On Groups which are Products of Weakly Totally Permutable Subgroups

Abstract

This work is a contribution to the theory of products of finite groups. A group G = AB is a weakly totally permutable product of subgroups A and B if every subgroup, U of A such that U ≤ A ∩ B or A ∩ B ≤ U, permutes with every subgroup of B and if every subgroup V of B such that V ≤ A ∩ B or A ∩ B ≤ V , permutes with every subgroup of A. It follows that a totally permutable product is a weakly totally permutable product. Some results on totally permutable products in the framework of formation theory are generalised. In particular it is shown that if the factors of a weakly totally permutable product are in F, then the product is also in F, where F is a formation containing U, the class of all finite supersoluble groups. It is also shown that the F-residual (and F-projector) of the product G is just the product of the F-residuals (and respectively F-projectors) of the factors A and B, when F is a saturated formation containing U. Moreover, it is shown that a weakly totally permutable product is an SC-group if and only if its factors are SC-groups. In the framework of Fitting classes some results are extended to weakly totally permutable products. Fischer classes containing U were proved to behave nicely with respect to forming products in totally permutable products. It is shown that a particular Fischer class, F N, where F is a Fitting class containing U and N is the class of all nilpotent groups, also behave nicely with respect to forming products in weakly totally permutable products.