Prime Power Indices in Factorised Groups

Let the group G = A B be the product of the subgroups A and B . We determine some structural properties of G when the p -elements in A ∪ B have prime power indices in G , for some prime p . More generally, we also consider the case that all prime power order elements in A ∪ B have prime power indices in G . In particular, when G = A = B , we obtain as a consequence some known results.