On finite-by-nilpotent groups

TL;DR

This paper proves that in groups where conjugacy classes of certain commutator values are uniformly bounded, the next lower central subgroup is finite with a bound depending on these parameters.

Contribution

It generalizes Neumann's theorem by establishing finiteness of higher lower central subgroups under bounded conjugacy conditions.

Findings

The (n+1)th lower central subgroup has finite order bounded by parameters m and n.

Generalizes classical results on BFC-groups to broader classes of groups.

Provides bounds on the size of certain subgroups based on conjugacy class sizes.

Abstract

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

Equations

$${\delta }_0=x_1,{\delta }_n=[{\delta }_{n-1}(x_1,\dots ,x_{2^{n-1}}),{\delta }_{n-1}(x_{2^{n-1}+1},\dots ,x_{2^n})].$$

$$[x^y,y^{-1},z^y][y^z,z^{-1},x^z][z^x,x^{-1},y^x]=1.$$

$$[y,x,z^y][z,y,x^z][x,z,y^x]=1.$$

$$|g^{{\gamma }_{k-1}(G)}|\le m^{n-k+2}|[{\gamma }_k(G),{\gamma }_n(G)]|.$$

$$|(gN{)}^{{\gamma }_d(G/N)}|\le m^{n-d+1}\text{for every γn−d+1-value gN∈G/N},$$

$$[x,y,z^x][z,x,y^z][y,z,x^y]=1.$$

$$[z,x]\in [{\gamma }_d(G),{\gamma }_{c-1}(G)]\le {\gamma }_{d-1+c}(G)={\gamma }_n(G)$$

$$[y,z]\in {\gamma }_{d+1}(G)\le {\gamma }_k(G),$$

$$1=[x,y,z^x][z,x,y^z][y,z,x^y]$$

$$[x,y{]}^{z^x}=[x,y](x^{-1}{)}^y(x^y{)}^{[y,z]}{y}^{-1}{y}^{[z,x]}.$$

$$|\{(x^y{)}^{[y,z]}\mid z\in {\gamma }_d(G)\}|\le m^{n-d-1+1}.$$

$$|\{[x,y{]}^{z^x}\mid z\in {\gamma }_d(G)\}|=|\{[x,y{]}^z\mid z\in {\gamma }_d(G)\}|\le m{m}^{n-d}=m^{n-d+1}$$

$$[[y_{ij},x],z]\in [{\gamma }_k(G),{\gamma }_n(G)]=1.$$

$$[W,K]\le [W,x_1]\cdots [W,x_s][W,M]\le [W,x_1]\cdots [W,x_s][W,a].$$

Full text

The first and third authors are members of INDAM. The fourth author was supported by CNPq-Brazil

1. Introduction

Given a group $G$ and an element $x\in G$, we write $x^{G}$ for the conjugacy class containing $x$. Of course, if the number of elements in $x^{G}$ is finite, we have $|x^{G}|=[G:C_{G}(x)]$. A group is said to be a BFC-group if its conjugacy classes are finite and of bounded size. One of the most famous of B. H. Neumann’s theorems says that in a BFC-group the commutator subgroup $G^{\prime}$ is finite [6]. It follows that if $|x^{G}|\leq m$ for each $x\in G$, then the order of $G^{\prime}$ is bounded by a number depending only on $m$. A first explicit bound for the order of $G^{\prime}$ was found by J. Wiegold [10], and the best known was obtained in [5] (see also [7] and [9]).

The recent articles [3] and [2] deal with groups $G$ in which conjugacy classes containing commutators are bounded. Recall that multilinear commutator words are words which are obtained by nesting commutators, but using always different variables. More formally, the group-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. Examples of multilinear commutators include the familiar lower central words $\gamma_{n}(x_{1},\dots,x_{n})=[x_{1},\dots,x_{n}]$ and derived words $\delta_{n}$, on $2^{n}$ variables, defined recursively by

[TABLE]

We let $w(G)$ denote the verbal subgroup of $G$ generated by all $w$-values. Of course, $\gamma_{n}(G)$ is the $n$th term of the lower central series of $G$ while $\delta_{n}(G)=G^{(n)}$ is the $n$th term of the derived series.

The following theorem was established in [2].

