1 Introduction

Given a group G, an element \(g\in G\) is called a (left) Engel element if for any \(x\in G\) there exists a positive integer \(n=n(x,g)\) such that \([x,_n g]=1\), where the commutator \([x,_n g]\) is defined inductively by the rules

$$\begin{aligned}{}[x,_1 g]=[x,g]=x^{-1}g^{-1}xg\quad \mathrm{and,\, for}\; n\ge 2,\quad [x,_n g]=[[x,_{n-1} g],g]. \end{aligned}$$

If n can be chosen independently of x, then g is called a (left) n-Engel element, or more generally a bounded (left) Engel element. The group G is an Engel group (resp. an n-Engel group) if all its elements are Engel (resp. n-Engel).

A celebrated result due to Zelmanov [24,25,26] refers to the positive solution of the Restricted Burnside Problem (RBP for short): every residually finite group of bounded exponent is locally finite. The group G is said to have a certain property locally if any finitely generated subgroup of G possesses that property. An interesting result in this context, due to Wilson [21], states that every n-Engel residually finite group is locally nilpotent. Another result that was deduced following the positive solution of the RBP is that given positive integers m, n, if G is a residually finite group in which for every \(x \in G\) there exists a positive integer \(q=q(x) \leqslant m\) such that \(x^q\) is n-Engel, then G is locally virtually nilpotent [1]. We recall that a group possesses a certain property virtually if it has a subgroup of finite index with that property. For more details concerning Engel elements in residually finite groups see [1,2,3, 17, 18].

One of the goals of the present article is to study residually finite groups in which some powers are bounded Engel elements. We establish the following result.

Theorem A

Let q be a prime. Let G be a residually finite group satisfying an identity. Suppose that for every \(x \in G\) there exists a q-power \(m=m(x)\) such that the element \(x^m\) is a bounded Engel element. Then G is locally virtually nilpotent.

A natural question arising in the context of the above theorem is whether the theorem remains valid with m allowed to be an arbitrary natural number rather than q-power. This is related to the conjecture that if G is a residually finite periodic group satisfying an identity, then the group G is locally finite [23, p. 400]. Note that the hypothesis that G satisfies an identity is really needed. For instance, it is well known that there are residually finite p-groups that are not locally finite [5]. In particular, these groups cannot be locally virtually nilpotent. Similar examples have been obtained independently by Grigorchuk, Gupta-Sidki and Sushchansky and are published in [6, 7, 19], respectively.

Recall that a group-word \(w=w(x_1,\dots ,x_s)\) is a nontrivial element of the free group \(F = F(x_1,\dots ,x_s)\) on free generators \(x_1,\dots ,x_s\). A word is a commutator word if it belongs to the commutator subgroup \(F'\). A non-commutator word u is a group-word such that the sum of the exponents of some variable involved in it is non-zero. A group-word w can be viewed as a function defined in any group G. The subgroup of G generated by the w-values is called the verbal subgroup of G corresponding to the word w. It is usually denoted by w(G). However, if k is a positive integer and \(w = x_1^k\), it is customary to write \(G^{k}\) rather than w(G).

There is a well-known quantitative version of Wilson’s theorem, that is, if G is a d-generator residually finite n-Engel group, then G has \(\{d,n\}\)-bounded nilpotency class. As usual, the expression “\(\{a,b,...\}\)-bounded” means “bounded from above by some function which depends only on parameters a, b, ...”. We establish the following related result.

Theorem B

Let d, n be positive integers and w a non-commutator word. Assume that G is a d-generator residually finite group in which all w-values are n-Engel. Then the verbal subgroup w(G) has \(\{d,n,w\}\)-bounded nilpotency class.

A non-quantitative version of the above theorem already exists in the literature. It was obtained in [3, Theorem C].

The paper is organized as follows. In the next section we describe some important ingredients of what are often called “Lie methods in group theory”. Theorems A and B are proved in Sects. 3 and 4, respectively. The proofs of the main results rely of Zelmanov’s techniques that led to the solution of the RBP [24,25,26], Lazard’s criterion for a pro-p group to be p-adic analytic [9], and a result of Nikolov and Segal [13] on verbal width in groups.

