A question on saturated formations of finite groups (Q1920858)

Let \(\mathfrak H\) be a saturated formation, \({\mathfrak H}=LF(H)\), where \(H\) is the canonical local definition of \(\mathfrak H\) and let \(\mathfrak F\) be a formation. Denote by \({\mathfrak H}\downarrow {\mathfrak F}\) the class of all groups whose \({\mathfrak H}\)-projectors belong to \(\mathfrak F\). Since \({\mathfrak H}\downarrow {\mathfrak F}={\mathfrak H}\downarrow ({\mathfrak H}\cap {\mathfrak F})\), it can be supposed in the sequel that \({\mathfrak F}\subseteq {\mathfrak H}\). If \(\mathfrak H\) is also saturated, we will denote \({\mathfrak F}=LF(F)\), where \(F\) is the canonical local definition of \(\mathfrak F\). The class \({\mathfrak X}={\mathfrak H}\downarrow {\mathfrak F}\) is a formation but in general \(\mathfrak X\) is not saturated. The saturation of the formation \({\mathfrak H}\downarrow {\mathfrak F}=(G\in {\mathfrak S}:\text{Proj}_{\mathfrak H}(G)\subseteq {\mathfrak F})\) was fully analyzed by Doerk. Our aim is to give some necessary conditions, in the general nonsoluble universe, for a saturated formation \(\mathfrak H\) to have the following property: \({\mathfrak H}\downarrow {\mathfrak F}\) is a saturated formation for all saturated formations \(\mathfrak F\) (Theorem C). Unfortunately, the conditions are not sufficient. As a preliminary result we give a criterion for \({\mathfrak H}\downarrow {\mathfrak F}\) to be saturated in terms of the canonical local definitions of \(\mathfrak H\) and \(\mathfrak F\) (Theorem A). Finally, we undertake the ``symmetric'' problem of characterizing the saturated formations \(\mathfrak F\) such that \({\mathfrak H}\downarrow {\mathfrak F}\) is saturated for all saturated formations \(\mathfrak H\). This is done in Theorem B.