Theorem 1.1**.**

Let $m$ be a positive integer and $w$ a multilinear commutator word. Suppose that $G$ is a group in which $|x^{G}|\leq m$ for any $w$-value $x$. Then the order of the commutator subgroup of $w(G)$ is finite and $m$-bounded.

Throughout the article we use the expression “$(a,b,\dots)$-bounded” to mean that a quantity is finite and bounded by a certain number depending only on the parameters $a,b,\dots$.

The present article grew out of the observation that a modification of the techniques developed in [3] and [2] can be used to deduce that if $|x^{G^{\prime}}|\leq m$ for each $x\in G$, then $\gamma_{3}(G)$ has finite $m$-bounded order. Naturally, one expects that a similar phenomenon holds for other terms of the lower central series of $G$. This is indeed the case.

Theorem 1.2**.**

Let $m,n$ be positive integers and $G$ a group. If $|x^{\gamma_{n}(G)}|\leq m$ for any $x\in G$, then $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order.

Using the concept of verbal conjugacy classes, introduced in [4], one can obtain a generalization of Theorem 1.2. Let $X_{n}=X_{n}(G)$ denote the set of $\gamma_{n}$-values in a group $G$. It was shown in [1] that if $|x^{X_{n}}|\leq m$ for each $x\in G$, then $|x^{\gamma_{n}(G)}|$ is $(m,n)$-bounded. Hence, we have

Corollary 1.3**.**

Let $m,n$ be positive integers and $G$ a group. If $|x^{X_{n}(G)}|\leq m$ for any $x\in G$, then $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order.

Observe that Neumann’s theorem can be obtained from Corollary 1.3 by specializing $n=1$. Another result which is straightforward from Corollary 1.3 is the following characterization of finite-by-nilpotent groups.

Theorem 1.4**.**

A group $G$ is finite-by-nilpotent if and only if there are positive integers $m,n$ such that $|x^{X_{n}}|\leq m$ for any $x\in G$.

2. Preliminary results

Recall that in any group $G$ the following “standard commutator identities” hold, when $x,y,z\in G$.

1. (1)

$[xy,z]=[x,z]^{y}[y,z]$ 2. (2)

$[x,yz]=[x,z][x,y]^{z}$ 3. (3)

$[x,y^{-1},z]^{y}[y,z^{-1},x]^{z}[z,x^{-1},y]^{x}=1$ (Hall-Witt identity); 4. (4)

$[x,y,z^{x}][z,x,y^{z}][y,z,x^{y}]=1.$

Note that the fourth identity follows from the third one. Indeed, we have

[TABLE]

Since $[x^{y},y^{-1}]=[y,x]$, it follows that

[TABLE]

Recall that $X_{i}$ denote the set of $\gamma_{i}$-values in a group $G$.

Lemma 2.1**.**

Let $k,n$ be integers with $2\leq k\leq n$ be integers and let $G$ be a group such that $[\gamma_{k}(G),\gamma_{n}(G)]$ is finite and $|x^{\gamma_{n}(G)}|\leq m$ for any $x\in G$. Then for every $g\in X_{n}$ we have

[TABLE]

Proof.

Let $N=[\gamma_{k}(G),\gamma_{n}(G)]$. It is sufficient to prove that in the quotient group $G/N$, for every integer $d$ with $k-1\leq d\leq n$

[TABLE]

since this implies that $g^{\gamma_{d}(G)}$ is contained at most $m^{n-d+1}$ cosets of $N$, whenever $g\in X_{{n-d+1}}$.

So in what follows we assume that $N=1$. The proof is by induction on $n-d$. The case $d=n$ is immediate from the hypotheses.

Let $c=n-d+1$. Choose $g\in X_{c}$ and write $g=[x,y]$ with $x\in X_{c-1}$ and $y\in G$. Let $z\in\gamma_{d}(G)$. We have

[TABLE]

Note that

[TABLE]

and

[TABLE]

whence $[z,x,y^{z}]=[z,x,y[y,z]]=[z,x,y].$ Thus,

[TABLE]

It follows that

[TABLE]

Since $x^{y}\in X_{c-1}$ and $[y,z]\in\gamma_{d+1}(G)$, by induction

[TABLE]

Moreover, $[z,x]\in\gamma_{n}(G)$ an so $|\{y^{[z,x]}\mid z\in\gamma_{d}(G)\}|\leq m$. Thus,

[TABLE]

as claimed. ∎

Let $H$ be a group generated by a set $X$ such that $X=X^{-1}$. Given an element $g\in H$, we write $l_{X}(g)$ for the minimal number $l$ with the property that $g$ can be written as a product of $l$ elements of $X$. Clearly, $l_{X}(g)=0$ if and only if $g=1$. We call $l_{X}(g)$ the length of $g$ with respect to $X$. The following result is Lemma 2.1 in [3].

