1 Introduction

Let n be a positive integer. We say that a group G is (left) n-Engel if it satisfies the identity \([y,{}_n\,x]\equiv 1\), where the word \( [x,_n y] \) is defined inductively by the rules

$$\begin{aligned}{}[x,_1y]=x^{-1}y^{-1}xy,\ \ \ [x,_ny]=[[x,_{n-1}y],y ] \ \ \hbox { for all } n\ge 2. \end{aligned}$$

A important theorem of Wilson [13, Theorem 2] says that finitely generated residually finite n-Engel groups are nilpotent. More specific properties of residually finite n-Engel groups can be found for example in a theorem of Burns and Medvedev (quoted below as Theorem 5) stating that there exist functions c(n) and e(n) such that any residually finite n-Engel group G has a nilpotent normal subgroup N of class at most c(n) such that the quotient group G/N has exponent dividing e(n) . The interested reader is referred to the survey [12] and references therein for further results on finite and residually finite Engel groups. The purpose of the present article is to provide the proof for the following theorem.

Theorem 1

Let G be a finitely generated residually finite group satisfying the identity \( [x,_ny^q]\equiv 1. \) Then there exists a function f(n) such that G has a nilpotent subgroup of finite index of class at most f(n) .

A group is called locally graded if every non-trivial finitely generated subgroup has a proper subgroup of finite index. The class of locally graded groups contains locally (soluble-by-finite) groups as well as residually finite groups. We can extend the Theorem 1 to the class of locally graded groups.

Corollary 1

Let G be a finitely generated locally graded group satisfying the identity \( [x,_ny^q]\equiv 1. \) Then there exists a function f(n) such that G has a nilpotent subgroup of finite index of class at most f(n) .

In the next section we describe the Lie-theoretic machinery that will be used in the proof of Theorem 1. The proof of the theorem and of the corollary is given in Sect. 3.

2 About Lie algebras

Let L be a Lie algebra over a field K and X a subset of L. By a commutator in elements of X we mean any element of L that can be obtained as a Lie product of elements of X with some system of brackets. If \(x_1,\ldots ,x_k,x, y\) are elements of L, we define inductively

$$\begin{aligned}{}[x_1]=x_1; [x_1,\ldots ,x_k]=[[x_1,\ldots ,x_{k-1}],x_k] \end{aligned}$$

and \([x,_0y]=x; [x,_my]=[[x,_{m-1}y],y],\) for all positive integers k, m. As usual, we say that an element \(a\in L\) is ad-nilpotent if there exists a positive integer n such that \([x,_na]=0\) for all \(x\in L\). Denote by F the free Lie algebra over K on countably many free generators \(x_1,x_2,\ldots \). Let \(f=f(x_1,x_2,\ldots ,x_n)\) be a non-zero element of F. The algebra L is said to satisfy the identity \(f \equiv 0\) if \(f(l_1,l_2,\ldots ,l_n) = 0\) for any \(l_1,l_2,\ldots ,l_n\in L\).

The next theorem represents the most general form of the Lie-theoretical part of the solution of the Restricted Burnside Problem [15, 17, 18]. It was announced by Zelmanov [15]. A detailed proof can be found in [18].

Theorem 2

Let L be a Lie algebra over a field and suppose that L satisfies a polynomial identity. If L can be generated by a finite set X such that every commutator in elements of X is ad-nilpotent, then L is nilpotent.

2.1 Associating a Lie ring to a group

Let G be a group. A series of subgroups

$$\begin{aligned} G=G_1\ge G_2\ge \dots \end{aligned}$$
(*)

is called an N-series if it satisfies \([G_i,G_j]\le G_{i+j}\) for all \(i,j\ge 1\). Obviously any N-series is central, i.e. \(G_i/G_{i+1}\le Z(G/G_{i+1})\) for any i. Let p be a prime. An N-series is called \(N_p\)-series if \(G_i^p\le G_{pi}\) for all i. Given an N-series \((*)\), let \(L^*(G)\) be the direct sum of the abelian groups \(L_i^*=G_i/G_{i+1}\), written additively. Commutation in G induces a binary operation \([\cdot ,\cdot ]\) in \(L^*(G)\). For homogeneous elements \(xG_{i+1}\in L_i^*,yG_{j+1}\in L_j^*\) the operation is defined by

