Introduction

A great deal of work has been done to investigate the situation that a group G can be written as a product of two subgroups K and H. N. Ito [9] has shown that whenever K and H are abelian then G is soluble. Also, if G is finite and H and K are nilpotent then G is soluble (Wielandt [24], Kegel [10]). In the 1970's and 1980's several special cases of this type were considered; for a good selection of the kind of results that were obtained one can look in Arnberg [1] and the references given there.

Now it is interesting to notice that if H is a subgroup of G then the natural way to combine H and G is to write G = AH where A is a left transversal to H in G. It is surprising that the influence of the properties of H and its transversals on the structure of G has been studied so little (of course, transversals have been used very efficiently in the construction of the transfer homomorphism). In this survey we shall consider the situation that G = AH = BH and the left transversals A and B are connected by the commutator condition [A, B] ≤ H. We investigate the solubility of G as well as the situation in some finite simple groups. The situation and the conditions that we study arise in a natural way from some problems in loop theory and quasigroup theory. Thus we also give some applications of our results in the final section of our survey.