On a class of supersoluble groups. (Q1413433)

It is well known that the product of two normal supersoluble subgroups need not be supersoluble in general. The author considers a class \(\mathfrak C\) of groups \(G\) which contain a normal subgroup \(N\) such that for some positive integer \(n\) every subgroup of the \(n\)th term of the lower central series of \(G/N'\) is normal in \(G/N'\). The author shows that every product of a normal finitely generated \(\mathfrak C\)-subgroup and a subnormal supersoluble subgroup is supersoluble. In particular, this implies that every product of a finite number of normal finitely generated \(\mathfrak C\)-subgroups is supersoluble.