Let $G$ be finite group, $x, y \in G$. Let $m=|\langle x \rangle ^G: \langle x \rangle|$, $n=|\langle y \rangle^G: \langle y \rangle|$. In general, we can not get that $$(*) \ \ \ \ mn=|\langle xy \rangle^G:\langle xy \rangle|.$$
So we need additional condition to make $(*)$ hold.
If we require that $\langle x \rangle^G \cap \langle y \rangle^G=1$, does $(*)$ hold? If not, which conditons can make it hold?
We also interested with the case that $G$ is a infinite group.
Any reference about the normal closure of the product of two element will be welcomed.