Weakly totally permutable products and Fitting classes

It is known that if G=AB is a product of its totally permutable subgroups A and B, then G∈F if and only if A∈F and B∈F when F is a Fischer class containing the class U of supersoluble groups. We show that this holds when G=AB is a weakly totally permutable product for a particular Fischer class, F⋄N, where F is a Fitting class containing the class U and N a class of nilpotent groups. We also extend some results concerning the U-hypercentre of a totally permutable product to a weakly totally permutable product. Keywords