Abstract
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 \(w(x,y)=1.\) Assume 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 G satisfies the \(w_{m,n}\)-property, then either w is the identity in G or |G| is bounded in terms of \(\gamma , m\) and n. We apply this result to the 2-Engel word.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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.
[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
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
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.
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.
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
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.
(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
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
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,
Combining (0.1) and 0.2, we deduce
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
This implies
\(\square \)
FormalPara Corollary 6Let \(w=[x,y,y]\) 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 ProofBy [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 \(\tau =\frac{2}{1-\delta }.\) \(\square \)
Data availibility
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Abdollahi, A., Azad, A., Mohammadi Hassanabadi, A., Zarrin, M.: B. H. Neumann’s question on ensuring commutativity of finite groups. Bull. Aust. Math. Soc. 74(1), 121–132 (2006)
Delizia, C., Jezernik, U., Moravec, P., Nicotera, C.: Gaps in probabilities of satisfying some commutator-like identities. Israel J. Math. 237(1), 115–140 (2020)
Gustafson, W.H.: What is the probability that two group elements commute? Amer. Math. Monthly 80, 1031–1034 (1973)
Kövari, T., Sós, V.T., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954)
Lucchini, A.: Applying the Kövári-Sós-Turán theorem to a question in group theory. Comm. Alg. 48(11), 4966–4968 (2020)
Neumann, B.H.: Ensuring commutativity of finite groups, Special issue on group theory. J. Aust. Math. Soc. 71(2), 233–234 (2001)
Pyber, L.: The number of pairwise non-commuting elements and the index of the centre in a finite group. J. London Math. Soc. 35(2), 287–295 (1987)
Swanepoel, K. J.: Favourite distances in 3-space, Electron. J. Combin. 27(2), Paper No. 2.17, 11 (2020)
Funding
Open access funding provided by Università degli Studi di Padova within the CRUI-CARE Agreement. The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lucchini, A. A Generalization of a Question Asked by B. H. Neumann. Results Math 77, 181 (2022). https://doi.org/10.1007/s00025-022-01709-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-022-01709-1