On finite products of nilpotent groups (Q1337817)

Let the finite group \(G = AB\) be the product of two nilpotent subgroups \(A\) and \(B\). When does a given subgroup of \(G\) have a conjugate \(S\) which inherits the product structure, i.e. satisfies \(S = (A \cap S) (B \cap S)\)? Here it is for instance shown that such a conjugate can always be found if the subgroup is (sub)abnormal in \(G\) or if it is a maximal \(\mathfrak F\)-subgroup of \(G\), where \(\mathfrak F\) is a saturated formation of finite groups. This generalizes a result of H. Heineken for \(\mathfrak F\)- projectors.