Minimal Non-F-Groups with Sylow Residual and the Unitary Groups PSU(3, p^{2n}): A Negative Answer to Kourovka Notebook Problem 15.98

In Problem 15.98 of the Kourovka Notebook (2002), L. A. Shemetkov asked whether, for a saturated formation \(\mathfrak{F}\) and a finite soluble minimal non-\(\mathfrak{F}\)-group \(G\) whose \(\mathfrak{F}\)-residual is a Sylow \(p\)-subgroup, this residual is always isomorphic to a factor group of the Sylow \(p\)-subgroup of \(\mathrm{PSU}(3,p^{2n})\). For the formation of nilpotent groups this is a theorem of Mazurov and Syskin (\(p=2\)) and of Zhurtov and Syskin on Schmidt groups. We show that the answer is no: there is a saturated formation \(\mathfrak{F}\) and a finite soluble minimal non-\(\mathfrak{F}\)-group \(G\) of order \(2304\) whose \(\mathfrak{F}\)-residual is a Sylow \(2\)-subgroup isomorphic to no factor group of a Sylow \(2\)-subgroup of \(\mathrm{PSU}(3,2^{2n})\); \(2304\) is the least order of such a counterexample. The obstruction is elementary: in a nonabelian factor group \(Q\) of the Sylow \(p\)-subgroup of \(\mathrm{PSU}(3,p^{2n})\) the commutators \([y,x]\), \(x\in Q\), fill \(Q'\) for every non-central \(y\), while our residual, a special group of order \(2^8\) with centre of order \(4\), has a non-central element with only two commutators. The example comes from an elementary construction, a cyclic group acting on a special \(p\)-group, which gives a second counterexample of order \(3^8\cdot2^4\) and shows that the answer stays no for subgroup-closed formations containing all supersoluble groups, for formations contained in \(\mathfrak{N}_p\mathfrak{N}_r\), and for the formations of metanilpotent and of nilpotent-by-abelian groups. For the formation of supersoluble groups the answer is yes; we include a proof, not claimed as new, which uses the theorem of Zhurtov and Syskin as stated in its review. The least-order statement does not use that theorem. No printed answer to the question was found in the sources we searched; a search that finds nothing is not a proof of novelty, and the construction may be known to specialists. This is an unrefereed note.