On formations with systems of hereditary subformations (Q1390364)

By P. Neumann's theorem every nilpotent formation \(\mathcal F\) is hereditary (i.e. if \(G\in{\mathcal F}\) and \(H\leq G\) then \(H\in{\mathcal F}\)). Hence it follows that every metanilpotent saturated formation is hereditary. A. N. Skiba proved the following extension of these results: If, for a (saturated) formation \(\mathcal F\), all its (saturated) subformations are hereditary, then the formation \(\mathcal F\) is nilpotent (respectively, metanilpotent). Here the following analogous result is established: Let \(\mathcal F\) be a (decidable) \(p\)-saturated formation. Any of its \(p\)-saturated subformations is (normally) hereditary if and only if every group from \(\mathcal F\) is an extension of a Sylow \(p\)-subgroup by means of a nilpotent group.