Generalised mutually permutable products and saturated formations

Abstract

[EN] We say that a group G = AB is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing A boolean AND B and B permutes with every subgroup of Acontaining A boolean AND B. We prove that some known results for mutually permutable products remain true for weakly mutually permutable ones. Moreover, if G' is nilpotent, A permutes with every Sylow subgroup of Band B permutes with every Sylow subgroup of A, we show that G(F) = A(F) B-F, where F is a saturated formation containing U, the class of supersoluble groups. This generalises the corresponding result on mutually permutable products.