We consider groups of the form \({G} = {AB}\) with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

我们认为，形式的基团\（{G} = {AB} \）具有两个局部循环子阿和乙。这些基团的结构是在A和B均为周期性或其中一个为周期性而另一个都不为周期性的情况下确定的。连同先前对A和B的案例的研究由于没有扭转，因此可以对所有两个局部循环子组的乘积进行完整的分类。作为一个应用，表明两个局部循环子群的周期积的Prüfer秩不超过3，并且该界限很明显。还证明了有限数量的成对可置换周期性局部循环亚组的乘积是局部超溶基。这将B.Huppert的一个著名定理推广为有限群。