In 2001 Bernhard H. Neumann asked the following question [6]: let G be a finite group and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between |G|, m, n guarantee that G is abelian?

A partial answer has been given by A. Abdollahi, A. Azad, A. Mohammadi Hassanabadi and M. Zarrin [1]. They proved that there exists a function \(f: {\mathbb {N}} \times {\mathbb {N}} \rightarrow {\mathbb {N}}\) with the following property. Let G be a finite group and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. If \({|G| > f(n,m)}\), then G is abelian. It follows from the proof that one can take \(f(n,m)={c^{m+n}\max \{m,n\}}\) where c is a constant appearing in the following theorem, proved by L. Pyber in 1987 [7]: if a finite group G contains at most n pairwise non-commuting elements, then \(|G/Z(G)|\le c^n\). An alternative elementary proof, with a better estimation of the function f can be obtained as a corollary of the following theorem.

FormalPara Theorem 1

[5,  Theorem 1]. Let \({\mathfrak {X}}\) be a class of groups and suppose that there exists a real positive number \(\gamma \) with the following property: if X is a finite group and the probability that two randomly chosen elements of X generate a group in \({\mathfrak {X}}\) is greater than \(\gamma ,\) then X is in \({\mathfrak {X}}\). Assume that a finite group G is such that for every two subsets M and N of cardinalities m and n,  respectively, there exist \(x \in M\) and \(y \in N\) such that \(\langle x, y \rangle \in {\mathfrak {X}}.\) If \(m\le n,\) then either \(G \in {\mathfrak {X}}\) or

$$\begin{aligned} |G|\le \left( \frac{2}{1-\gamma }\right) ^m(n-1). \end{aligned}$$
FormalPara Corollary 2

Let G be a finite group and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. If \(m\le n\) and G is not abelian, then

$$\begin{aligned} |G|\le \left( \frac{16}{3}\right) ^m(n-1). \end{aligned}$$
FormalPara Proof

W. H. Gustafson [3] proved that if G is a finite non-abelian group, then the probability that a randomly chosen pair of elements of G commutes is at most \(\frac{5}{8}.\) So the statement follows immediately applying Theorem 1 to the class of finite abelian groups and taking \(\gamma =\frac{5}{8}.\) \(\square \)

In this short note we show that, essentially with the same arguments, it can be proved a result similar to Theorem 1, but involving words in place of classes, and in particular, a result similar to Corollary 2, involving the 2-Engel word in place of the commutator word.

FormalPara Definition 3

Let \(w \in F_2\) be a word and let m and n be two positive integers. We say that a finite group G has the \(w_{m,n}\)-property if however a set M of m elements and a set N of n elements of the group is chosen, there exist at least one element \(x \in M\) and at least one element \(y \in N\) such that \(w(x,y)=1.\)

Our main result is the following.

FormalPara Theorem 4

Assume that a word \(w \in F_2\) has the property that there exists a constant \(\gamma < 1\) such that whenever w is not the identity in a finite group X, then the probability that \(w(x_1,x_2)=1\) in X is at most \(\gamma .\) If \(m\le n\) and a finite group G satisfies the \(w_{m,n}\)-property, then either w is the identity in G or

$$\begin{aligned} |G|\le \left( \frac{2}{1-\gamma }\right) ^m(n-1). \end{aligned}$$

The Proof of Theorem 1 relies on the Kövári-Sós-Turán theorem [4]. The Proof of Theorem 4 is quite similar, but requires a version of Kövári-Sós-Turán theorem for direct graphs (see for example [8,  Sect. 3]). Let \(\vec K_{r,s}\) be the complete bipartite directed graph in which the vertex set is a disjoint union \(A \cup B\) with \(|A| = r\) and \(|B| = s,\) and an arc is directed from each vertex of A to each vertex of B.

FormalPara Theorem 5

(Kövári-Sós-Turán). Let \(\vec \Gamma = (V, \vec E)\) be a directed graph with \(|V|= t\). Suppose that \(\vec \Gamma \) does not contain a copy of \(\vec K_{r,s}. \) Then

$$\begin{aligned} |\vec E|\le (s-1)^{\frac{1}{r}}t^{2-\frac{1}{r}}+(r-1)t. \end{aligned}$$
FormalPara Proof of Theorem 4

Suppose that G satisfies the \(w_{m,n}\)-property. Consider the direct graph \(\Gamma _w(G)\) whose vertices are the elements of G and in which there is an edge \(x_1\mapsto x_2\) if and only if \(w(x_1,x_2)\ne 1.\) If w is not the identity in G, then the probability that two vertices of \(\Gamma _{w}(G)\) are joined by an edge is at least \(1-\gamma \), so, denoting by \(\eta \) the number of edges of \(\Gamma _{w}(G)\), we must have

$$\begin{aligned} \eta \ge {(1-\gamma )|G|^2}. \end{aligned}$$
(0.1)

On the other hand, since G satisfies the \(w_{m,n}\)-property, the graph \(\Gamma _w(G)\) cannot contain \(\vec {K}_{m,n}\) as a subgraph. By Theorem 5,

$$\begin{aligned} \eta \le {(n-1)^{1/m}|G|^{2-1/m}+(m-1)|G|}. \end{aligned}$$
(0.2)

Combining (0.1) and 0.2, we deduce

$$\begin{aligned} \left( \frac{n-1}{|G|}\right) ^{1/m}+\frac{n-1}{|G|}\ge \left( \frac{n-1}{|G|}\right) ^{1/m}+\frac{m-1}{|G|}\ge 1-\gamma . \end{aligned}$$
(0.3)

We may assume \(|G|\ge n-1\). This implies \(\left( \frac{n-1}{|G|}\right) ^{1/m}\ge \frac{n-1}{|G|}\) and therefore it follows from (0.3) that

$$\begin{aligned} \left( \frac{n-1}{|G|}\right) ^{1/m}\ge \frac{1-\gamma }{2}. \end{aligned}$$
(0.4)

This implies

$$\begin{aligned} |G|\le \left( \frac{2}{1-\gamma }\right) ^m(n-1). \end{aligned}$$

\(\square \)

FormalPara Corollary 6

Let \(w=[x,y,y]\) be the 2-Engel word. There exists a constant \(\tau \) such that if \(m\le n\) and G satisfies the \(w_{m,n}\)-property, then either w is the identity in G or \(|G|\le \tau ^m(n-1).\)

FormalPara Proof

By [2], there exists a constant \(\delta \) such that if [x, y, y] is not the identity in G,  then the probability that \([g_1,g_2,g_2]=1\) in G is at most \(\delta .\) By Theorem 4, we may take \(\tau =\frac{2}{1-\delta }.\) \(\square \)