Characterization of the soluble radical by a sequence of words

It is shown that there is a sequence (wn(x,y)) of elements of the free group on x, y with the following property: if G is a finite group (or more generally, a linear group), then the largest soluble normal subgroup of G consists of all elements g such that wn(g,h)=1 for all h∈G and all sufficiently large indices n.