On a generalisation of totally permutable products of finite groups

A group G=AB is a weakly totally permutable product of subgroups A and B if for every subgroup, U of A such that U⩽A∩B or A∩B⩽U, permutes with every subgroup of B and if for every subgroup V of B such that V⩽A∩B or A∩B⩽V, permutes with every subgroup of A. Results on totally permutable products are extended to weakly totally permutable products. In particular it is shown that for a weakly totally permutable product if the factors are in F, then the product is also in F, where F is a formation containing U, the class of finite supersoluble groups. It is also shown that F-residuals and F-projectors behave nicely in weakly totally permutable products when F is a saturated formation containing U.