On multiply local formations with systems of normally hereditary subformations (Q1357957)

A formation \({\mathfrak F}\) is said to be \(n\)-multiply local (\(n>0\)) [cf. the book of \textit{A. N. Skiba}, Algebra of formations, Belaruskaya Navuka, Minsk (1997)]\ if it has a local formation function such that any of its non-empty values is an \((n-1)\)-multiply local formation. Any formation of finite groups is assumed to be \(0\)-multiply local. The main result of this paper is the following Theorem. Let \({\mathfrak F}\) be an \(n\)-multiply local formation, and let \(p\) be a prime number. For any \(n\)-multiply local subformation \({\mathfrak M}\) from \({\mathfrak F}\) the formation \({\mathfrak N}_p{\mathfrak M}\) is \(S\)-closed if and only if \({\mathfrak F}\subseteq{\mathfrak N}_p{\mathfrak N}^{n+1}\).