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

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