11.16 (1990)

Let $V_r$ be the class of all finite groups $G$ satisfying a law $[x, \phantom{}_r y] = [x, \phantom{}_s y]$ for some $s = s(G) > r$. Here $[x, \phantom{}_1 y] = [x, y]$ and $[x, \phantom{}_{i+1} y] = [[x, \phantom{}_i y], y]$.
$\qquad$ a) Is there a function $f$ such that every soluble group in $V_r$ has Fitting length $< f(r)$? For $r < 3$ see (R. Brandl, Bull. Austral. Math. Soc., 28 (1983), 101–110).
$\qquad$ b) Is it true that $V_r$ contains only finitely many nonabelian simple groups? This is true for $r < 4$.