection of x on P . Set X = 1 2 i i i {x | x ∈ X}. Thus, X is the set of all x where x ranges over the set i i i X. By Lemma 2.2 each set X consists of n-Engel elements. It is clear i that P is generated by the set X . Since each Sylow subgroup P can i i i be generated by at most m elements, Lemma 2.3 implies that P can be i generatedby at most melements fromX . Ineach set X choose m (not i i necessarily distinct) elements x ,...,x such that P = hx ,...,x i. i1 im i i1 im For every j = 1,...,m we let y be the product of x , where i ranges j ij through the positive integers. By Lemma 2.2, the elements y are n- j Engel. It is clear that G = hy ,...,y i and so the lemma follows. 1 m (cid:3) Lemma 2.5. Let a be an Engel element of a pronilpotent group G. Then there exist a positive integer n and a finite set of primes π such that [Oπ′(G),na] = 1. Proof. For each positive integer i we set X = {b ∈ G; [b, a] = 1}. i i The sets X are closed in G and cover G. Therefore, by Baire’s cate- i gory theorem [2, p. 200], at least one of these sets contains non-empty interior. Hence, there exist a positive integer n, an open subgroup H and b ∈ G such that [bh, a] = 1 for every h ∈ H. Let m be the index n of H in G, and let π be the set of primes dividing m. Let J be the product of the Sylow subgroups of G corresponding to primes in π and K = Oπ′(H) = Oπ′(G). So G = J ×K and hence without loss of gen- erality we can assume that b ∈ J. Note that, 1 = [bx, a] = [b, a][x, a] n n n for every x ∈ K. Therefore [x, a] = 1 for any x ∈ K. The result n (cid:3) follows. ON PROFINITE GROUPS WITH ENGEL-LIKE CONDITIONS 5 Lemma 2.6. Let p be a prime and k a positive integer. Assume that G is a profinite group such that G = Op′(G)P, where P is a p-Sylow subgroup. Suppose that Op′(G) = hg1,...,gsi and P = hh1,...,hti. Then [Op′(G),P] is generated by conjugates of commutators [gi,khj] for 1 ≤ i ≤ s and 1 ≤ j ≤ t. Proof. Let D be the normal closure of all commutators [g , h ], where i k j 1 ≤ i ≤ s and 1 ≤ j ≤ t. By Lemma 2.1 this is the same as the normal closure of all commutators [gi,hj]. It is clear that D ≤ [Op′(G),P] and so we can pass to the quotient G/D and assume that D = 1. In this case we have G = hg1,...,gsi × hh1,...,hti and so [Op′(G),P] ≤ D. (cid:3) The lemma follows. Lemma 2.7. Let G be a profinite group and k,n positive integers with (n,p) = 1 for each p ∈ π(G). Let X be the set of all γ -values in G and k Y the set of n-th powers of γ -values. Then X = Y. k Proof. A simple inverse limit argument shows that without loss of gen- erality we can assume that G is finite. Since (|G|,n) = 1, it follows that |Y| = |X|. By [1] Y ⊆ X. Hence X = Y. (cid:3) Lemma 2.8. Let G be a profinite group in which every γ -value has k finite p-power order. If G is generated by p-elements, then G is a pro-p group. If G is generated by finitely many p-elements, then G is a finite p-group. Proof. Let Q be any finite quotient of G. By Lemma 3.3 of [9] we have γ (Q) ≤ O (Q). Since Q is generated by p-elements, it follows that Q k p is a finite p-groupand hence G is a pro-p group. In particular, Theorem 1.2 of [8] tells us that γ (G) is locally finite. Suppose now that G is k generated by finitely many p-elements. Then any nilpotent quotient of G is finite and therefore γ (G) is open. It follows that γ (G) is finitely k k generated. We already know that γ (G) is locally finite and so now we k conclude that γ (G) is finite. The lemma follows. (cid:3) k The following result is a straightforward corollary of Theorem 1.4 in [5]. Theorem 2.9. Let k be a positive integer and G a finitely generated profinite group. Then every element in γ (G) is a product of finitely k many γ -values. k 3. Associated Lie algebras In the present section we will describe the Lie theoretical tools that are employed in the proofs of our results. 6 RAIMUNDOBASTOS ANDPAVEL SHUMYATSKY Let Lbea Liealgebra over afield k. Weuse theleft normed notation: thus if l ,l ,...,l are elements of L, then 1 2 n [l ,l ,...,l ] = [...[[l ,l ],l ],...,l ]. 