On some products of groups

Abstract A classical result of Reinhold Baer states that a group G = XN , which is the product of two normal supersoluble subgroups X and N , is supersoluble if and only if G ʹ is nilpotent. This result has been weakened in [6] for a finite group G : in fact, we do not need that both X and N are normal, but only that N is normal and X permutes with every maximal subgroup of each Sylow subgroup of N . In our paper, we improve the result mentioned above by showing that we only need X to permute with the maximal subgroups of the non-cyclic Sylow subgroups of N . Also, we extend this result (and several others) to relevant classes of infinite groups. The central idea behind our results stems from grasping the key aspects of what happens in [6]. It turns out that tensor products play a very crucial role, and it is precisely this shift of perspective that makes it possible not only to improve those results but also extend them to infinite groups.