Lemma 2.2**.**

Let $H$ be a group generated by a set $X=X^{-1}$ and let $K$ be a subgroup of finite index $m$ in $H$. Then each coset $Kb$ contains an element $g$ such that $l_{X}(g)\leq m-1$.

In the sequel the above lemma will be used in the situation where $H=\gamma_{n}(G)$ and $X=X_{n}$ is the set of $\gamma_{n}$-values in $G$. Therefore we will write $l(g)$ to denote the smallest number such that the element $g\in\gamma_{n}(G)$ can be written as a product of as many $\gamma_{n}$-values.

Recall that if $G$ is a group, $a\in G$ and $H$ is a subgroup of $G$, then $[H,a]$ denotes the subgroup of $G$ generated by all commutators of the form $[h,a]$, where $h\in H$. It is well-known that $[H,a]$ is normalized by $a$ and $H$.

Lemma 2.3**.**

Let $k,m,n\geq 2$ and let $G$ be a group in which $|x^{\gamma_{n}(G)}|\leq m$ for any $x\in G$. Suppose that $[\gamma_{k}(G),\gamma_{n}(G)]$ is finite. Then for every $x\in\gamma_{k-1}(G)$ the order of $[\gamma_{n}(G),x]$ is bounded in terms of $m$, $n$ and $|[\gamma_{k}(G),\gamma_{n}(G)]|$ only.

Proof.

By Neumann’s theorem $\gamma_{n}(G)^{\prime}$ has $m$-bounded order, so the statement is true for $k\geq n+1$. Therefore we deal with the case $k\leq n$. Without loss of generality we can assume that $[\gamma_{k}(G),\gamma_{n}(G)]=1$.

Let $x\in\gamma_{k-1}(G)$. Since $|x^{\gamma_{n}(G)}|\leq m$, the index of $C_{\gamma_{n}(G)}(x)$ in $\gamma_{n}(G)$ is at most $m$ and by Lemma 2.2 we can choose elements $y_{1},\dots,y_{m}\in X_{n}$ such that $l(y_{i})\leq m-1$ and $[\gamma_{n}(G),x]$ is generated by the commutators $[y_{i},x]$. For each $i=1,\dots,m$ write $y_{i}=y_{i\,1}\cdots y_{i\,{m-1}}$, where $y_{i\,j}\in X_{n}$. The standard commutator identities show that $[y_{i},x]$ can be written as a product of conjugates in $\gamma_{n}(G)$ of the commutators $[y_{ij},x]$. Since $[y_{ij},x]\in\gamma_{k}(G)$, for any $z\in\gamma_{n}(G)$ we have that

[TABLE]

Therefore $[y_{i},x]$ can be written as a product of the commutators $[y_{ij},x]$.

Let $T=\langle x,y_{ij}\mid 1\leq i,j\leq m\rangle$. It is clear that $[\gamma_{n}(G),x]\leq T^{\prime}$ and so it is sufficient to show that $T^{\prime}$ has finite $(m,n)$-bounded order. Observe that $T\leq\gamma_{k-1}(G)$. By Lemma 2.1, $C_{\gamma_{k-1}(G)}(y_{ij})$ has $(m,n)$-bounded index in $\gamma_{k-1}(G)$. It follows that $C_{T}(\{y_{ij}\mid 1\leq i,j\leq m\})$ has $(m,n)$-bounded index in $T$. Moreover, $T\leq\langle x\rangle\gamma_{n}(G)$ and $|x^{\gamma_{n}(G)}|\leq m$, whence $|T:C_{T}(x)|\leq m$. Therefore the centre of $T$ has $(m,n)$-bounded index in $T$. Thus, Schur’s theorem [8, 10.1.4] tells us that $T^{\prime}$ has finite $(m,n)$-bounded order, as required. ∎

The next lemma can be seen as a development related to Lemma 2.4 in [3] and Lemma 4.5 in [10]. It plays a central role in our arguments.