$$\begin{aligned}{}[xG_{i+1},yG_{j+1}]=[x,y]G_{i+j+1}\in L_{i+j}^* \end{aligned}$$

and extended to arbitrary elements of \(L^*(G)\) by linearity. It is easy to check that the operation is well-defined and that \(L^*(G)\) with the operations \(+\) and \([\cdot ,\cdot ]\) is a Lie ring. If all quotients \(G_i/G_{i+1}\) of an N-series \((*)\) have prime exponent p then \(L^*(G)\) can be viewed as a Lie algebra over the field with p elements. In the important case where the series \((*)\) is the p-dimension central series (also known under the name of Zassenhaus-Jennings-Lazard series) of G we write \(D_i=D_i(G)=\prod _{jp^k\ge i} \gamma _j(G)^{p^k}\) for the i-th term of the series of G, L(G) for the corresponding associated Lie algebra over the field with p elements and \(L_p(G)\) for the subalgebra generated by the first homogeneous component \(D_1/D_2\) in L(G). Observe that the p-dimension central series is an \(N_p\)-series (see [5, p. 250] for details).

The nilpotency of \(L_p(G)\) has strong influence in the structure of a finitely generated pro-p group G. The proof of the following theorem can be found in [4, 1.(k) and 1.(o) in Interlude A].

Theorem 3

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 following lemma was established in [6, p. 131].

Lemma 1

For any \(x\in G\) we have \((ad\,{\tilde{x}})^p=ad\,(\widetilde{x^p})\).

Corollary 2

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 theorem is a particular case of a result that was established by Wilson and Zelmanov in [14].

Theorem 4

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

3 Proof of the main theorem

The following useful result is a consequence of [13, Lemma 2.1] (see also [11, Lemma 3.5] for details).

Lemma 2

Let G be a finitely generated residually finite-nilpotent group. For each prime p let \( J_p \) be the intersection of all normal subgroups of G of finite p-power index. If \( G/J_p \) has a nilpotent subgroup of finite index of class at most c for each p, then G also has a nilpotent subgroup of finite index of class at most c.

Proof

It follows from proof of [11, Lemma 3.5] that there exists a finite set of primes \( \pi \) such that G embeds in the direct product \(\prod _{p\in \pi } G/J_{p}. \) We will identify G with its images in direct product. By hypothesis, for any \( p\in \pi \), \( G/J_{p} \) contains a nilpotent subgroup of finite index \(H_p \) with class at most c. Set \( H=\cap _{p\in \pi } {H_p} \). Thus, \( G\cap H \) has finite index in G and has nilpotency class at mos c, which completes the proof.

Recall that a group is locally graded if every non-trivial finitely generated subgroup has a proper subgroup of finite index. Note that the quotient of a locally graded group need not be locally graded, since free groups are locally graded (see [10, 6.1.9]), but no finitely generated infinite simple group is locally graded. However, the following results give a sufficient conditions for a quotient to be locally graded (see [7] for details).

Lemma 3

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

Let p be a prime and q be a positive integer. A finite p-group G is said to be powerful if and only if \( [G,G]\le G^p \) for \( p\ne 2 \) (or \([G,G]\le G^4 \) for \( p=2 \)), where \( G^q \) denotes the subgroup of G generated by all qth powers. While considering a pro-p group G we shall be interested only in closed subgroups. So by the commutator subgroup \( G'=[G,G] \) we mean the closed commutator subgroup, \( G^q \) means the closed subgroup generated by the qth powers. Similarly to powerful finite p-groups, we may define the powerful pro-p groups. For more details we refer the reader to [7, Chapters 2 and 3 ]. In [1] the following useful result for powerful finite n-Engel p-group was established.

Lemma 4

There exists a function s(n) such that any powerful finite n-Engel p-group is nilpotent of class at most s(n) .

The proof of Theorem 1 will requires the following lemma.

Lemma 5

Let s(n) be as in Lemma 4. If G is a finitely generated powerful pro-p group satisfying the identity \( [x,_ny^q]\equiv 1, \) then \( G^q \) has nilpotency class at most s(n) .