2 Associated Lie algebras

Let L be a Lie algebra over a field \({\mathbb {K}}\). We use the left normed notation: thus if \(l_1,l_2,\dots ,l_n\) are elements of L, then

$$\begin{aligned}{}[l_1,l_2,\dots ,l_n]=[\dots [[l_1,l_2],l_3],\dots ,l_n]. \end{aligned}$$

We recall that an element \(a\in L\) is called ad-nilpotent if there exists a positive integer n such that \([x,{}_na]=0\) for all \(x\in L\). When n is the least integer with the above property then we say that a is ad-nilpotent of index n.

Let \(X\subseteq L\) be any subset of L. By a commutator of elements in X, we mean any element of L that can 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 \({\mathbb {K}}\) on countably many free generators \(x_1,x_2,\dots \). Let \(f=f(x_1,x_2,\dots ,x_n)\) be a non-zero element of F. The algebra L is said to satisfy the identity \(f\equiv 0\) if \(f(l_1,l_2,\dots ,l_n)=0\) for any \(l_1,l_2,\dots ,l_n\in L\). In this case we say that L is PI. Now, we recall an important theorem of Zelmanov [23, Theorem 3] that has many applications in group theory.

Theorem 2.1

Let L be a Lie algebra over a field generated by a finite set. Assume that L is PI and that each commutator in the generators is ad-nilpotent. Then L is nilpotent.

2.1 On Lie algebras associated with groups

Let G be a group and p a prime. Let us denote by \(D_i=D_i(G)\) the i-th dimension subgroup of G in characteristic p. These subgroups form a central series of G known as the Zassenhaus–Jennings–Lazard series (see [8, p. 250] for more details). Set \(L(G)=\bigoplus D_i/D_{i+1}\). Then L(G) can naturally be viewed as a Lie algebra over the field \({{\mathbb {F}}}_p\) with p elements.

The subalgebra of L(G) generated by \(D_1/D_2\) will be denoted by \(L_p(G)\). The nilpotency of \(L_p(G)\) has strong influence in the structure of a finitely generated pro-p group G. According to Lazard [10] the nilpotency of \(L_p(G)\) is equivalent to G being p-adic analytic (for details see [10, A.1 in Appendice and Sections 3.1 and 3.4 in Ch. III] or [4, 1.(k) and 1.(o) in Interlude A]).

Theorem 2.2

Let G be a finitely generated pro-p group. If \(L_p(G)\) is nilpotent, then G is p-adic analytic.

Let \(x\in G\) and let \(i=i(x)\) be the largest positive integer such that \(x\in D_i\) (here, \( D_i \) is a term of the p-dimensional central series to G). We denote by \({\tilde{x}}\) the element \(xD_{i+1}\in L(G)\). We now quote two results providing sufficient conditions for \({\tilde{x}}\) to be ad-nilpotent. The first lemma was established in [9, p. 131].

Lemma 2.3

For any \(x\in G\) we have \((ad\,{{{\tilde{x}}}})^p=ad\,(\widetilde{x^p})\). Consequently, if x is of finite order t then \({\tilde{x}}\) is ad-nilpotent of index at most t.

Corollary 2.4

Let x be an element of a group G for which there exists a positive integer m such that \(x^m\) is n-Engel. Then \({\tilde{x}}\) is ad-nilpotent.

The following result was established by Wilson and Zelmanov in [22].

Lemma 2.5

Let G be a group satisfying an identity. Then for each prime number p the Lie algebra \(L_p(G)\) is PI.

3 Proof of Theorem A

Recall that a group is locally graded if every nontrivial finitely generated subgroup has a proper subgroup of finite index. Interesting classes of groups (e.g., locally finite groups, locally nilpotent groups, residually finite groups) are locally graded (see [11, 12] for more details).

It is easy to see that a quotient of a locally graded group need not be locally graded (see for instance [14, 6.19]). However, the next result gives a sufficient condition for a quotient to be locally graded [11].

Lemma 3.1

Let G be a locally graded group and N a normal locally nilpotent subgroup of G. Then G / N is locally graded.

In [23], Zelmanov has shown that if G is a residually finite p-group which satisfies a nontrivial identity, then G is locally finite. Next, we extend this result to the class of locally graded groups.

Lemma 3.2

Let p be a prime. Let G be a locally graded p-group which satisfies an identity. Then G is locally finite.

