Generalised mutually permutable products and saturated formations (Q2068185)

Recall that a finite group \(G=AB\) is termed the \textit{mutually permutable} product of subgroups \(A\) and \(B\) if \(A\) permutes with every subgroup of \(B\) and \(B\) permutes with every subgroup of \(A\). Let \(G=AB\) be a finite group with nilpotent derived subgroup. As given in [ the first and the third author, J. Algebra 218, No. 2, 563--572 (1999; Zbl 0939.20018)], if \(G\) is the mutually permutable product of \(A\) and \(B\), then \(G^\mathfrak{F}=A^\mathfrak{F}B^\mathfrak{F}\) holds for any saturated formation \(\mathfrak{F}\) containing the class \(\mathfrak{U}\) of all supersoluble groups. The authors extend the result on mutually permutable product and call \(G\) the \textit{weakly mutually permutable} product of \(A\) and \(B\) if \(A\) and \(B\) each permutes with every subgroup containing \(A\cap B\) of the other one. The paper concentrates on such finite groups \(G\) with \(G'\) nilpotent and shows that if a formation \(\mathfrak{F}\) is equal to \(\mathfrak{U}\) or is a saturated Fitting formation containing \(\mathfrak{U}\), then \(G^\mathfrak{F}=\langle A^\mathfrak{F},B^\mathfrak{F}\rangle\). The authors also confirm that \(G^\mathfrak{F}=A^\mathfrak{F}B^\mathfrak{F}\) provided that \(G\) has a nilpotent derived subgroup and \(A\) permutes with every Sylow subgroup of \(B\) and \(B\) permutes with every Sylow subgroup of \(A\).