1 2 n 1 2 3 n An element a ∈ L is called ad-nilpotent if there exists a positive integer n such that [x, a] = 0 for all x ∈ L. If n is the least integer with n the above property then we say that a is ad-nilpotent of index n. Let X ⊆ L be any subset of L. By a commutator in elements of X we mean any element of L that could be obtained from elements of X by means of repeated operation of commutation with an arbitrary system of brackets including the elements of X. Denote by F the free Lie algebra over k on countably many free generators x ,x ,.... Let f = 1 2 f(x ,x ,...,x ) be a non-zero element of F. The algebra L is said to 1 2 n satisfy the identity f = 0 if f(l ,l ,...,l ) = 0 for any l ,l ,...,l ∈ L. 1 2 n 1 2 n In this case we say that L is PI. We are now in a position to quote a theorem of Zelmanov [17, III(0.4)] which has numerous important applications to group theory. Theorem 3.1. Let L be a Lie algebra generated by a ,a ,...,a . As- 1 2 m sume that L is PI and that each commutator in the generators is ad- nilpotent. Then L is nilpotent. Let G be a group and p a prime. We denote by D = D (G) the i-th i i dimension subgroup of G in characteristic p. These subgroups form a central series of G known as the Zassenhaus-Jennings-Lazard series. Set L(G) = LD /D . Then L(G) can naturally be viewed as a i i+1 Lie algebra over the field F with p elements. The subalgebra of L p generated by D /D will be denoted by L (G). The following result is 1 2 p due to Lazard [4]. Theorem 3.2. Let G be a finitely generated pro-p group. If L (G) is p nilpotent, then G is p-adic analytic. Let x ∈ G, and let i = i(x) be the largest integer such that x ∈ D . i We denote by x˜ the element xD ∈ L(G). We now cite two results i+1 providing sufficient conditions for x˜ to be ad-nilpotent. The following lemma is immediate from the proof of Lemma in [12, Section 3]. Lemma 3.3. Let x be an Engel element of a profinite group G. Then x˜ is ad-nilpotent. Lemma 3.4. (Lazard, [3, page 131]) For any x ∈ G we have (adx˜)p = ad(xp). In particular, if xq = 1, then x˜ is ad-nilpotent of index at most e q. ON PROFINITE GROUPS WITH ENGEL-LIKE CONDITIONS 7 We note that q in Lemma 3.4 does not need to be a p-power. In fact it is easy to see that if ps is the maximal p-power dividing q, then x˜ is ad-nilpotent of index at most ps. Combining Lemma 3.3 and Lemma 3.4, one obtains Lemma 3.5. Let x be an element of a profinite group G for which there exists a natural number q such that xq is Engel. Then x˜ is ad-nilpotent. Let H be a subgroup of G and a ,...,a ∈ G. Let w = w(x ,...,x ) 1 n 1 n beanontrivial element ofthefreegroupwithfreegeneratorsx ,...,x . 1 n Following [12] we say that the law w = 1 is satisfied on the cosets a H,...,a H if w(a h ,...,a h ) = 1 for any h ,...,h ∈ H. In [12] 1 n 1 1 n n 1 n Wilson and Zelmanov proved the following theorem. Theorem 3.6. If G is a group which has a subgroup H of finite index and elements a ,...,a such that a law w = 1 is satisfied on the cosets 1 n a H,...,a H, then for each prime p the Lie algebra L (G) is PI. 1 n p 4. Proofs of the main results Throughout the rest of the paper k stands for an arbitrary but fixed positive integer and G¯ = G × ... × G (k + 1 factors). First we will establish that if under the hypothesis of Theorem 1.2 all numbers q are coprime with the primes in π(G), then γ (G) is locally nilpotent. We k start with the case where G is a pro-p group. Proposition 4.1. Let G be a finitely generated pro-p group such that for any γ -value x ∈ G there exists a natural p′-number q such that xq k is Engel. Then γ (G) is locally nilpotent. k Proof. Since any element of γ (G) can be expressed as a product of k finitely many γ -values (Theorem 2.9), it is sufficient to show that any k subgroupK generatedbyfinitelymanyγ -valuesa ,...,a isnilpotent. k 1 s Let q1,...,qs be p′-numbers such that the elements aq11,...,aqss are Engel in K. It is clear that K = haq1,...,aqsi. For each pair of positive 1 s ′ integers i,j, where j is a p-number, we set S = {(g ,g ,...,g ) ∈ K¯; [g , [g ,...,g ]j] = 1}. i,j 1 2 k+1 1 i 2 k+1 Since the sets S are closed in K¯ and cover K¯, by Baire’s category i,j theorem at least one of them contains non-empty interior. Therefore there exist an open subgroup H of K, elements b,b ,...,b ∈ K 1 k and