On Shumyatsky's Question on Soluble Subgroups and the Generalized Fitting Height: A Partial Answer for Groups with Chief Factors of Type PSL(2,q) and Sz(q)

Let \(h^*(G)\) be the generalized Fitting height of a finite group \(G\), and \(h(K)\) the Fitting height of a soluble group \(K\). P. Shumyatsky asked (Kourovka Notebook, Problem 19.82) whether every finite group \(G\) has a soluble subgroup \(K\) with \(h^*(G)=h(K)\), that is, whether \(h^*(G)\) never exceeds the largest Fitting height \(s(G)\) of a soluble subgroup; Khukhro and Shumyatsky proved \(\lambda(G)\le s(G)\) for the nonsoluble length \(\lambda(G)\), which is at most \(h^*(G)\) (Israel J. Math. 2015). We give a partial answer: yes for every finite group whose nonabelian chief factors are of type \(\mathrm{PSL}(2,q)\) (\(q\ge4\)) or \(\mathrm{Sz}(q)\), provided that no \(\mathrm{PSL}(2,q)\)-factor with \(q\) odd and \(q\equiv\pm1\pmod 8\) induces a group containing \(\mathrm{PGL}(2,q)\); the general question stays open. For these groups, if \(O_p(G)=1\), some soluble \(K\) has \(h(K/O_p(K))\ge h^*(G)\). Extending the method of Khukhro and Shumyatsky, we exclude a minimal counterexample by a witness system, a condition on the automorphisms induced on a minimal normal subgroup \(S^m\), and construct witness systems for \(\mathrm{PSL}(2,q)\) and \(\mathrm{Sz}(q)\) from unipotent subgroups, Zsigmondy primes and the Lang–Steinberg theorem. The excluded case is a limit of the method, not a counterexample; the criterion is only sufficient and fails for some simple groups; \(A_n\) (\(n\ge7\)), the other groups of Lie type and the sporadic groups are not treated. No printed answer to the question was found. This is an unrefereed note.