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Published online by Cambridge University Press: 11 January 2010
The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G', the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble.
Further studies have been done by several authors within the framework of formation theory.
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