A generalization of a question asked by B. H. Neumann

TL;DR

This paper generalizes a question by B. H. Neumann regarding properties of finite groups related to specific words and their identities, providing bounds and applications to the 2-Engel word.

Contribution

It introduces a generalized framework for the $w_{m,n}$-property in finite groups and establishes bounds on group order based on probabilistic and combinatorial conditions.

Findings

If $w$ is not an identity, the probability that $w(x_1,x_2)=1$ is at most $ ext{gamma}$.

Groups satisfying the $w_{m,n}$-property are either $w$-identities or have bounded order.

The results are applied specifically to the 2-Engel word.

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 of $x \in M$ and at least one element of $y \in M$ such that $w(x,y)=1.$ Assume that there exists a constant $\gamma < 1$ such that whenever $w$ is not an identity in a finite group $X$, then the probability that $w(x_1,x_2)=1$ in $X$ is at most $\gamma.$ If $m\leq n$ and $G$ satisfies the $w_{m,n}$-property, then either $w$ is an identity in $G$ or $|G|$ is bounded in terms of $\gamma, m$ and $n$. We apply this result to the 2-Engel word.