integers n,q such that the cosets bH,b H,...,b H satisfy the law 1 k [y, [x ,...,x ]q] ≡ 1. n 1 k By Theorem 3.6 L = L (K) is PI. Let a˜ ,...,a˜ be the homoge- p 1 s neous elements of L corresponding to a ,...,a . Since for any group 1 s 8 RAIMUNDOBASTOS ANDPAVEL SHUMYATSKY commutator h in a ,...,a there exists a p′-number q such that hq 1 s is Engel, Lemma 3.5 shows that any Lie commutator in a˜ ,...,a˜ is 1 s ad-nilpotent. Zelmanov’s Theorem 3.1 now tells us that L is nilpotent. ThereforeK isp-adicanalytic(Theorem3.2).Obviously K cannot con- tain a subgroup isomorphic to the free discrete group of rank two, so by the Tits’ Alternative [10] K has a soluble subgroup of finite index. Choose a dense subgroup D of K which is generated (as an abstract group) by finitely many Engel elements. By Plotkin’s theorem [6, The- orem 7.34] D is nilpotent. It follows that K is nilpotent, as well. The (cid:3) proof is complete. Theorem 4.2. Let π be a set of primes and G a finitely generated ′ pro-π group such that for any γ -value x ∈ G there exists a natural k π-number q such that xq is Engel. Then γ (G) is locally nilpotent. k Proof. It will be convenient first to prove the theorem under the addi- tional hypothesis that G is pronilpotent. For each pair of positive integers i,j, where j is a π-number, we set S = {(g ,g ,...,g ) ∈ G¯; [g , [g ,...,g ]j] = 1}. i,j 1 2 k+1 1 i 2 k+1 Arguingas intheproofofProposition4.1 we deduce that thereexist an open subgroup H of G, elements b,b ,...,b ∈ G and integers n,q such 1 k that the cosets bH,b H,...,b H satisfy the law [y, [x ,...,x ]q] ≡ 1. 1 k n 1 k Let m be the index of H in G. We write J for the product of the Sylow subgroups P ,...,P of G corresponding to the primes dividing 1 r m and K for the product of the Sylow subgroups of G corresponding to the primes not dividing m. Since G = J × K and since G = JH, without loss of generality we can assume that b,b ,...,b ∈ J and 1 k it follows that K satisfies the law [y, [x ,...,x ]q] ≡ 1. Lemma 2.7 n 1 k shows that K actually satisfies the law [y, [x ,...,x ]] ≡ 1. Hence n 1 k γ (K) is locally nilpotent by [8, Theorem 1.1]. Further, Proposition 4.1 k shows that γ (P ) is locally nilpotent for every i = 1,...,r. We have k i γ (G) = γ (P )×...×γ (P )×γ (K). Thus, it follows that in the case k k 1 k r k where G is pronilpotent γ (G) is locally nilpotent. k Now we drop the assumption that G is pronilpotent. Since any ele- ment of γ (G) can be expressed as a product of finitely many γ -values k k (Theorem 2.9), it is sufficient to prove that any subgroup generated by finitely many γ -values a ,...,a is nilpotent. Set H = ha ,...,a i. We k 1 t 1 t need to show that H is nilpotent. Let q ,...,q be positive π-numbers 1 t such that aq1,...,aqt are Engel. It is clear that H = haq1,...,aqti. 1 t 1 t Since all Engel elements of a finite group lie in the Fitting subgroup [7, 12.3.7], it follows that H is pronilpotent. Hence, by the previous ON PROFINITE GROUPS WITH ENGEL-LIKE CONDITIONS 9 paragraph, γ (H) is locally nilpotent. The theorem now follows from k (cid:3) the result of Plotkin just as in the proof of Proposition 4.1. Now we will deal with the situation where the numbers q are not necessarily coprime with the primes in π(G). Proposition 4.3. Let k be positive integer and G a finitely generated pronilpotent group in which for every γ -value x there is an integer q k such that xq is Engel. Then γ (G) is locally virtually nilpotent. k Proof. For each pair of positive integers i,j we set S = {(g ,...,g ) ∈ G¯; [g , [g ,...,g ]j] = 1}. i,j 1 k+1 1 i 2 k+1 Arguing as in the proof of Proposition 4.1 we deduce that there exist an open subgroup H of G, elements b,b ,...,b ∈ G and integers n,q 1 k such that [bh, [b h ,...,b h ]q] = 1 n 1 1 k k for every h,h ,...,h ∈ H. Let m be the index of H in G. We write 1 k J for the product of the Sylow subgroups of G corresponding to the primes dividing m and K for the product of the Sylow subgroups of G corresponding to the primes not dividing m. Since G = J×K and since G = JH, without loss of generality we can assume that b,b ,...,b ∈ J 1 k and it follows that K satisfies the law [y, [x ,...,x ]q] ≡ 