Proof

Choose arbitrarily a finitely generated subgroup H of G. Let R be the finite residual of H, i.e., the intersection of all subgroups of finite index in H. If \(R=1\), then H is a finitely generated residually finite group. By Zelmanov’s result [23, Theorem 4], H is finite. So it suffices to show that H is residually finite. We argue by contradiction and suppose that \(R \ne 1\). By the above argument, H / R is finite and thus R is finitely generated. As R is locally graded we have that R contains a proper subgroup of finite index in H, which gives a contradiction. Since H be chosen arbitrarily, we now conclude that G is locally finite, as well. The proof is complete. \(\square \)

We denote by \({\mathcal {N}}\) the class of all finite nilpotent groups. The following result is a straightforward corollary of [21, Lemma 2.1] (see [15, Lemma 3.5] for details).

Lemma 3.3

Let G be a finitely generated residually-\({\mathcal {N}}\) group. For each prime p, let \(R_p\) be the intersection of all normal subgroups of G of finite p-power index. If \(G/R_p\) is nilpotent for each prime p, then G is nilpotent.

We are now in a position to prove Theorem A.

Proof of Theorem A

Recall that G is a residually finite group satisfying an identity in which for every \(x \in G\) there exists a q-power \(m=m(x)\) such that the element \(x^m\) is a bounded Engel element. We need to prove that every finitely generated subgroup of G is virtually nilpotent.

Firstly, we prove that all bounded Engel elements (in G) are contained in the Hirsch-Plotkin radical of G. Let H be a subgroup generated by finitely many bounded Engel elements in G, say \(H = \langle h_1, \ldots , h_t \rangle \), where \(h_i\) is a bounded Engel element in G for every \(i=1,\ldots ,t\). Since finite groups generated by Engel elements are nilpotent [14, 12.3.7], we can conclude that H is residually-\({\mathcal {N}}\). As a consequence of Lemma 3.3, we can assume that G is residually-(finite p-group) for some prime p. Let \(L=L_p(H)\) be the Lie algebra associated with the Zassenhaus–Jennings–Lazard series

$$\begin{aligned} H=D_1\ge D_2\ge \cdots \end{aligned}$$

of H. Then L is generated by \({\tilde{h}}_i=h_i D_2\), \(i=1,2,\dots ,t\). Let \({\tilde{h}}\) be any Lie-commutator in \({\tilde{h}}_i\) and h be the group-commutator in \(h_i\) having the same system of brackets as \({\tilde{h}}\). Since for any group commutator h in \(h_1\dots ,h_t\) there is a q-power \(m=m(h)\) and a positive integer \(n=n(h)\) such that \(h^m\) is n-Engel, Corollary 2.4 shows that any Lie commutator in \(\tilde{h}_1\dots ,{{\tilde{h}}}_t\) is ad-nilpotent. On the other hand, H satisfies an identity and therefore, by Lemma 2.5, L satisfies some non-trivial polynomial identity. According to Theorem 2.1L is nilpotent. Let \({\hat{H}}\) denote the pro-p completion of H. Then \(L_p({\hat{H}})=L\) is nilpotent and \({\hat{H}}\) is a p-adic analytic group by Theorem 2.2. By [4, 1.(n) and 1.(o) in Interlude A]), \({\hat{H}}\) is linear, and so therefore is H. Clearly H cannot have a free subgroup of rank 2 and so, by Tits’ Alternative [20], H is virtually soluble. By [14, 12.3.7], H is nilpotent. Since \(h_1,\dots ,h_t\) have been chosen arbitrarily, we now conclude that all bounded Engel elements are in the Hirsch-Plotkin radical of G.

Let H be a finitely generated subgroup of G, and K be the subgroup generated by all bounded Engel elements (in G) contained in H. Now, we need to prove that K is a nilpotent subgroup of finite index in H. By the previous paragraph, K is locally nilpotent. By Lemma 3.1, H / K is a locally graded q-group. Since G satisfies a nontrivial identity, by Lemma 3.2, H / K is finite and so, K is finitely generated. From this we deduce that K is nilpotent. The proof is complete. \(\square \)

4 Proof of Theorem B

Combining the positive solution of the RBP with the result [3, Theorem C] one can show that if u is a non-commutator word and G is a finitely generated residually finite group in which all u-values are n-Engel, then the verbal subgroup u(G) is nilpotent. This section is devoted to obtain a quantitative version of the aforementioned result.

