On finite groups in which commutators are covered by Engel subgroups

Pavel Shumyatsky; Danilo Silveira

doi:10.1515/jgth-2019-0002

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On finite groups in which commutators are covered by Engel subgroups

Published/Copyright: July 12, 2019

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From the journal Journal of Group Theory Volume 22 Issue 6

Abstract

Let m,n be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups G1,…,Gm whose union contains all w-values in G. Assume further that all elements of the subgroups G1,…,Gm are n-Engel in G. It is shown that the verbal subgroup w⁢(G) is s-Engel for some {m,n,w}-bounded number s.

1 Introduction

Given a group-word w=w⁢(x1,…,xk), we think of it primarily as a function of k variables defined on any group G. We denote by w⁢(G) the verbal subgroup of G corresponding to the word w, that is, the subgroup generated by the w-values in G. When the set of all w-values in G is contained in a union of subgroups, we wish to know whether the properties of the covering subgroups have impact on the structure of the verbal subgroup w⁢(G). The reader can consult the articles [1, 2, 4, 5, 6, 15] for results on countable coverings of w-values in profinite groups.

The purpose of this paper is to prove the following result.

Theorem 1.1.

Let m,n be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups G1,…,Gm whose union contains all w-values in G. Furthermore, assume that all elements of the subgroups G1,…,Gm are n-Engel in G. Then w⁢(G) is s-Engel for some {m,n,w}-bounded number s.

Here and throughout the article, we use the expression “{a,b,…}-bounded” to abbreviate “bounded from above in terms of a,b,… only”.

Recall that multilinear commutators are words which are obtained by nesting commutators, but using always different variables. More formally, the word w⁢(x)=x in one variable is a multilinear commutator; if u and v are multilinear commutators involving different variables, then the word w=[u,v] is a multilinear commutator, and all multilinear commutators are obtained in this way. The number of variables involved in a multilinear commutator w is called the weight of w.

Also, recall that a group G is called an Engel group if, for every x,y∈G, the equation [y,x,x,…,x]=1 holds, where x is repeated in the commutator sufficiently many times depending on x and y. The long commutators [y,x,…,x], where x occurs i times, are denoted by [y,xi]. An element x∈G is (left) n-Engel if [y,xn]=1 for all y∈G. A group G is n-Engel if [y,xn]=1 for all x,y∈G. Currently, finite n-Engel groups are understood fairly well. A theorem of Zorn says that finite Engel groups are nilpotent (see [12, Theorem 12.3.4]). More specific properties of finite n-Engel groups can be found for example in a theorem of Burns and Medvedev quoted as Theorem 3.5 in Section 3 of this paper. The interested reader is referred to the survey [17] and references therein for further results on finite and residually finite Engel groups.

In the next section, we describe the Lie-theoretic machinery that will be used in the proof of Theorem 1.1. The proof of the theorem is given in Section 3.

2 Associating a Lie ring to a group

There are several well-known ways to associate a Lie ring to a group G (see [8, 9, 13]). For the reader’s convenience, we will briefly describe the construction that we are using in the present paper.

A series of subgroups

($*$)G=G1≥G2≥⋯

is called an N-series if it satisfies [Gi,Gj]≤Gi+j for all i and j. Obviously, any N-series is central, i.e., Gi/Gi+1≤Z⁢(G/Gi+1) for any i. Given an N-series ($*$), let L*⁢(G) be the direct sum of the abelian groups Li*=Gi/Gi+1, written additively. Commutation in G induces a binary operation [⋅,⋅] in L*⁢(G). For homogeneous elements x⁢Gi+1∈Li*, y⁢Gj+1∈Lj*, the operation is defined by

[x⁢Gi+1,y⁢Gj+1]=[x,y]⁢Gi+j+1∈Li+j*

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 [⋅,⋅] is a Lie ring.

In this paper, we use the above construction in the cases where ($*$) is either the lower central series of G or the p-dimension central series, also known under the name of Zassenhaus–Jennings–Lazard series (see [8, p. 250] for details). In the former case, we denote the associated Lie ring by L⁢(G). In the latter case, L*⁢(G) can be viewed as a Lie algebra over the field with p elements. We write Lp⁢(G) for the subalgebra generated by the first homogeneous component G1/G2. Usually, nilpotency of L*⁢(G) has a strong effect on the structure of G. In particular, L⁢(G) is nilpotent of class c if and only if the group G is nilpotent of class c. Nilpotency of Lp⁢(G) also leads to strong conclusions about G. The proof of the following theorem can be found in [10].

