A Generalization of a Question Asked by B. H. Neumann

Abstract

AbstractLet $$w \in F_2$$ w ∈ F 2 be a word and let m and n be two positive integers. We say that a finite group G has the $$w_{m,n}$$ 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 $$x \in M$$ x ∈ M and at least one element $$y \in N$$ y ∈ N such that $$w(x,y)=1.$$ w ( x , y ) = 1 . Assume that there exists a constant $$\gamma < 1$$ γ < 1 such that whenever w is not the identity in a finite group X, then the probability that $$w(x_1,x_2)=1$$ w ( x 1 , x 2 ) = 1 in X is at most $$\gamma .$$ γ . If $$m\le n$$ m ≤ n and G satisfies the $$w_{m,n}$$ w m , n -property, then either w is the identity in G or |G| is bounded in terms of $$\gamma , m$$ γ , m and n. We apply this result to the 2-Engel word.