A group G is called radicable if for each element x of G and for each positive integer n there exists an element y of G such that x=yn. A group G is reduced if it has no non-trivial radicable subgroups. Let the hyper-((locally nilpotent) or finte) group G=AB be the product of two periodic hyper-(abelian or finite) subgroups A and B. Then the following hold: (i) G is periodic. (ii) If the Sylow p-subgroups of A and B are Chernikov (respectively: finite, trivial), then the p-component of every abelian normal section of G is Chernikov (respectively: finite, trivial).