1. Let π be n 1 k the set of prime divisors of q. Then γk(Oπ′(K)) is locally nilpotent by Theorem 4.2. Now it suffices to show that γ (P) is locally virtually nilpotent for k every Sylow subgroup P corresponding to a prime dividing mq. There is no loss of generality in assuming that G is pro-p group. Since every element of γ (G) is a product of finitely many γ -values, it is sufficient k k to prove that any subgroup generated by finitely many γ -values is vir- k tually nilpotent. Let a ,...,a be γ -values in G and M = ha ,...,a i. 1 t k 1 t Our argument will now imitate parts of the proof of Proposition 4.1. By Theorem 3.6 L = L (M) is PI. Let a˜ ,...,a˜ be the homogeneous p 1 s elements of L corresponding to a ,...,a . Since for any group com- 1 s mutator h in a ,...,a there exists a positive integer q such that hq 1 s is Engel, Lemma 3.5 shows that any Lie commutator in a˜ ,...,a˜ is 1 s ad-nilpotent. Theorem 3.1 now tells us that L is nilpotent. Therefore, M is p-adic analytic (Lemma 3.2). Obviously M cannot contain a sub- group isomorphic to the free discrete group of rank two, so by the Tits’ Alternative [10] M has a soluble subgroup of finite index. Let E be the closed subgroup of M generated by all Engel elements in G con- tained in M. Lemma 2.8 implies that M/E is finite and therefore E is finitely generated. Since E is generated by all Engel elements con- tained in M and since E is finitely generated, Lemma 2.3 implies that 10 RAIMUNDOBASTOS ANDPAVEL SHUMYATSKY E is generated by finitely many Engel elements. Choose an abstract dense subgroup D in E generated by finitely many Engel elements. By Plotkin’s theorem [6, Theorem 7.34], D is nilpotent. Hence, also E is (cid:3) nilpotent. The result follows. The proof of Theorem 1.1 is now easy. Proof of Theorem 1.1. Recall that G is a profinite group in which for every element x ∈ G there exists a natural number q = q(x) such that xq is Engel. We need to show that finitely generated subgroups of G are virtually nilpotent. Let H be a finitely generated closed subgroup of G, and let K be the subgroup generated by all Engel elements in H. By the hypothe- sis H/K is a finitely generated periodic profinite group. According to Zelmanov’s Theorem [16], H/K is finite. In particular, K is a finitely generated pronilpotent group. By Proposition 4.3 applied with k = 1, K is virtually nilpotent. This completes the proof. (cid:3) The proof of Theorem 1.2 will be only somewhat more complicated. Proof of Theorem 1.2. Recall that G is a finitely generated profinite group in which some p-power of any γ -value is Engel. We wish to show k that finitely generated subgroups of γ (G) are virtually nilpotent. k The case where k = 1 is covered by Theorem 1.1. So we will assume thatk ≥ 2. Since anyelement ofγ (G) canbeexpressed asa product of k finitely many γ -values (Theorem 2.9), it is sufficient to prove that any k subgroup generated by finitely many γ -values a ,...,a is virtually k 1 t nilpotent. Let H = ha ,...,a i, and let K be the closed subgroup of H 1 t generated by all Engel elements of H. By Lemma 2.8 H/K is a finite p- group and so K is finitely generated. We will show that K is nilpotent. It is clear that K is pronilpotent and Op′(H) = Op′(K). It follows that Op′(H) is finitely generated. Choose a Sylow p-subgroup P in H. Since H/Op′(H) is isomorphic with P, we conclude that P is finitely generated. Let P = hh1,...,hri and Op′(H) = hg1,...,gsi. By Lemma 2.6 [Op′(H),P] is generated by conjugates of [gi,k−1hj]. Of course every element [gj,k−1hl] is a γk-value. Let qjl be positive p-powers such that [gj,k−1hl]qjl are Engel, for all j ∈ {1,...,s} and l ∈ {1,...,r}. ′ For each i = 1,...,t write a = b c , where b is a generator of the p- i i i i part of the procyclic subgroup ha i and c is a generator of the p-part i i of the subgroup haii. Since H/[Op′(H),P] is pronilpotent, it follows that Op′(H) is generated by [Op′(H),P] and the elements b1,...,bt. Let q1,...,qt be p-powers such that aiqi is Engel, for any i = 1,...,t. Thus, all elements b q1,...,b qt are Engel (Lemma 2.2). Further, since 1 t all qi and qjl are p-powers, it follows that Op′(H) is generated by