Some permutability properties related to \(\mathfrak F\)-hypercentrally embedded subgroups of finite groups (Q1398156)

Let \(F\) be a saturated formation. The subgroup \(T\) of \(G\) is \(F\)-hypercentrally embedded if every \(G\)-chief factor of the interval \([T^G,T_G]\) is \(F\)-central. The authors show that the set of \(F\)-hypercentrally embedded subgroups is a lattice whenever \(F\) is subgroup closed and saturated (Theorem 2). Main result is Theorem 4: If (i) \(F\) is saturated and contains the formation \(N\) of nilpotent groups, (ii) \(T\) is an \(S\)-permutable subgroup of a soluble group \(G\), then \(T\) is \(F\)-hypercentrally embedded in \(G\) iff \(TD=DT\) for some \(F\)-normalizer \(D\) of \(G\).