Proof

Since G satisfies the identity \( [x,_ny^q]\equiv 1, \) we can deduce from [4, Corollary 3.5] that \( H=G^q=\{x^q\ | \ x\in G\} \) is a powerful n-Engel pro-p group. According to [4, Corollary 3.3], H is the inverse limit of an inverse system of powerful finite p-groups \( H_\lambda \). Lemma 4 implies that any group \( H_\lambda \) has class at most s(n) , and so, H has class at most s(n) as well. Finally, by a result due to Zelmanov [16, Theorem 1] saying that any torsion profinite group is locally finite we get that the quotient group G/H is finite. This completes the proof. \(\square \)

The proof of Theorem 1 will also require the following result, due to Burns and Medvedev [3].

Theorem 5

There exist functions c(n) and e(n) such that any residually finite n-Engel group G has a nilpotent normal subgroup N of class at most c(n) such that G/N has exponent dividing e(n) . \(\square \)

We are now ready to embark on the proof of our main result.

Proof of Theorem 1

For any positive integer n let s(n) and c(n) be as in Lemma 4 and Theorem 5, respectively. Set \( f(n)=\max \{s(n),c(n)\} \). Since G satisfies the identity \( [x,_ny^q]\equiv 1 \) we can deduce from [2, Theorem A] that \( H=G^q \) is locally nilpotent. According to Lemma 3, G/H is locally graded. By Zelmanov’s solution of the Restricted Burnside Problem [15, 17, 18], locally graded groups of finite exponent are locally finite (see for example [8, Theorem 1]), and so G/H is finite. Thus H is finitely generated and so it is nilpotent.

By Lemma 2, we can assume that H is residually (finite p-group) for some prime p. If p does not divides q, then H is finitely generated residually finite n-Engel group. By Theorem 5, H contains a nilpotent normal subgroup N of class at most f(n) such that the quotient group G/N has exponent dividing e(n) . Thus, we can see that G/N is finite. Thereby, in what follows we can assume that H is residually (finite p-group), where p divides q.

Set \( H=\langle h_1,\ldots ,h_t\rangle . \) Let \( L=L_p(H) \) be the Lie algebra associated with the p-dimensional central series 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 \(h_1\dots ,h_t\) we have that \(h^q\) is n-Engel, Corollary 2 shows that any Lie commutator in \(\tilde{h}_1\dots ,\tilde{h}_t\) is ad-nilpotent. Since H satisfies the identity \( [x,_ny^q]\equiv 1 \), by Theorem 4, L satisfies some non-trivial polynomial identity. According to Theorem 2L is nilpotent.

Let \(\hat{H}\) be the pro-p completion of H, that is, the inverse limit of all quotients of H which are finite p-groups. Notice that \(\hat{H}\) is finitely generated, being H finitely generated.

Since the finite p-quotients of H are the same as the finite p-quotients of \( \hat{H} \) by (a) and (d) of [9, Proposition 3.2.2], we get that \(L_p(\hat{H})=L\). Hence, \(L_p(\hat{H})\) is nilpotent and so, \(\hat{H}\) is a p-adic analytic group by Theorem 3.

By [4, 1.(a) and 1.(o) in Interlude A], \(\hat{H}\) is virtually powerful, that is, \(\hat{H}\) has a powerful subgroup K of finite index. By Lemma 5, \( K^q \) has class at most f(n). Furthermore, it follows from [16, Theorem 1] that group \(K/K^q \) is finite. Finally, since H is residually-p, it embeds in \( \hat{H}. \) Thus, \(H\cap K^q\) is a nilpotent subgroup of finite index in G of class at most f(n) . This completes the proof. \(\square \)

Proof of Corollary 1

Let f(n) be as in Theorem 1. It follows from [2, Theorem C] that \( H=G^q \) is locally nilpotent. By Lemma 3, G/H is a locally graded group. By Zelmanov’s solution of the Restricted Burnside Problem, locally graded groups of finite exponent are locally finite. Thus, G/H is finite and so, H is a finitely generated nilpotent group. Since polycyclic groups are residually finite [10, 5.4.17], we can deduce from Theorem 1 that H contains a subgroup of finite index and of class at most f(n) . The proof is complete.