Theorem 2.1.

Let P be a d-generated finite p-group, and suppose that Lp⁢(G) is nilpotent of class c. Then P has a powerful characteristic subgroup of {p,c,d}-bounded index.

Recall that powerful p-groups were introduced by Lubotzky and Mann in [11]. They have many nice properties, some of which are listed in the next section.

Thus criteria of nilpotency of Lie algebras provide effective tools for applications in group theory.

Let X be a subset of a Lie algebra 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,y are elements of L, we define inductively

[x,y0]=x and [x,yi]=[[x,yi-1],y] for all positive integers⁢i.

As usual, we say that an element a∈L is ad-nilpotent if there exists a positive integer n such that [x,an]=0 for all x∈L. If n is the least integer with the above property, then we say that a is ad-nilpotent of index n.

The next theorem is a deep result of Zelmanov with many applications to group theory. It was announced by Zelmanov in [19, 20]. A detailed proof was published in [21].

Theorem 2.2.

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.

Theorem 2.2 admits the following quantitative version (see, for instance, [10]).

Theorem 2.3.

Let L be a Lie algebra over a field K. Assume that L is generated by m elements such that each commutator in the generators is ad-nilpotent of index at most n. Suppose that L satisfies a polynomial identity f≡0. Then L is nilpotent of {f,K,m,n}-bounded class.

As usual, γi⁢(L) denotes the ith term of the lower central series of L. The following Lie-ring variation on the theme of Theorem 2.2 is a particular case of [16, Proposition 2.6].

Theorem 2.4.

Let L be a Lie ring satisfying a polynomial identity f≡0. Assume that L is generated by m elements such that every commutator in the generators is ad-nilpotent of index at most n. Then there exist positive integers e and c depending only on f, m and n such that e⁢γc⁢(L)=0.

3 Proof of the main theorem

It will be convenient first to prove Theorem 1.1 in the particular case where w=δk is a derived word. Recall that the derived words δk, on 2k variables, are defined recursively by

δ0=x1,δk=[δk-1⁢(x1,…,x2k-1),δk-1⁢(x2k-1+1,…,x2k)] for⁢k≥1.

The verbal subgroup corresponding to the word δk in a group G is the familiar kth term of the derived series of G denoted by G(k).

Lemma 3.1.

Let m,n,k be positive integers, and let G be a finite group with subgroups G1,…,Gm whose union contains all δk-values in G. If all elements of the subgroups G1,…,Gm are n-Engel in G, then G(k) is s-Engel for some {k,m,n}-bounded number s.

A subset X of a group G is called commutator-closed if [x,y]∈X whenever x,y∈X. The fact that, in any group, the set of all δk-values is commutator-closed will be used without explicit reference.

The proof of Lemma 3.1 will require the following two lemmas which were obtained in [1, Lemma 3.1] and [16, Lemma 4.1], respectively.

Lemma 3.2.

Let G be a nilpotent group generated by a commutator-closed subset X which is contained in a union of finitely many subgroups G1,G2,…,Gm. Then G=G1⁢G2⁢⋯⁢Gm.

Lemma 3.3.

Let G be a group generated by m elements which are n-Engel. If G is soluble with derived length d, then G is nilpotent of {d,m,n}-bounded class.

The proof of Lemma 3.1 requires the concept of powerful p-groups. A finite p-group P is said to be powerful if and only if [P,P]≤Pp for p≠2 (or [P,P]≤P4 for p=2), where Pi denotes the subgroup of P generated by all ith powers. If P is a powerful p-group, then the subgroups γi⁢(P), P(i) and Pi are also powerful. Moreover, for given positive integers n1,…,nj, it follows, by repeated applications of [11, Propositions 1.6 and 4.1.6], that

[Pn1,…,Pnj]≤γj⁢(P)n1⁢⋯⁢nj.

Furthermore, if a powerful p-group P is generated by d elements, then any subgroup of P can be generated by at most d elements and P is a product of d cyclic subgroups. For more details, we refer the reader to [9, Chapter 11].

Proof of Lemma 3.1.

By the hypothesis, each δk-value is n-Engel in G. Hence Zorn’s theorem [12, Theorem 12.3.4] implies that G(k) is nilpotent. Replacing Gi by Gi∩G(k) if necessary, we can assume that all subgroups Gi are contained in G(k). Then, by Lemma 3.2, G(k)=G1⁢G2⁢⋯⁢Gm.

