On the formations of finite solvable groups with property \({\mathcal{P}}_2 \) (Q6641629)

If \(G\) is a finite group, then \(G\) is the product of its pairwise permutable subgroups \(A_{1}, A_{2}, \ldots, A_{n} \leq G\) whenever \(G = A_{1}A_{2} \ldots A_{n}\) and \(A_{i}A_{j}=A_{j}A_{i}\) for all \(i, j \in \{1,2, \ldots, n\}\). Let \(\mathfrak{F}\) and \(\mathfrak{X}\) be two classes of groups and \(k\) a positive integer, then \(\mathfrak{F}\) has property \(\mathcal{P}_{k}\) for \(\mathfrak{X}\)-groups whenever \(\mathfrak{F}\) contains every \(\mathfrak{X}\)-group \(G\) expressible as the product of some subgroups such that \(A_{i_{1}}A_{i_{2}} \ldots A_{i_{k}}\in \mathfrak{F}\) for every choice of indices \(i_{1}, i_{2}, \ldots, i_{k} \in \{1,2,\ldots,n\}\).\N\NIn this paper, the authors describe all \(Z\)-saturated \(s_{F}\)-closed formations and Fischer formations of solvable groups with property \(\mathcal{P}_{2}\). In particular, the set of all such formations coincides with the set of hereditary Shemetkov formations in the class \(\mathfrak{S}\) of all finite solvable groups. The authors also describe the hereditary saturated formations \(\mathfrak{X}\) with every saturated subformation having property \(\mathcal{P}_{2}\) for \(\mathfrak{X}\).