Lemma 2.4**.**

Let $k,n\geq 2$. Assume that $|x^{\gamma_{n}(G)}|\leq m$ for any $x\in G$. Suppose that $[\gamma_{k}(G),\gamma_{n}(G)]$ is finite. Then the order of $[\gamma_{k-1}(G),\gamma_{n}(G)]$ is bounded in terms of $m$, $n$ and $|[\gamma_{k}(G),\gamma_{n}(G)]|$ only.

Proof.

Without loss of generality we can assume that $[\gamma_{k}(G),\gamma_{n}(G)]=1$. Let $W=\gamma_{n}(G)$. Choose an element $a\in X_{k-1}$ such that the number of conjugates of $a$ in $W$ is maximal possible, that is, $r=|a^{W}|\geq|g^{W}|$ for all $g\in X_{k-1}$.

By Lemma 2.2 we can choose $b_{1},\dots,b_{r}\in W$ such that $l(b_{i})\leq m-1$ and $a^{W}=\{a^{b_{i}}|i=1,\dots,r\}$. Let $K=\gamma_{k-1}(G)$. Set $M=(C_{K}(\langle b_{1},\dots,b_{r}\rangle))_{K}$ (i.e. $M$ is the intersection of all $K$-conjugates of $C_{K}(\langle b_{1},\dots,b_{r}\rangle)$). Since $l(b_{i})\leq m-1$ and, by Lemma 2.1, $C_{K}(x)$ has $(m,n)$-bounded index in $K$ for each $x\in X_{n}$, the subgroup $C_{K}(\langle b_{1},\dots,b_{r}\rangle)$ has $(m,n)$-bounded index in $K$, so also $M$ has $(m,n)$-bounded index in $K$.

Let $v\in M$. Note that $(va)^{b_{i}}=va^{b_{i}}$ for each $i=1,\dots,r$. Therefore the elements $va^{b_{i}}$ form the conjugacy class $(va)^{W}$ because they are all different and their number is the allowed maximum. So, for an arbitrary element $h\in W$ there exists $b\in\{b_{1},\dots,b_{r}\}$ such that $(va)^{h}=va^{b}$ and hence $v^{h}a^{h}=va^{b}$. Therefore $[h,v]=v^{-h}v=a^{h}a^{-b}$ and so $[h,v]^{a}=a^{-1}a^{h}a^{-b}a=[a,h][b,a]\in[W,a].$ Thus $[W,v]^{a}\leq[W,a]$ and so $[W,M]\leq[W,a].$

Let $x_{1},\dots,x_{s}$ be a set of coset representatives of $M$ in $K$. As $[W,x_{i}]$ is normalized by $W$ for each $i$, it follows that

[TABLE]

Since $s$ is $(m,n)$-bounded and by Lemma 2.3 the orders of all subgroups $[W,x_{i}]$ and $[W,a]$ are bounded in terms of $m$ and $n$ only, the result follows. ∎

Proof of Theorem 1.2..

Let $G$ be a group in which $|x^{\gamma_{n}(G)}|\leq m$ for any $x\in G$. We need to show that $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order. We will show that the order of $[\gamma_{k}(G),\gamma_{n}(G)]$ is finite and $(m,n)$-bounded for $k=n,n-1,\dots,1$. This is sufficient for our purposes since $[\gamma_{1}(G),\gamma_{n}(G)]=\gamma_{n+1}(G)$. We argue by backward induction on $k$. The case $k=n$ is immediate from Neumann’s theorem so we assume that $k\leq n-1$ and the order of $[\gamma_{k+1}(G),\gamma_{n}(G)]$ is finite and $(m,n)$-bounded. Lemma 2.4 now shows that also the order of $[\gamma_{k}(G),\gamma_{n}(G)]$ is finite and $(m,n)$-bounded, as required. ∎

Proof of Corollary 1.3..

Let $G$ be a group in which $|x^{X_{n}(G)}|\leq m$ for any $x\in G$. We wish to show that $\gamma_{n+1}(G)$ has finite $(m,n)$-bounded order. Theorem 1.2 of [1] tells us that $|x^{\gamma_{n}(G)}|$ is $(m,n)$-bounded. The result is now immediate from Theorem 1.2. ∎

Proof of Theorem 1.4..

In view of Corollary 1.3 the theorem is self-evident since a group $G$ is finite-by-nilpotent if and only if some term of the lower central series of $G$ is finite. ∎