On finite groups in which commutators are covered by Engel subgroups

TL;DR

This paper proves that in finite groups where all commutator values are contained within certain Engel subgroups, the associated verbal subgroup exhibits bounded Engel properties depending on the initial parameters.

Contribution

It establishes a bounded Engel property for verbal subgroups in finite groups under specific conditions involving commutator values and Engel subgroups.

Findings

Verbal subgroup w(G) is s-Engel for some bounded s

Conditions relate union of subgroups covering all w-values

Results depend on parameters m, n, w

Abstract

Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.

Equations

$$[x{G}_{i+1},y{G}_{j+1}]=[x,y]G_{i+j+1}\in L_{i+j}^{\ast }$$

$$[x{,}_{0}y]=x\text{ and }[x{,}_{i}y]=[[x{,}_{i-1}y],y]\text{ for all positive integers }i.$$

$$[P^{n_1},\dots ,P^{n_j}]\le {\gamma }_j(P{)}^{n_1\cdots n_j}.$$

$$[R_1,R_1]\le [N^{(k)},N^{(k)}{]}^{e_1^{2^k}{e}_1^{2^k}}\le (N^{(k)}{)}^{p{e}_1^{2^k}{e}_1^{2^k}}=R_1^{p{e}_1^{2^k}}.$$

$$[R_1,R_1]\le R_1^{4e_1^{2^k}}.$$

$${\gamma }_{c_1}(R_1{)}^{e_1}\le {\gamma }_{c_1+1}(R_1)=[R_1^{\prime }{,}_{c_1-1}{R}_1]\le [R_1^{p{e}_1^{2^k}}{,}_{c_1-1}{R}_1]\le {\gamma }_{c_1}(R_1{)}^{p{e}_1^{2^k}}$$

$${\gamma }_{c_1}(R_1{)}^{e_1}\le {\gamma }_{c_1}(R_1{)}^{4e_1^{2^k}}.$$

$$H=⟨a_1,\dots ,a_r,b_1,\dots ,b_r,m_1,m_2⟩.$$

Full text

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

1. Introduction

Given a group-word $w=w(x_{1},\dots,x_{k})$, 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 $G_{1},\ldots,G_{m}$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_{1},\ldots,G_{m}$ 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 “$\{a,b,\dots\}$-bounded” to abbreviate “bounded from above in terms of $a,b,\dots$ 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 $x,y\in G$ the equation $[y,x,x,\dots,x]=1$ holds, where $x$ is repeated in the commutator sufficiently many times depending on $x$ and $y$. The long commutators $[y,x,\dots,x]$, where $x$ occurs $i$ times, are denoted by $[y,{}_{i}\,x]$. An element $x\in G$ is (left) $n$-Engel if $[y,{}_{n}\,x]=1$ for all $y\in G$. A group $G$ is $n$-Engel if $[y,{}_{n}\,x]=1$ for all $x,y\in G$. Currently, finite $n$-Engel groups are understood fairly well. A theorem of Baer says that finite Engel groups are nilpotent (see [12, Theorem 12.3.7]). 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 refered 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, 14]). For the reader’s convenience we will briefly describe the construction that we are using in the present paper.

A series of subgroups

[TABLE]

is called an $N$-series if it satisfies $[G_{i},G_{j}]\leq G_{i+j}$ for all $i,j$. Obviously any $N$-series is central, i.e. $G_{i}/G_{i+1}\leq Z(G/G_{i+1})$ for any $i$. 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 $[,]$ in $L^{*}(G)$. For homogeneous elements $xG_{i+1}\in L_{i}^{*},yG_{j+1}\in L_{j}^{*}$ the operation is defined by

[TABLE]

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 $L_{p}(G)$ for the subalgebra generated by the first homogeneous component $G_{1}/G_{2}$. Usually nilpotency of $L^{*}(G)$ has 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 $L_{p}(G)$ 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 $L_{p}(G)$ 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

[TABLE]

As usual, we say that an element $a\in L$ is ad-nilpotent if there exists a positive integer $n$ such that $[x,_{n}a]=0$ for all $x\in L$. 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 $f\equiv 0$. Then $L$ is nilpotent of $\{f,K,m,n\}$-bounded class.

