Only finite groups are considered here. The well-known fact that the product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that formations, even saturated, need not be closed under the product of normal subgroups. Clearly, any formation is, however, closed under direct products and even under central products. Therefore it is interesting to study factorized groups whose subgroup factors are connected by certain permutability properties. In fact, the following question can be formulated. Let the group G = HK be the product of subgroups H and K of G which lie in a formation ℱ. What is a relationship between the factors H and K -weaker than their elementwise permutability in the case of a direct product- which will guarantee G ∈ ℱ? In [1] the following results are proved:

Theorem 1Let G = HK be a group which is the product of two supersoluble subgroups H and K. If every subgroup of H permutes with every subgroup of K (we say in this case that H and K are totally permutable subgroups of G), then G is supersoluble.