The proof of Theorem B require the following lemmas.

Lemma 4.1

Let d, m, n positive integers. Let G be a d-generator residually finite group in which \(x^m\) is n-Engel for every \(x\in G \). Then the subgroup \(G^m\) has \( \{d,m,n\} \)-bounded nilpotency class.

Proof

Let \(H=G^m\). By [3, Theorem C], H is locally nilpotent. Moreover, Lemma 3.1 ensures us that the quotient group G / H is locally graded. By Zelmanov’s solution of the RBP, locally graded groups of finite exponent are locally finite (see for example [12, Theorem 1]), and so G / H is finite of \(\{d,m\}\)-bounded order. We can deduce from [14, Theorem 6.1.8(ii)] that H has \(\{d,m\}\)-boundedly many generators. In particular, H is nilpotent. In order to complete the proof, we need to show that H has \(\{d,m,n\}\)-bounded class yet.

Note that there exists a family of normal and finite index subgroups \(\{N_i\}_{i\in {\mathcal {I}}}\) in G which are all contained in H such that H is isomorphic to a subgroup of the Cartesian product of the finite quotients \(H/N_i\). We show that all quotients have \(\{d,m,n\}\)-bounded class. Indeed, we have \(H/N_i=(G/N_i)^m\). Note that H is \(\{d,m\}\)-boundedly generated. Thus, by [13, Theorem 1], \(H/N_i\) is \(\{d,m\}\)-boundedly generated where any generator is an m-th power which is an n-Engel element. By [17, Lemma 2.2], there exists a number c depending only on \(\{d,m,n\}\) such that each factor \(H/N_i\) has nilpotency class at most c. So H is of nipotency class at most c, as well. The proof is complete. \(\square \)

A well known theorem of Gruenberg says that a soluble group generated by finitely many Engel elements is nilpotent (see [14, 12.3.3]). We will require a quantitative version of this theorem whose proof can be found in [16, Lemma 4.1].

Lemma 4.2

Let G be a group generated by m elements which are n-Engel and suppose that G is soluble with derived length d. Then G is nilpotent of \(\{d,m,n\}\)-bounded class.

For the reader’s convenience we restate Theorem B.

Theorem B

Let d, n be positive integers and w a non-commutator word. Assume that G is a d-generator residually finite group in which all w-values are n-Engel. Then the verbal subgroup w(G) has \(\{d,n,w\}\)-bounded nilpotency class.

Proof

Let \(w = w(x_1,...,x_r)\) be a non-commutator word. We may assume that the sum of the exponents of \(x_1\) is \(k \ne 0\). Substitute 1 for \(x_2,...,x_r\) and an arbitrary element \(g \in G\) for \(x_1\). We see that \(g^k\) is a w-value for every \(g \in G\). Thus every k-th power is n-Engel in G. Lemma 4.1 ensures that \(G^k\) has \(\{d,n,w,\}\)-bounded nilpotency class.

Following an argument similar to that used in the proof of Lemma 4.1 we can deduce that the verbal subgroup w(G) is nilpotent. By Zelmanov’s solution of the RBP, locally graded groups of finite exponent are locally finite (see for example [12, Theorem 1]), and so \(G/G^k\) is finite of \(\{d,w\}\)-bounded order. Thus, the verbal subgroup w(G) has \(\{d,m,w\}\)-bounded derived length.

Note that there exists a family of normal and finite index subgroups \(\{N_i\}_{i\in {\mathcal {I}}}\) in G that are all contained in w(G) such that w(G) is isomorphic to a subgroup of the Cartesian product of the finite quotients \(w(G)/N_i\). We show that all quotients \(w(G)/N_i\) have \(\{d,n,w\}\)-bounded class. Indeed, we have \(w(G)/N_i=w(G/N_i)\). We also have w(G) is \(\{d,w\}\)-boundedly generated. By [13, Theorem 3] each quotient \(w(G)/N_i\) is \(\{d,w\}\)-boundedly generated by w-values which are n-Engel elements. Since w(G) has \(\{d,m,w\}\)-bounded derived length, according to Lemma 4.2 we can deduce that \(w(G)/N_i\) has \(\{d,n,w\}\)-bounded nilpotency class Thus, w(G) has \(\{d,n,w\}\)-bounded nilpotency class, as well. This completes the proof. \(\square \)