On normally embedded projectors in finite soluble groups (Q1903032)

Let \(NE(G)\) be the set of all normally embedded subgroups of a group \(G\). The authors start from the question: what saturated formations \(\mathcal F\) of characteristic \(\pi\) satisfy the following property: (a) In each group \(G\), the set of primes dividing the order of the \(\mathcal F\)-projectors of \(G\) coincides with the set of primes dividing the order of the Hall \(\pi\)-subgroups of \(G\). This problem is closely related to the question of characterizing the saturated formations \(\mathcal F\) for which the class \(N_{\mathcal F}\), of all soluble groups \(G\) such that every \(\mathcal F\)-projector of \(G\) is in \(NE(G)\), is a saturated formation. The answer to the above mentioned problems is given by proving that the saturated formations \(\mathcal F\) satisfying the property (a) are exactly those for which \(N_{\mathcal F}\) is a saturated formation. Further, the authors characterize those saturated formations \(\mathcal F\) for which the formation \(S_{\mathcal F}\), of all soluble groups \(G\) such that any \(\mathcal F\)-normalizer of \(G\) is in \(NE(G)\), is saturated. A useful description of the canonical local definition of the saturation of \(S_{\mathcal F}\) and a study of the saturation of the formation \(\mathcal X\) of the \(A\)-groups are also given.