Choose arbitrary elements a,b∈G(k). It is sufficient to show that the subgroup 〈a,b〉 is nilpotent of {m,n,k}-bounded class. Write a=a1⁢⋯⁢am, b=b1⁢⋯⁢bm, where ai and bi belong to Gi. Let H be the subgroup generated by the elements ai,bi for i=1,…,m. Since the subgroup 〈a,b〉 is contained in H, it is enough to show that H is nilpotent of {m,n,k}-bounded class. Observe that the generators ai,bi of H are n-Engel elements. Thus, in view of Lemma 3.3, it is sufficient to prove that H has {m,n,k}-bounded derived length. Since H is nilpotent, we need to show that each Sylow p-subgroup P of H has {m,n,k}-bounded derived length.

Obviously, P can be generated by 2⁢m elements, each of which is n-Engel. Set R=P(k). We will now prove that R can be generated by {m,n,k}-boundedly many, say r, elements. Note that, by the Burnside Basis Theorem [12, Theorem 5.3.2], it is sufficient to show that the Frattini quotient R/Φ⁢(R) has {m,n,k}-boundedly many generators. The quotient P/Φ⁢(R) has derived length at most k+1. Thus Lemma 3.3 implies that P/Φ⁢(R) has {m,n,k}-bounded nilpotency class. It follows that R/Φ⁢(R) can be generated with {m,n,k}-boundedly many elements. This is also true for R.

Next, we will show that R has {m,n,k}-bounded derived length. By the Burnside Basis Theorem, R is generated by rδk-values which are n-Engel elements. Let L1=L⁢(R) be the Lie ring associated to R using the lower central series. The proof of [18, Theorem 1] shows that, whenever a group K satisfies a commutator identity f≡1, the Lie ring L⁢(K) satisfies the linearized version of “the same” identity [y,δkn⁢(x1,…,x2k)]≡0. Since the identity [y,δkn⁢(x1,…,x2k)]≡1 is satisfied in R, it follows that the Lie ring L1 satisfies the linearized version of the identity [y,δkn⁢(x1,…,x2k)]≡0. Further, each commutator in the generators of L1 corresponding to δk-values in R is ad-nilpotent of index at most n. By Theorem 2.4, there exist positive integers e and c, depending only on k, m and n, such that e⁢γc⁢(L1)=0. If p is not a divisor of e, we have γc⁢(L1)=0, and so the group R is nilpotent of class at most c-1. In what follows, we assume that p is a divisor of e. Note that, in this case, p is bounded in terms of k, m and n.

Let L2=Lp⁢(R) be the Lie algebra associated to R using the p-dimensional series. Applying Theorem 2.3, we deduce that L2 is nilpotent with {m,n,k}-bounded nilpotency class. Hence, by Theorem 2.1, R has a powerful subgroup N of {m,n,k}-bounded index. It is now sufficient to show that N has {m,n,k}-bounded derived length.

Since the index of N in R is {m,n,k}-bounded, it follows that N can be generated with {m,n,k}-boundedly many elements, say t. Taking into account that N is powerful, we deduce that all subgroups of N can be generated by at most t elements, and the kth derived subgroup N(k) is also powerful. We now look at the Lie ring L⁢(N(k)) associated to N(k).

By Theorem 2.4, there exist positive integers e1,c1, depending only on k, m and n, such that e1⁢γc1⁢(L⁢(N(k)))=0. Since P is a p-group, we can assume that e1 is a p-power. Set

($**$)R1=(N(k))e12k=(Ne1)(k).

Note that if p≠2, then

[R1,R1]≤[N(k),N(k)]e12k⁢e12k≤(N(k))p⁢e12k⁢e12k=R1p⁢e12k.

If p=2, then we have

[R1,R1]≤R14⁢e12k.

Since e1⁢γc1⁢(L⁢(R1))=0, we deduce that γc1⁢(R1)e1≤γc1+1⁢(R1). Taking into account that R1 is powerful, if p≠2, we obtain that

γc1⁢(R1)e1≤γc1+1⁢(R1)=[R1′,R1c1-1]≤[R1p⁢e12k,R1c1-1]≤γc1⁢(R1)p⁢e12k.

If p=2, we obtain that

γc1⁢(R1)e1≤γc1⁢(R1)4⁢e12k.

