On Brandl's Question on Finite Simple Groups of Bounded Engel Depth: A Reduction to the Suzuki Groups

For \(r\ge1\) let \(V_r\) be the class of finite groups \(G\) that satisfy a law \([x,{}_r\,y]=[x,{}_s\,y]\) for some \(s=s(G)>r\). R. Brandl asked (Kourovka Notebook, Problem 11.16 b)) whether \(V_r\) contains only finitely many nonabelian simple groups. Brandl proved that \(V_2\) contains none (1983), and Brandl and Nikolova proved that for \(q\ge4\) the group \(\mathrm{PSL}(2,q)\) lies in \(V_3\) only for \(q\in\{4,5,8\}\) (1986). We give a partial answer to part b): for every \(r\), all but finitely many nonabelian finite simple groups in \(V_r\) are Suzuki groups, so b) holds if and only if the Engel depth of \(\mathrm{Sz}(q)\) tends to infinity; this last statement and part a) stay open. Here the Engel depth of \(G\) is the least \(r\) with \(G\in V_r\); part a), on the Fitting length of soluble groups in \(V_r\), is not treated. Extending the method of Brandl and Nikolova, we prove that \(\mathrm{PSL}(2,q)\notin V_r\) for all \(q>Q_r\), with an explicit \(Q_r\): the Fricke trace recursion turns long Engel chains into rational points of an absolutely irreducible plane curve, which the Weil bound provides. Alternating, classical and exceptional groups other than \(\mathrm{Sz}(q)\) have alternating sections of unbounded degree or sections \(\mathrm{PSL}(2,q)\), and the classification of finite simple groups completes the reduction. For an odd prime \(p\), every proper section of \(\mathrm{Sz}(2^p)\) has Engel depth at most \(3\), so no argument through sections can reach these groups. As evidence only, we compute the Engel depths \(62\) of \(\mathrm{Sz}(8)\) and \(527\) of \(\mathrm{Sz}(32)\), and lower bounds from explicit pairs up to \(792{,}500\) for \(\mathrm{Sz}(8192)\). No printed answer to part b) was found in the sources searched; we claim no priority. This is an unrefereed note.