As usual, $\gamma_{i}(L)$ denotes the $i$th 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 $f\equiv 0$. 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 $e\gamma_{c}(L)=0$.

3. Proof of the main theorem

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

$\delta_{0}=x_{1}\text{ and }\delta_{k}=[\delta_{k-1}(x_{1},\ldots,x_{2^{k-1}}),\delta_{k-1}(x_{2^{k-1}+1},\ldots,x_{2^{k}})]$ for $k\geq 1$.

The verbal subgroup corresponding to the word $\delta_{k}$ in a group $G$ is the familiar $k$th 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 $G_{1},\ldots,G_{m}$ whose union contains all $\delta_{k}$-values in $G$. If all elements of the subgroups $G_{1},\ldots,G_{m}$ 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 $[x,y]\in X$ whenever $x,y\in X$. The fact that in any group the set of all $\delta_{k}$-values is commutator-closed will be used without explicit references.

The proof of Lemma 3.1 will require the following two lemmas which were obtained in [1, Lemma 3.1] and [16, 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 $G_{1},G_{2},\ldots,G_{m}$. Then $G=G_{1}G_{2}\cdots G_{m}$.

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 $[P,P]\leq P^{p}$ for $p\neq 2$ (or $[P,P]\leq P^{4}$ for $p=2$), where $P^{i}$ denotes the subgroup of $P$ generated by all $i$th powers. If $P$ is a powerful $p$-group, then the subgroups $\gamma_{i}(P),P^{(i)}$ and $P^{i}$ are also powerful. Moreover, for given positive integers $n_{1},\ldots,n_{j}$, it follows, by repeated applications of [11, Propositions 1.6 and 4.1.6], that

[TABLE]

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 $\delta_{k}$-value is $n$-Engel in $G$. Hence, Baer’s theorem [12, Theorem 12.3.7] implies that $G^{(k)}$ is nilpotent. Replacing if necessary $G_{i}$ by $G_{i}\cap G^{(k)}$, we can assume that all subgroups $G_{i}$ are contained in $G^{(k)}$. Then, by Lemma 3.2, $G^{(k)}=G_{1}G_{2}\cdots G_{m}$.

Choose arbitrary elements $a,b\in G^{(k)}$. It is sufficient to show that the subgroup $\langle a,b\rangle$ is nilpotent of $\{m,n,k\}$-bounded class. Write $a=a_{1}\cdots a_{m}$ and $b=b_{1}\cdots b_{m}$, where $a_{i}$ and $b_{i}$ belong to $G_{i}$. Let $H$ be the subgroup generated by the elements $a_{i},b_{i}$ for $i=1,\ldots,m.$ Since the subgroup $\langle a,b\rangle$ is contained in $H$, it is enough to show that $H$ is nilpotent of $\{m,n,k\}$-bounded class. Observe that the generators $a_{i},b_{i}$ 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 Burnside Basis Theorem [12, Theorem 5.3.2], it is sufficient to show that the Frattini quotient $R/\Phi(R)$ has $\{m,n,k\}$-boundedly many generators. The quotient $P/\Phi(R)$ has derived length at most $k+1$. Thus, Lemma 3.3 implies that $P/\Phi(R)$ has $\{m,n,k\}$-bounded nilpotency class. It follows that $R/\Phi(R)$ 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 $r$ $\delta_{k}$-values which are $n$-Engel elements. Let $L_{1}=L(R)$ be the Lie ring associated to $R$ using the lower central series. The proof of [18, Theorem 1] shows that since $R$ satisfies the identity $[y,_{n}\delta_{k}(x_{1},\ldots,x_{2^{k}})]\equiv 1$, the Lie ring $L_{1}$ satisfies the linearized version of the identity $[y,_{n}\delta_{k}(x_{1},\ldots,x_{2^{k}})]\equiv 0$. Further, each commutator in the generators of $L_{1}$ corresponding to $\delta_{k}$-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 $e\gamma_{c}(L_{1})=0$. If $p$ is not a divisor of $e$, we have $\gamma_{c}(L_{1})=0$ 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 $L_{2}=L_{p}(R)$ be the Lie algebra associated to $R$ using the $p$-dimensional series. Applying Theorem 2.3 we deduce that $L_{2}$ 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 $k$th 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 $e_{1},c_{1}$ depending only on $k,m$ and $n$, such that $e_{1}\gamma_{c_{1}}(L(N^{(k)}))=0$. Since $P$ is a $p$-group, we can assume that $e_{1}$ is a $p$-power. Set $R_{1}=(N^{(k)})^{e_{1}^{2^{k}}}=(N^{e_{1}})^{(k)}$.