Hence γc1⁢(R1)e1=1. Since γc1⁢(R1) is powerful and generated by at most t elements, we conclude that γc1⁢(R1) is a product of at most t cyclic subgroups. Hence the order of γc1⁢(R1) is at most e1t. It follows that the derived length of R1 is {k,m,n}-bounded. Recall that N(k) is a powerful p-group and ($**$). It follows that the derived length of N(k) is {k,m,n}-bounded. Hence the derived length of P is {k,m,n}-bounded, as required. The proof is now complete. ∎

The next lemma is well known (see, for example, [14, Lemma 4.1] for a proof).

Lemma 3.4.

Let G be a group and w a multilinear commutator word of weight k. Then every δk-value in G is a w-value.

The proof of Theorem 1.1 will require the following result, due to Burns and Medvedev [3].

Theorem 3.5.

Let n be a positive integer. There exist constants c and e depending only on n such that if G is a finite n-Engel group, then the exponent of γc⁢(G) divides e.

Another useful result which we will need is the next theorem [7, Theorem B].

Theorem 3.6.

Let w be a multilinear commutator word, and let G be a soluble group. Then there exists a series of subgroups from 1 to w⁢(G) such that

all subgroups of the series are normal in G,
every section of the series is abelian and can be generated by w-values all of whose powers are also w-values.

Furthermore, the length of this series only depends on the word w and on the derived length of G.

Corollary 3.7.

Assume the hypotheses of Theorem 1.1, and suppose additionally that G is soluble with derived length k. Then each element of w⁢(G) can be written as a product of {k,m}-boundedly many elements from the subgroups G1,…,Gm.

Proof.

Let 1=A0≤A1≤…≤Au=w⁢(G) be a series as in Theorem 3.6. Arguing by induction on u, it is sufficient to show that each element of A1 can be written as a product of {k,m}-boundedly many elements from the subgroups G1,…,Gm. Since A1 is abelian and generated by w-values, each of which lies in some Gi, we deduce that A1 is the product of subgroups of the form A1∩Gi. The result follows. ∎

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

Recall that w is a multilinear commutator word. Since each w-value in G is n-Engel, Zorn’s theorem implies that the verbal subgroup w⁢(G) is nilpotent. Let d be the weight of the word w. Combining Lemmas 3.4 and 3.1, we deduce that G(d) is u-Engel for some {m,n,d}-bounded number u. Theorem 3.5 shows that there exists an {m,n,d}-bounded number c such that γc⁢(G(d)) has {m,n,d}-bounded exponent. It follows that there is an {m,n,d}-bounded number k such that M=G(k) has {m,n,d}-bounded exponent.

Choose arbitrary elements a,b∈w⁢(G). We will show that the subgroup 〈a,b〉 is nilpotent of {m,n,w}-bounded class. Corollary 3.7 shows that any element in w⁢(G)/M can be written as a product of {m,n,w}-boundedly many, say r, w-values. Thus we can write a=a1⁢⋯⁢ar⁢m1 and b=b1⁢⋯⁢br⁢m2, where ai,bi are w-values for i=1,…,r and m1,m2 belong to M. Let H be the subgroup generated by all these elements, that is,

H=〈a1,…,ar,b1,…,br,m1,m2〉.

Note that the subgroup 〈a,b〉 is contained in H, and therefore it is sufficient to show that H is nilpotent of {m,n,w}-bounded class.

Set N=M∩H, and let H¯ be the quotient group H/Φ⁢(N). Note that the image of N in H¯ is an abelian group, and so the images of m1,m2 in H¯ are 2-Engel. Note also that the derived length of H¯ is at most k+1. Lemma 3.3 yields that the nilpotency class of H¯ is {m,n,w}-bounded. Thus we get that the image of N in H¯ has {m,n,w}-boundedly many generators. Of course, this is true also for N. Recall that the exponent of N is {m,n,w}-bounded, and so we obtain from the positive solution of the restricted Burnside problem [19, 20] that the order of N is {m,n,w}-bounded. Since H is nilpotent, there is an {m,n,w}-bounded number t such that N is contained in the tth term Zt⁢(H) of the upper central series of H. Consequently, H is nilpotent of {m,n,w}-bounded class, as required. The proof is now complete. ∎

Communicated by Evgenii I. Khukhro

Funding statement: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.

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Received: 2019-01-03

Revised: 2019-06-02

Published Online: 2019-07-12

Published in Print: 2019-11-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2019-0002