16.77 (2006)

It is known that in every Noetherian group the nilpotent radical coincides with the collection of all Engel elements (R. Baer, Math. Ann., 133 (1957), 256–270; B. I. Plotkin, Izv. Vyssh. Uchebn. Zaved. Mat., 1958, no. 1(2), 130–135 (Russian)). It would be nice to find a similar characterization of the solvable radical of a finite group. More precisely, let $u = u(x, y)$ be a sequence of words satisfying 15.75. We say that an element $g \in G$ is $u$-Engel if there exists $n = n(g)$ such that $u_n(x, g) = 1$ for every element $x \in G$. Does there exist a sequence $u = u(x, y)$ such that the solvable radical of a finite group coincides with the set of all $u$-Engel elements?