Note that if $p\neq 2$, then

[TABLE]

If $p=2$, then we have

[TABLE]

Since $e_{1}\gamma_{c_{1}}(L(R_{1}))=0$, we deduce that $\gamma_{c_{1}}(R_{1})^{e_{1}}\leq\gamma_{c_{1}+1}(R_{1})$. Taking into account that $R_{1}$ is powerful, if $p\neq 2$ we obtain that

[TABLE]

If $p=2$, we obtain that

[TABLE]

Hence, $\gamma_{c_{1}}(R_{1})^{e_{1}}=1$. Since $\gamma_{c_{1}}(R_{1})$ is powerful and generated by at most $t$ elements, we conclude that $\gamma_{c_{1}}(R_{1})$ is a product of at most $t$ cyclic subgroups. Hence the order of $\gamma_{c_{1}}(R_{1})$ is at most $e_{1}^{t}$. It follows that the derived length of $R_{1}$ is $\{k,m,n\}$-bounded. Recall that $N^{(k)}$ is a powerful $p$-group and $R_{1}=(N^{(k)})^{e_{1}^{2^{k}}}$. 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 [13, 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 $\delta_{k}$-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 $\gamma_{c}(G)$ 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 $G_{1},\dots,G_{m}$.

Proof.

Let $1=A_{0}\leq A_{1}\leq\dots\leq A_{u}=w(G)$ be a series as in Theorem 3.6. Arguing by induction on $u$ it is sufficient to show that each element of $A_{1}$ can be written as a product of $\{k,m\}$-boundedly many elements from the subgroups $G_{1},\dots,G_{m}$. Since $A_{1}$ is abelian and generated by $w$-values each of which lies in some $G_{i}$, we deduce that $A_{1}$ is the product of subgroups of the form $A_{1}\cap G_{i}$. 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, Baer’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 $\gamma_{c}(G^{(d)})$ 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 $a,b\in w(G)$. We will show that the subgroup $\langle a,b\rangle$ 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 $a=a_{1}\cdots a_{r}m_{1}$ and $b=b_{1}\cdots b_{r}m_{2}$, where $a_{i},b_{i}$ are $w$-values for $i=1,\ldots,r$ and $m_{1},m_{2}$ belong to $M$. Let $H$ be the subgroup generated by all these elements, that is,

[TABLE]

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

Set $N=M\cap H$ and let $\bar{H}$ be the quotient group $H/\Phi(N)$. Note that the image of $N$ in $\bar{H}$ is an abelian group, and so the images of $m_{1},m_{2}$ in $\bar{H}$ are 2-Engel. Note also that the derived length of $\bar{H}$ is at most $k+1$. Lemma 3.3 yields that the nilpotency class of $\bar{H}$ is $\{m,n,w\}$-bounded. Thus, we get that the image of $N$ in $\bar{H}$ 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 $t$th term $Z_{t}(H)$ of the upper central series of $H$. Consequently $H$ is nilpotent of $\{m,n,w\}$-bounded class, as required. The proof is now complete. ∎