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